3.30.82 \(\int \frac {(d+c x^2) (a x+\sqrt {-b+a^2 x^2})^{5/4}}{(-b+a^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=384 \[ \frac {4 \sqrt [8]{b} c x \sqrt [4]{\frac {\sqrt {a^2 x^2-b}}{\sqrt {b}}+\frac {a x}{\sqrt {b}}}}{5 a^2}+\frac {\sqrt {a^2 x^2-b} \sqrt [4]{\frac {\sqrt {a^2 x^2-b}+a x}{\sqrt {b}}} \left (4 a^2 \sqrt [8]{b} c x^2-5 a^2 \sqrt [8]{b} d-9 b^{9/8} c\right )}{5 a^3 \left (a x-\sqrt {b}\right ) \left (a x+\sqrt {b}\right )}-\frac {5 \left (a^2 d+b c\right ) \tan ^{-1}\left (\sqrt [4]{\frac {\sqrt {a^2 x^2-b}+a x}{\sqrt {b}}}\right )}{2 a^3 b^{3/8}}-\frac {5 \left (a^2 d+b c\right ) \tanh ^{-1}\left (\sqrt [4]{\frac {\sqrt {a^2 x^2-b}+a x}{\sqrt {b}}}\right )}{2 a^3 b^{3/8}}-\frac {5 (-1)^{3/4} \left (a^2 d+b c\right ) \tanh ^{-1}\left (\sqrt [4]{-1} \sqrt [4]{\frac {\sqrt {a^2 x^2-b}+a x}{\sqrt {b}}}\right )}{2 a^3 b^{3/8}}-\frac {5 \sqrt [4]{-1} \left (a^2 d+b c\right ) \tanh ^{-1}\left ((-1)^{3/4} \sqrt [4]{\frac {\sqrt {a^2 x^2-b}+a x}{\sqrt {b}}}\right )}{2 a^3 b^{3/8}} \]

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Rubi [B]  time = 1.74, antiderivative size = 869, normalized size of antiderivative = 2.26, number of steps used = 35, number of rules used = 17, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {6742, 2122, 288, 329, 301, 211, 1165, 628, 1162, 617, 204, 212, 206, 203, 2120, 463, 459} \begin {gather*} \frac {4 c \left (a x+\sqrt {a^2 x^2-b}\right )^{5/4}}{5 a^3}+\frac {2 b c \left (a x+\sqrt {a^2 x^2-b}\right )^{5/4}}{a^3 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )}+\frac {2 d \left (a x+\sqrt {a^2 x^2-b}\right )^{5/4}}{a \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )}-\frac {5 b^{5/8} c \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 a^3}-\frac {5 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {5 b^{5/8} c \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a^3}-\frac {5 d \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a b^{3/8}}+\frac {5 b^{5/8} c \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}+1\right )}{2 \sqrt {2} a^3}+\frac {5 d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}+1\right )}{2 \sqrt {2} a b^{3/8}}-\frac {5 b^{5/8} c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 a^3}-\frac {5 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {5 b^{5/8} c \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{4 \sqrt {2} a^3}-\frac {5 d \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{4 \sqrt {2} a b^{3/8}}+\frac {5 b^{5/8} c \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{4 \sqrt {2} a^3}+\frac {5 d \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{4 \sqrt {2} a b^{3/8}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + c*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(5/4))/(-b + a^2*x^2)^(3/2),x]

[Out]

(4*c*(a*x + Sqrt[-b + a^2*x^2])^(5/4))/(5*a^3) + (2*b*c*(a*x + Sqrt[-b + a^2*x^2])^(5/4))/(a^3*(b - (a*x + Sqr
t[-b + a^2*x^2])^2)) + (2*d*(a*x + Sqrt[-b + a^2*x^2])^(5/4))/(a*(b - (a*x + Sqrt[-b + a^2*x^2])^2)) - (5*b^(5
/8)*c*ArcTan[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)])/(2*a^3) - (5*d*ArcTan[(a*x + Sqrt[-b + a^2*x^2])^(1/4)
/b^(1/8)])/(2*a*b^(3/8)) - (5*b^(5/8)*c*ArcTan[1 - (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/b^(1/8)])/(2*Sqr
t[2]*a^3) - (5*d*ArcTan[1 - (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/b^(1/8)])/(2*Sqrt[2]*a*b^(3/8)) + (5*b^
(5/8)*c*ArcTan[1 + (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/b^(1/8)])/(2*Sqrt[2]*a^3) + (5*d*ArcTan[1 + (Sqr
t[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/b^(1/8)])/(2*Sqrt[2]*a*b^(3/8)) - (5*b^(5/8)*c*ArcTanh[(a*x + Sqrt[-b +
a^2*x^2])^(1/4)/b^(1/8)])/(2*a^3) - (5*d*ArcTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)])/(2*a*b^(3/8)) - (
5*b^(5/8)*c*Log[b^(1/4) - Sqrt[2]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/
(4*Sqrt[2]*a^3) - (5*d*Log[b^(1/4) - Sqrt[2]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + Sqrt[a*x + Sqrt[-b + a
^2*x^2]]])/(4*Sqrt[2]*a*b^(3/8)) + (5*b^(5/8)*c*Log[b^(1/4) + Sqrt[2]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4)
+ Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/(4*Sqrt[2]*a^3) + (5*d*Log[b^(1/4) + Sqrt[2]*b^(1/8)*(a*x + Sqrt[-b + a^2*
x^2])^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/(4*Sqrt[2]*a*b^(3/8))

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 301

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(
a/b), 2]]}, Dist[s/(2*b), Int[x^(m - n/2)/(r + s*x^(n/2)), x], x] - Dist[s/(2*b), Int[x^(m - n/2)/(r - s*x^(n/
2)), x], x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LeQ[n/2, m] && LtQ[m, n] &&  !GtQ[a/b, 0]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 463

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> -Simp[((b*c - a*
d)^2*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b^2*e*n*(p + 1)), x] + Dist[1/(a*b^2*n*(p + 1)), Int[(e*x)^m*(a + b
*x^n)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + b^2*c^2*n*(p + 1) + a*b*d^2*n*(p + 1)*x^n, x], x], x] /; FreeQ[{a,
b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 2120

Int[(x_)^(p_.)*((g_) + (i_.)*(x_)^2)^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :>
Dist[(1*(i/c)^m)/(2^(2*m + p + 1)*e^(p + 1)*f^(2*m)), Subst[Int[x^(n - 2*m - p - 2)*(-(a*f^2) + x^2)^p*(a*f^2
+ x^2)^(2*m + 1), x], x, e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0
] && EqQ[c*g - a*i, 0] && IntegersQ[p, 2*m] && (IntegerQ[m] || GtQ[i/c, 0])

Rule 2122

Int[((g_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dis
t[(1*(i/c)^m)/(2^(2*m + 1)*e*f^(2*m)), Subst[Int[(x^n*(d^2 + a*f^2 - 2*d*x + x^2)^(2*m + 1))/(-d + x)^(2*(m +
1)), x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, d, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0] && E
qQ[c*g - a*i, 0] && IntegerQ[2*m] && (IntegerQ[m] || GtQ[i/c, 0])

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx &=\int \left (\frac {d \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}}+\frac {c x^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}}\right ) \, dx\\ &=c \int \frac {x^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx+d \int \frac {\left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx\\ &=\frac {c \operatorname {Subst}\left (\int \frac {\sqrt [4]{x} \left (b+x^2\right )^2}{\left (-b+x^2\right )^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{a^3}+\frac {(4 d) \operatorname {Subst}\left (\int \frac {x^{9/4}}{\left (-b+x^2\right )^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{a}\\ &=\frac {2 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {2 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {c \operatorname {Subst}\left (\int \frac {\sqrt [4]{x} \left (3 b^2+2 b x^2\right )}{-b+x^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{2 a^3 b}+\frac {(5 d) \operatorname {Subst}\left (\int \frac {\sqrt [4]{x}}{-b+x^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{2 a}\\ &=\frac {4 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{5 a^3}+\frac {2 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {2 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {(5 b c) \operatorname {Subst}\left (\int \frac {\sqrt [4]{x}}{-b+x^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{2 a^3}+\frac {(10 d) \operatorname {Subst}\left (\int \frac {x^4}{-b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{a}\\ &=\frac {4 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{5 a^3}+\frac {2 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {2 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {(10 b c) \operatorname {Subst}\left (\int \frac {x^4}{-b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{a^3}-\frac {(5 d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{a}+\frac {(5 d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{a}\\ &=\frac {4 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{5 a^3}+\frac {2 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {2 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {(5 b c) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{a^3}+\frac {(5 b c) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{a^3}-\frac {(5 d) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b}-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{2 a \sqrt [4]{b}}-\frac {(5 d) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b}+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{2 a \sqrt [4]{b}}+\frac {(5 d) \operatorname {Subst}\left (\int \frac {\sqrt [4]{b}-x^2}{\sqrt {b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{2 a \sqrt [4]{b}}+\frac {(5 d) \operatorname {Subst}\left (\int \frac {\sqrt [4]{b}+x^2}{\sqrt {b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{2 a \sqrt [4]{b}}\\ &=\frac {4 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{5 a^3}+\frac {2 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {2 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {5 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {5 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {\left (5 b^{3/4} c\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b}-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{2 a^3}-\frac {\left (5 b^{3/4} c\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b}+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{2 a^3}+\frac {\left (5 b^{3/4} c\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{b}-x^2}{\sqrt {b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{2 a^3}+\frac {\left (5 b^{3/4} c\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{b}+x^2}{\sqrt {b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{2 a^3}-\frac {(5 d) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [8]{b}+2 x}{-\sqrt [4]{b}-\sqrt {2} \sqrt [8]{b} x-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{4 \sqrt {2} a b^{3/8}}-\frac {(5 d) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [8]{b}-2 x}{-\sqrt [4]{b}+\sqrt {2} \sqrt [8]{b} x-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{4 \sqrt {2} a b^{3/8}}+\frac {(5 d) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b}-\sqrt {2} \sqrt [8]{b} x+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{4 a \sqrt [4]{b}}+\frac {(5 d) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b}+\sqrt {2} \sqrt [8]{b} x+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{4 a \sqrt [4]{b}}\\ &=\frac {4 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{5 a^3}+\frac {2 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {2 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {5 b^{5/8} c \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a^3}-\frac {5 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {5 b^{5/8} c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a^3}-\frac {5 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {5 d \log \left (\sqrt [4]{b}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{4 \sqrt {2} a b^{3/8}}+\frac {5 d \log \left (\sqrt [4]{b}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{4 \sqrt {2} a b^{3/8}}-\frac {\left (5 b^{5/8} c\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [8]{b}+2 x}{-\sqrt [4]{b}-\sqrt {2} \sqrt [8]{b} x-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{4 \sqrt {2} a^3}-\frac {\left (5 b^{5/8} c\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [8]{b}-2 x}{-\sqrt [4]{b}+\sqrt {2} \sqrt [8]{b} x-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{4 \sqrt {2} a^3}+\frac {\left (5 b^{3/4} c\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b}-\sqrt {2} \sqrt [8]{b} x+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{4 a^3}+\frac {\left (5 b^{3/4} c\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b}+\sqrt {2} \sqrt [8]{b} x+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{4 a^3}+\frac {(5 d) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a b^{3/8}}-\frac {(5 d) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a b^{3/8}}\\ &=\frac {4 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{5 a^3}+\frac {2 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {2 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {5 b^{5/8} c \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a^3}-\frac {5 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {5 d \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a b^{3/8}}+\frac {5 d \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a b^{3/8}}-\frac {5 b^{5/8} c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a^3}-\frac {5 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {5 b^{5/8} c \log \left (\sqrt [4]{b}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{4 \sqrt {2} a^3}-\frac {5 d \log \left (\sqrt [4]{b}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{4 \sqrt {2} a b^{3/8}}+\frac {5 b^{5/8} c \log \left (\sqrt [4]{b}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{4 \sqrt {2} a^3}+\frac {5 d \log \left (\sqrt [4]{b}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{4 \sqrt {2} a b^{3/8}}+\frac {\left (5 b^{5/8} c\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a^3}-\frac {\left (5 b^{5/8} c\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a^3}\\ &=\frac {4 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{5 a^3}+\frac {2 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {2 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {5 b^{5/8} c \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a^3}-\frac {5 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {5 b^{5/8} c \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a^3}-\frac {5 d \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a b^{3/8}}+\frac {5 b^{5/8} c \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a^3}+\frac {5 d \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a b^{3/8}}-\frac {5 b^{5/8} c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a^3}-\frac {5 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {5 b^{5/8} c \log \left (\sqrt [4]{b}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{4 \sqrt {2} a^3}-\frac {5 d \log \left (\sqrt [4]{b}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{4 \sqrt {2} a b^{3/8}}+\frac {5 b^{5/8} c \log \left (\sqrt [4]{b}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{4 \sqrt {2} a^3}+\frac {5 d \log \left (\sqrt [4]{b}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{4 \sqrt {2} a b^{3/8}}\\ \end {align*}

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Mathematica [F]  time = 0.65, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((d + c*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(5/4))/(-b + a^2*x^2)^(3/2),x]

[Out]

Integrate[((d + c*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(5/4))/(-b + a^2*x^2)^(3/2), x]

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IntegrateAlgebraic [A]  time = 4.11, size = 446, normalized size = 1.16 \begin {gather*} \frac {\sqrt {-b+a^2 x^2} \left (-9 b^{9/8} c-5 a^2 \sqrt [8]{b} d+4 a^2 \sqrt [8]{b} c x^2\right ) \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{5 a^3 \left (-\sqrt {b}+a x\right ) \left (\sqrt {b}+a x\right )}+\frac {4 \sqrt [8]{b} c x \sqrt [4]{\frac {a x}{\sqrt {b}}+\frac {\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{5 a^2}-\frac {5 \left (b c+a^2 d\right ) \tan ^{-1}\left (\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{2 a^3 b^{3/8}}+\frac {5 \left (b c+a^2 d\right ) \tan ^{-1}\left (\frac {-1+\sqrt {\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{\sqrt {2} \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}\right )}{2 \sqrt {2} a^3 b^{3/8}}-\frac {5 \left (b c+a^2 d\right ) \tanh ^{-1}\left (\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{2 a^3 b^{3/8}}+\frac {5 \left (b c+a^2 d\right ) \tanh ^{-1}\left (\frac {1+\sqrt {\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{\sqrt {2} \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}\right )}{2 \sqrt {2} a^3 b^{3/8}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

IntegrateAlgebraic[((d + c*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(5/4))/(-b + a^2*x^2)^(3/2),x]

[Out]

(Sqrt[-b + a^2*x^2]*(-9*b^(9/8)*c - 5*a^2*b^(1/8)*d + 4*a^2*b^(1/8)*c*x^2)*((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b]
)^(1/4))/(5*a^3*(-Sqrt[b] + a*x)*(Sqrt[b] + a*x)) + (4*b^(1/8)*c*x*((a*x)/Sqrt[b] + Sqrt[-b + a^2*x^2]/Sqrt[b]
)^(1/4))/(5*a^2) - (5*(b*c + a^2*d)*ArcTan[((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4)])/(2*a^3*b^(3/8)) + (5*(
b*c + a^2*d)*ArcTan[(-1 + Sqrt[(a*x + Sqrt[-b + a^2*x^2])/Sqrt[b]])/(Sqrt[2]*((a*x + Sqrt[-b + a^2*x^2])/Sqrt[
b])^(1/4))])/(2*Sqrt[2]*a^3*b^(3/8)) - (5*(b*c + a^2*d)*ArcTanh[((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4)])/(
2*a^3*b^(3/8)) + (5*(b*c + a^2*d)*ArcTanh[(1 + Sqrt[(a*x + Sqrt[-b + a^2*x^2])/Sqrt[b]])/(Sqrt[2]*((a*x + Sqrt
[-b + a^2*x^2])/Sqrt[b])^(1/4))])/(2*Sqrt[2]*a^3*b^(3/8))

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fricas [B]  time = 0.76, size = 5796, normalized size = 15.09

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+d)*(a*x+(a^2*x^2-b)^(1/2))^(5/4)/(a^2*x^2-b)^(3/2),x, algorithm="fricas")

[Out]

-1/40*(100*sqrt(2)*(a^5*x^2 - a^3*b)*((a^16*d^8 + 8*a^14*b*c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 +
70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)/(a^24*b^3))^(1/8)*a
rctan(1/3125*(3125*a^16*d^8 + 25000*a^14*b*c*d^7 + 87500*a^12*b^2*c^2*d^6 + 175000*a^10*b^3*c^3*d^5 + 218750*a
^8*b^4*c^4*d^4 + 175000*a^6*b^5*c^5*d^3 + 87500*a^4*b^6*c^6*d^2 + 25000*a^2*b^7*c^7*d + 3125*b^8*c^8 + sqrt(2)
*sqrt(-9765625*sqrt(2)*(a^25*b^2*d^5 + 5*a^23*b^3*c*d^4 + 10*a^21*b^4*c^2*d^3 + 10*a^19*b^5*c^3*d^2 + 5*a^17*b
^6*c^4*d + a^15*b^7*c^5)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*((a^16*d^8 + 8*a^14*b*c*d^7 + 28*a^12*b^2*c^2*d^6 + 5
6*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)
/(a^24*b^3))^(5/8) + 9765625*(a^20*d^10 + 10*a^18*b*c*d^9 + 45*a^16*b^2*c^2*d^8 + 120*a^14*b^3*c^3*d^7 + 210*a
^12*b^4*c^4*d^6 + 252*a^10*b^5*c^5*d^5 + 210*a^8*b^6*c^6*d^4 + 120*a^6*b^7*c^7*d^3 + 45*a^4*b^8*c^8*d^2 + 10*a
^2*b^9*c^9*d + b^10*c^10)*sqrt(a*x + sqrt(a^2*x^2 - b)) + 9765625*(a^22*b*d^8 + 8*a^20*b^2*c*d^7 + 28*a^18*b^3
*c^2*d^6 + 56*a^16*b^4*c^3*d^5 + 70*a^14*b^5*c^4*d^4 + 56*a^12*b^6*c^5*d^3 + 28*a^10*b^7*c^6*d^2 + 8*a^8*b^8*c
^7*d + a^6*b^9*c^8)*((a^16*d^8 + 8*a^14*b*c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d
^4 + 56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)/(a^24*b^3))^(1/4))*a^9*b*((a^16*d^8
+ 8*a^14*b*c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a^
4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)/(a^24*b^3))^(3/8) - 3125*sqrt(2)*(a^19*b*d^5 + 5*a^17*b^2*c*d^4 + 1
0*a^15*b^3*c^2*d^3 + 10*a^13*b^4*c^3*d^2 + 5*a^11*b^5*c^4*d + a^9*b^6*c^5)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*((a
^16*d^8 + 8*a^14*b*c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3
+ 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)/(a^24*b^3))^(3/8))/(a^16*d^8 + 8*a^14*b*c*d^7 + 28*a^12*b^2
*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*
d + b^8*c^8)) + 100*sqrt(2)*(a^5*x^2 - a^3*b)*((a^16*d^8 + 8*a^14*b*c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*
c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)/(a^24*b^3)
)^(1/8)*arctan(-(a^16*d^8 + 8*a^14*b*c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 +
56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8 - sqrt(2)*sqrt(sqrt(2)*(a^25*b^2*d^5 + 5*a
^23*b^3*c*d^4 + 10*a^21*b^4*c^2*d^3 + 10*a^19*b^5*c^3*d^2 + 5*a^17*b^6*c^4*d + a^15*b^7*c^5)*(a*x + sqrt(a^2*x
^2 - b))^(1/4)*((a^16*d^8 + 8*a^14*b*c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 +
56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)/(a^24*b^3))^(5/8) + (a^20*d^10 + 10*a^18*
b*c*d^9 + 45*a^16*b^2*c^2*d^8 + 120*a^14*b^3*c^3*d^7 + 210*a^12*b^4*c^4*d^6 + 252*a^10*b^5*c^5*d^5 + 210*a^8*b
^6*c^6*d^4 + 120*a^6*b^7*c^7*d^3 + 45*a^4*b^8*c^8*d^2 + 10*a^2*b^9*c^9*d + b^10*c^10)*sqrt(a*x + sqrt(a^2*x^2
- b)) + (a^22*b*d^8 + 8*a^20*b^2*c*d^7 + 28*a^18*b^3*c^2*d^6 + 56*a^16*b^4*c^3*d^5 + 70*a^14*b^5*c^4*d^4 + 56*
a^12*b^6*c^5*d^3 + 28*a^10*b^7*c^6*d^2 + 8*a^8*b^8*c^7*d + a^6*b^9*c^8)*((a^16*d^8 + 8*a^14*b*c*d^7 + 28*a^12*
b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c
^7*d + b^8*c^8)/(a^24*b^3))^(1/4))*a^9*b*((a^16*d^8 + 8*a^14*b*c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d
^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)/(a^24*b^3))^(3/
8) + sqrt(2)*(a^19*b*d^5 + 5*a^17*b^2*c*d^4 + 10*a^15*b^3*c^2*d^3 + 10*a^13*b^4*c^3*d^2 + 5*a^11*b^5*c^4*d + a
^9*b^6*c^5)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*((a^16*d^8 + 8*a^14*b*c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^
3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)/(a^24*b^3))^
(3/8))/(a^16*d^8 + 8*a^14*b*c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^
5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)) - 25*sqrt(2)*(a^5*x^2 - a^3*b)*((a^16*d^8 + 8*a^1
4*b*c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c
^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)/(a^24*b^3))^(1/8)*log(9765625*sqrt(2)*(a^25*b^2*d^5 + 5*a^23*b^3*c*d^4 + 1
0*a^21*b^4*c^2*d^3 + 10*a^19*b^5*c^3*d^2 + 5*a^17*b^6*c^4*d + a^15*b^7*c^5)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*((
a^16*d^8 + 8*a^14*b*c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^
3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)/(a^24*b^3))^(5/8) + 9765625*(a^20*d^10 + 10*a^18*b*c*d^9 +
45*a^16*b^2*c^2*d^8 + 120*a^14*b^3*c^3*d^7 + 210*a^12*b^4*c^4*d^6 + 252*a^10*b^5*c^5*d^5 + 210*a^8*b^6*c^6*d^
4 + 120*a^6*b^7*c^7*d^3 + 45*a^4*b^8*c^8*d^2 + 10*a^2*b^9*c^9*d + b^10*c^10)*sqrt(a*x + sqrt(a^2*x^2 - b)) + 9
765625*(a^22*b*d^8 + 8*a^20*b^2*c*d^7 + 28*a^18*b^3*c^2*d^6 + 56*a^16*b^4*c^3*d^5 + 70*a^14*b^5*c^4*d^4 + 56*a
^12*b^6*c^5*d^3 + 28*a^10*b^7*c^6*d^2 + 8*a^8*b^8*c^7*d + a^6*b^9*c^8)*((a^16*d^8 + 8*a^14*b*c*d^7 + 28*a^12*b
^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^
7*d + b^8*c^8)/(a^24*b^3))^(1/4)) + 25*sqrt(2)*(a^5*x^2 - a^3*b)*((a^16*d^8 + 8*a^14*b*c*d^7 + 28*a^12*b^2*c^2
*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d +
b^8*c^8)/(a^24*b^3))^(1/8)*log(-9765625*sqrt(2)*(a^25*b^2*d^5 + 5*a^23*b^3*c*d^4 + 10*a^21*b^4*c^2*d^3 + 10*a^
19*b^5*c^3*d^2 + 5*a^17*b^6*c^4*d + a^15*b^7*c^5)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*((a^16*d^8 + 8*a^14*b*c*d^7
+ 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8
*a^2*b^7*c^7*d + b^8*c^8)/(a^24*b^3))^(5/8) + 9765625*(a^20*d^10 + 10*a^18*b*c*d^9 + 45*a^16*b^2*c^2*d^8 + 120
*a^14*b^3*c^3*d^7 + 210*a^12*b^4*c^4*d^6 + 252*a^10*b^5*c^5*d^5 + 210*a^8*b^6*c^6*d^4 + 120*a^6*b^7*c^7*d^3 +
45*a^4*b^8*c^8*d^2 + 10*a^2*b^9*c^9*d + b^10*c^10)*sqrt(a*x + sqrt(a^2*x^2 - b)) + 9765625*(a^22*b*d^8 + 8*a^2
0*b^2*c*d^7 + 28*a^18*b^3*c^2*d^6 + 56*a^16*b^4*c^3*d^5 + 70*a^14*b^5*c^4*d^4 + 56*a^12*b^6*c^5*d^3 + 28*a^10*
b^7*c^6*d^2 + 8*a^8*b^8*c^7*d + a^6*b^9*c^8)*((a^16*d^8 + 8*a^14*b*c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c
^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)/(a^24*b^3))
^(1/4)) - 200*(a^5*x^2 - a^3*b)*((a^16*d^8 + 8*a^14*b*c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a
^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)/(a^24*b^3))^(1/8)*arctan
((sqrt((a^20*d^10 + 10*a^18*b*c*d^9 + 45*a^16*b^2*c^2*d^8 + 120*a^14*b^3*c^3*d^7 + 210*a^12*b^4*c^4*d^6 + 252*
a^10*b^5*c^5*d^5 + 210*a^8*b^6*c^6*d^4 + 120*a^6*b^7*c^7*d^3 + 45*a^4*b^8*c^8*d^2 + 10*a^2*b^9*c^9*d + b^10*c^
10)*sqrt(a*x + sqrt(a^2*x^2 - b)) + (a^22*b*d^8 + 8*a^20*b^2*c*d^7 + 28*a^18*b^3*c^2*d^6 + 56*a^16*b^4*c^3*d^5
+ 70*a^14*b^5*c^4*d^4 + 56*a^12*b^6*c^5*d^3 + 28*a^10*b^7*c^6*d^2 + 8*a^8*b^8*c^7*d + a^6*b^9*c^8)*((a^16*d^8
+ 8*a^14*b*c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a
^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)/(a^24*b^3))^(1/4))*a^9*b*((a^16*d^8 + 8*a^14*b*c*d^7 + 28*a^12*b^2
*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*
d + b^8*c^8)/(a^24*b^3))^(3/8) - (a^19*b*d^5 + 5*a^17*b^2*c*d^4 + 10*a^15*b^3*c^2*d^3 + 10*a^13*b^4*c^3*d^2 +
5*a^11*b^5*c^4*d + a^9*b^6*c^5)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*((a^16*d^8 + 8*a^14*b*c*d^7 + 28*a^12*b^2*c^2*
d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b
^8*c^8)/(a^24*b^3))^(3/8))/(a^16*d^8 + 8*a^14*b*c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4
*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)) + 50*(a^5*x^2 - a^3*b)*((a^16
*d^8 + 8*a^14*b*c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 +
28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)/(a^24*b^3))^(1/8)*log(3125*a^15*b^2*((a^16*d^8 + 8*a^14*b*c*d^
7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 +
8*a^2*b^7*c^7*d + b^8*c^8)/(a^24*b^3))^(5/8) + 3125*(a^10*d^5 + 5*a^8*b*c*d^4 + 10*a^6*b^2*c^2*d^3 + 10*a^4*b
^3*c^3*d^2 + 5*a^2*b^4*c^4*d + b^5*c^5)*(a*x + sqrt(a^2*x^2 - b))^(1/4)) - 50*(a^5*x^2 - a^3*b)*((a^16*d^8 + 8
*a^14*b*c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a^4*b
^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)/(a^24*b^3))^(1/8)*log(-3125*a^15*b^2*((a^16*d^8 + 8*a^14*b*c*d^7 + 28*
a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*
b^7*c^7*d + b^8*c^8)/(a^24*b^3))^(5/8) + 3125*(a^10*d^5 + 5*a^8*b*c*d^4 + 10*a^6*b^2*c^2*d^3 + 10*a^4*b^3*c^3*
d^2 + 5*a^2*b^4*c^4*d + b^5*c^5)*(a*x + sqrt(a^2*x^2 - b))^(1/4)) - 8*(4*a^3*c*x^3 - 4*a*b*c*x + (4*a^2*c*x^2
- 5*a^2*d - 9*b*c)*sqrt(a^2*x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^(1/4))/(a^5*x^2 - a^3*b)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+d)*(a*x+(a^2*x^2-b)^(1/2))^(5/4)/(a^2*x^2-b)^(3/2),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (c \,x^{2}+d \right ) \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {5}{4}}}{\left (a^{2} x^{2}-b \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+d)*(a*x+(a^2*x^2-b)^(1/2))^(5/4)/(a^2*x^2-b)^(3/2),x)

[Out]

int((c*x^2+d)*(a*x+(a^2*x^2-b)^(1/2))^(5/4)/(a^2*x^2-b)^(3/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (c x^{2} + d\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {5}{4}}}{{\left (a^{2} x^{2} - b\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+d)*(a*x+(a^2*x^2-b)^(1/2))^(5/4)/(a^2*x^2-b)^(3/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + d)*(a*x + sqrt(a^2*x^2 - b))^(5/4)/(a^2*x^2 - b)^(3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{5/4}\,\left (c\,x^2+d\right )}{{\left (a^2\,x^2-b\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x + (a^2*x^2 - b)^(1/2))^(5/4)*(d + c*x^2))/(a^2*x^2 - b)^(3/2),x)

[Out]

int(((a*x + (a^2*x^2 - b)^(1/2))^(5/4)*(d + c*x^2))/(a^2*x^2 - b)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x + \sqrt {a^{2} x^{2} - b}\right )^{\frac {5}{4}} \left (c x^{2} + d\right )}{\left (a^{2} x^{2} - b\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+d)*(a*x+(a**2*x**2-b)**(1/2))**(5/4)/(a**2*x**2-b)**(3/2),x)

[Out]

Integral((a*x + sqrt(a**2*x**2 - b))**(5/4)*(c*x**2 + d)/(a**2*x**2 - b)**(3/2), x)

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