3.5.52 \(\int \frac {x^{7/2} (c+d x^2)^3}{(a+b x^2)^2} \, dx\) [452]

Optimal. Leaf size=409 \[ \frac {(5 b c-17 a d) (b c-a d)^2 \sqrt {x}}{2 b^5}+\frac {d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^{5/2}}{10 b^4}+\frac {d^2 (39 b c-17 a d) x^{9/2}}{18 b^3}+\frac {17 d^3 x^{13/2}}{26 b^2}-\frac {x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{21/4}}-\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{21/4}}+\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} b^{21/4}}-\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} b^{21/4}} \]

[Out]

1/10*d*(17*a^2*d^2-39*a*b*c*d+27*b^2*c^2)*x^(5/2)/b^4+1/18*d^2*(-17*a*d+39*b*c)*x^(9/2)/b^3+17/26*d^3*x^(13/2)
/b^2-1/2*x^(5/2)*(d*x^2+c)^3/b/(b*x^2+a)+1/8*a^(1/4)*(-17*a*d+5*b*c)*(-a*d+b*c)^2*arctan(1-b^(1/4)*2^(1/2)*x^(
1/2)/a^(1/4))/b^(21/4)*2^(1/2)-1/8*a^(1/4)*(-17*a*d+5*b*c)*(-a*d+b*c)^2*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/
4))/b^(21/4)*2^(1/2)+1/16*a^(1/4)*(-17*a*d+5*b*c)*(-a*d+b*c)^2*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^
(1/2))/b^(21/4)*2^(1/2)-1/16*a^(1/4)*(-17*a*d+5*b*c)*(-a*d+b*c)^2*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)
*x^(1/2))/b^(21/4)*2^(1/2)+1/2*(-17*a*d+5*b*c)*(-a*d+b*c)^2*x^(1/2)/b^5

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Rubi [A]
time = 0.31, antiderivative size = 409, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {477, 478, 584, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {d x^{5/2} \left (17 a^2 d^2-39 a b c d+27 b^2 c^2\right )}{10 b^4}+\frac {\sqrt [4]{a} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (5 b c-17 a d) (b c-a d)^2}{4 \sqrt {2} b^{21/4}}-\frac {\sqrt [4]{a} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (5 b c-17 a d) (b c-a d)^2}{4 \sqrt {2} b^{21/4}}+\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} b^{21/4}}-\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} b^{21/4}}+\frac {\sqrt {x} (5 b c-17 a d) (b c-a d)^2}{2 b^5}+\frac {d^2 x^{9/2} (39 b c-17 a d)}{18 b^3}-\frac {x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {17 d^3 x^{13/2}}{26 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^(7/2)*(c + d*x^2)^3)/(a + b*x^2)^2,x]

[Out]

((5*b*c - 17*a*d)*(b*c - a*d)^2*Sqrt[x])/(2*b^5) + (d*(27*b^2*c^2 - 39*a*b*c*d + 17*a^2*d^2)*x^(5/2))/(10*b^4)
+ (d^2*(39*b*c - 17*a*d)*x^(9/2))/(18*b^3) + (17*d^3*x^(13/2))/(26*b^2) - (x^(5/2)*(c + d*x^2)^3)/(2*b*(a + b
*x^2)) + (a^(1/4)*(5*b*c - 17*a*d)*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(
21/4)) - (a^(1/4)*(5*b*c - 17*a*d)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(
21/4)) + (a^(1/4)*(5*b*c - 17*a*d)*(b*c - a*d)^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(
8*Sqrt[2]*b^(21/4)) - (a^(1/4)*(5*b*c - 17*a*d)*(b*c - a*d)^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/(8*Sqrt[2]*b^(21/4))

Rule 210

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

Rule 217

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 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 478

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

Rule 584

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

Rule 631

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 642

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 1176

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 1179

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]

Rubi steps

\begin {align*} \int \frac {x^{7/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {x^8 \left (c+d x^4\right )^3}{\left (a+b x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\text {Subst}\left (\int \frac {x^4 \left (c+d x^4\right )^2 \left (5 c+17 d x^4\right )}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 b}\\ &=-\frac {x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\text {Subst}\left (\int \left (\frac {(5 b c-17 a d) (b c-a d)^2}{b^4}+\frac {d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^4}{b^3}+\frac {d^2 (39 b c-17 a d) x^8}{b^2}+\frac {17 d^3 x^{12}}{b}+\frac {-5 a b^3 c^3+27 a^2 b^2 c^2 d-39 a^3 b c d^2+17 a^4 d^3}{b^4 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 b}\\ &=\frac {(5 b c-17 a d) (b c-a d)^2 \sqrt {x}}{2 b^5}+\frac {d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^{5/2}}{10 b^4}+\frac {d^2 (39 b c-17 a d) x^{9/2}}{18 b^3}+\frac {17 d^3 x^{13/2}}{26 b^2}-\frac {x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac {\left (a (5 b c-17 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 b^5}\\ &=\frac {(5 b c-17 a d) (b c-a d)^2 \sqrt {x}}{2 b^5}+\frac {d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^{5/2}}{10 b^4}+\frac {d^2 (39 b c-17 a d) x^{9/2}}{18 b^3}+\frac {17 d^3 x^{13/2}}{26 b^2}-\frac {x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac {\left (\sqrt {a} (5 b c-17 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 b^5}-\frac {\left (\sqrt {a} (5 b c-17 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 b^5}\\ &=\frac {(5 b c-17 a d) (b c-a d)^2 \sqrt {x}}{2 b^5}+\frac {d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^{5/2}}{10 b^4}+\frac {d^2 (39 b c-17 a d) x^{9/2}}{18 b^3}+\frac {17 d^3 x^{13/2}}{26 b^2}-\frac {x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac {\left (\sqrt {a} (5 b c-17 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 b^{11/2}}-\frac {\left (\sqrt {a} (5 b c-17 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 b^{11/2}}+\frac {\left (\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} b^{21/4}}+\frac {\left (\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} b^{21/4}}\\ &=\frac {(5 b c-17 a d) (b c-a d)^2 \sqrt {x}}{2 b^5}+\frac {d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^{5/2}}{10 b^4}+\frac {d^2 (39 b c-17 a d) x^{9/2}}{18 b^3}+\frac {17 d^3 x^{13/2}}{26 b^2}-\frac {x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} b^{21/4}}-\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} b^{21/4}}-\frac {\left (\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{21/4}}+\frac {\left (\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{21/4}}\\ &=\frac {(5 b c-17 a d) (b c-a d)^2 \sqrt {x}}{2 b^5}+\frac {d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^{5/2}}{10 b^4}+\frac {d^2 (39 b c-17 a d) x^{9/2}}{18 b^3}+\frac {17 d^3 x^{13/2}}{26 b^2}-\frac {x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{21/4}}-\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{21/4}}+\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} b^{21/4}}-\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} b^{21/4}}\\ \end {align*}

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Mathematica [A]
time = 0.57, size = 301, normalized size = 0.74 \begin {gather*} \frac {\frac {4 \sqrt [4]{b} \sqrt {x} \left (-9945 a^4 d^3+117 a^3 b d^2 \left (195 c-68 d x^2\right )+13 a^2 b^2 d \left (-1215 c^2+1404 c d x^2+68 d^2 x^4\right )+a b^3 \left (2925 c^3-12636 c^2 d x^2-2028 c d^2 x^4-340 d^3 x^6\right )+12 b^4 x^2 \left (195 c^3+117 c^2 d x^2+65 c d^2 x^4+15 d^3 x^6\right )\right )}{a+b x^2}-585 \sqrt {2} \sqrt [4]{a} (b c-a d)^2 (-5 b c+17 a d) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+585 \sqrt {2} \sqrt [4]{a} (b c-a d)^2 (-5 b c+17 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{4680 b^{21/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^(7/2)*(c + d*x^2)^3)/(a + b*x^2)^2,x]

[Out]

((4*b^(1/4)*Sqrt[x]*(-9945*a^4*d^3 + 117*a^3*b*d^2*(195*c - 68*d*x^2) + 13*a^2*b^2*d*(-1215*c^2 + 1404*c*d*x^2
+ 68*d^2*x^4) + a*b^3*(2925*c^3 - 12636*c^2*d*x^2 - 2028*c*d^2*x^4 - 340*d^3*x^6) + 12*b^4*x^2*(195*c^3 + 117
*c^2*d*x^2 + 65*c*d^2*x^4 + 15*d^3*x^6)))/(a + b*x^2) - 585*Sqrt[2]*a^(1/4)*(b*c - a*d)^2*(-5*b*c + 17*a*d)*Ar
cTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 585*Sqrt[2]*a^(1/4)*(b*c - a*d)^2*(-5*b*c + 17
*a*d)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(4680*b^(21/4))

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Maple [A]
time = 0.15, size = 327, normalized size = 0.80 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(d*x^2+c)^3/(b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

-2/b^5*(-1/13*d^3*x^(13/2)*b^3+2/9*a*b^2*d^3*x^(9/2)-1/3*b^3*c*d^2*x^(9/2)-3/5*a^2*b*d^3*x^(5/2)+6/5*a*b^2*c*d
^2*x^(5/2)-3/5*b^3*c^2*d*x^(5/2)+4*a^3*d^3*x^(1/2)-9*a^2*b*c*d^2*x^(1/2)+6*a*b^2*c^2*d*x^(1/2)-b^3*c^3*x^(1/2)
)+2*a/b^5*((-1/4*a^3*d^3+3/4*a^2*b*c*d^2-3/4*a*b^2*c^2*d+1/4*b^3*c^3)*x^(1/2)/(b*x^2+a)+1/32*(17*a^3*d^3-39*a^
2*b*c*d^2+27*a*b^2*c^2*d-5*b^3*c^3)*(a/b)^(1/4)/a*2^(1/2)*(ln((x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x-(
a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*
x^(1/2)-1)))

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Maxima [A]
time = 0.50, size = 499, normalized size = 1.22 \begin {gather*} \frac {{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \sqrt {x}}{2 \, {\left (b^{6} x^{2} + a b^{5}\right )}} - \frac {{\left (\frac {2 \, \sqrt {2} {\left (5 \, b^{3} c^{3} - 27 \, a b^{2} c^{2} d + 39 \, a^{2} b c d^{2} - 17 \, a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (5 \, b^{3} c^{3} - 27 \, a b^{2} c^{2} d + 39 \, a^{2} b c d^{2} - 17 \, a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (5 \, b^{3} c^{3} - 27 \, a b^{2} c^{2} d + 39 \, a^{2} b c d^{2} - 17 \, a^{3} d^{3}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (5 \, b^{3} c^{3} - 27 \, a b^{2} c^{2} d + 39 \, a^{2} b c d^{2} - 17 \, a^{3} d^{3}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )} a}{16 \, b^{5}} + \frac {2 \, {\left (45 \, b^{3} d^{3} x^{\frac {13}{2}} + 65 \, {\left (3 \, b^{3} c d^{2} - 2 \, a b^{2} d^{3}\right )} x^{\frac {9}{2}} + 351 \, {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{\frac {5}{2}} + 585 \, {\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} \sqrt {x}\right )}}{585 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(d*x^2+c)^3/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

1/2*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*sqrt(x)/(b^6*x^2 + a*b^5) - 1/16*(2*sqrt(2)*(5*b^3
*c^3 - 27*a*b^2*c^2*d + 39*a^2*b*c*d^2 - 17*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*s
qrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(5*b^3*c^3 - 27*a*b^2*c^2*d + 39*a^
2*b*c*d^2 - 17*a^3*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)
))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(5*b^3*c^3 - 27*a*b^2*c^2*d + 39*a^2*b*c*d^2 - 17*a^3*d^3)*log(sq
rt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(5*b^3*c^3 - 27*a*b^2*c^2*d +
39*a^2*b*c*d^2 - 17*a^3*d^3)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))*a
/b^5 + 2/585*(45*b^3*d^3*x^(13/2) + 65*(3*b^3*c*d^2 - 2*a*b^2*d^3)*x^(9/2) + 351*(b^3*c^2*d - 2*a*b^2*c*d^2 +
a^2*b*d^3)*x^(5/2) + 585*(b^3*c^3 - 6*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 4*a^3*d^3)*sqrt(x))/b^5

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2014 vs. \(2 (317) = 634\).
time = 0.79, size = 2014, normalized size = 4.92 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(d*x^2+c)^3/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

1/4680*(2340*(b^6*x^2 + a*b^5)*(-(625*a*b^12*c^12 - 13500*a^2*b^11*c^11*d + 128850*a^3*b^10*c^10*d^2 - 718060*
a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d^4 - 6477048*a^6*b^7*c^7*d^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*
b^5*c^5*d^7 + 12631455*a^9*b^4*c^4*d^8 - 7783756*a^10*b^3*c^3*d^9 + 3168018*a^11*b^2*c^2*d^10 - 766428*a^12*b*
c*d^11 + 83521*a^13*d^12)/b^21)^(1/4)*arctan((sqrt(b^10*sqrt(-(625*a*b^12*c^12 - 13500*a^2*b^11*c^11*d + 12885
0*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d^4 - 6477048*a^6*b^7*c^7*d^5 + 11369148*a^
7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4*c^4*d^8 - 7783756*a^10*b^3*c^3*d^9 + 3168018*a^11*
b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*d^12)/b^21) + (25*b^6*c^6 - 270*a*b^5*c^5*d + 1119*a^2*b^4*c^
4*d^2 - 2276*a^3*b^3*c^3*d^3 + 2439*a^4*b^2*c^2*d^4 - 1326*a^5*b*c*d^5 + 289*a^6*d^6)*x)*b^16*(-(625*a*b^12*c^
12 - 13500*a^2*b^11*c^11*d + 128850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d^4 - 647
7048*a^6*b^7*c^7*d^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4*c^4*d^8 - 778375
6*a^10*b^3*c^3*d^9 + 3168018*a^11*b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*d^12)/b^21)^(3/4) + (5*b^19
*c^3 - 27*a*b^18*c^2*d + 39*a^2*b^17*c*d^2 - 17*a^3*b^16*d^3)*sqrt(x)*(-(625*a*b^12*c^12 - 13500*a^2*b^11*c^11
*d + 128850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d^4 - 6477048*a^6*b^7*c^7*d^5 + 1
1369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4*c^4*d^8 - 7783756*a^10*b^3*c^3*d^9 + 316
8018*a^11*b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*d^12)/b^21)^(3/4))/(625*a*b^12*c^12 - 13500*a^2*b^1
1*c^11*d + 128850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d^4 - 6477048*a^6*b^7*c^7*d
^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4*c^4*d^8 - 7783756*a^10*b^3*c^3*d^9
+ 3168018*a^11*b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*d^12)) + 585*(b^6*x^2 + a*b^5)*(-(625*a*b^12*
c^12 - 13500*a^2*b^11*c^11*d + 128850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d^4 - 6
477048*a^6*b^7*c^7*d^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4*c^4*d^8 - 7783
756*a^10*b^3*c^3*d^9 + 3168018*a^11*b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*d^12)/b^21)^(1/4)*log(b^5
*(-(625*a*b^12*c^12 - 13500*a^2*b^11*c^11*d + 128850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 2603151*a^5*
b^8*c^8*d^4 - 6477048*a^6*b^7*c^7*d^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4
*c^4*d^8 - 7783756*a^10*b^3*c^3*d^9 + 3168018*a^11*b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*d^12)/b^21
)^(1/4) - (5*b^3*c^3 - 27*a*b^2*c^2*d + 39*a^2*b*c*d^2 - 17*a^3*d^3)*sqrt(x)) - 585*(b^6*x^2 + a*b^5)*(-(625*a
*b^12*c^12 - 13500*a^2*b^11*c^11*d + 128850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d
^4 - 6477048*a^6*b^7*c^7*d^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4*c^4*d^8
- 7783756*a^10*b^3*c^3*d^9 + 3168018*a^11*b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*d^12)/b^21)^(1/4)*l
og(-b^5*(-(625*a*b^12*c^12 - 13500*a^2*b^11*c^11*d + 128850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 26031
51*a^5*b^8*c^8*d^4 - 6477048*a^6*b^7*c^7*d^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*
a^9*b^4*c^4*d^8 - 7783756*a^10*b^3*c^3*d^9 + 3168018*a^11*b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*d^1
2)/b^21)^(1/4) - (5*b^3*c^3 - 27*a*b^2*c^2*d + 39*a^2*b*c*d^2 - 17*a^3*d^3)*sqrt(x)) + 4*(180*b^4*d^3*x^8 + 29
25*a*b^3*c^3 - 15795*a^2*b^2*c^2*d + 22815*a^3*b*c*d^2 - 9945*a^4*d^3 + 20*(39*b^4*c*d^2 - 17*a*b^3*d^3)*x^6 +
52*(27*b^4*c^2*d - 39*a*b^3*c*d^2 + 17*a^2*b^2*d^3)*x^4 + 468*(5*b^4*c^3 - 27*a*b^3*c^2*d + 39*a^2*b^2*c*d^2
- 17*a^3*b*d^3)*x^2)*sqrt(x))/(b^6*x^2 + a*b^5)

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Sympy [F(-1)] Timed out
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(x**(7/2)*(d*x**2+c)**3/(b*x**2+a)**2,x)

[Out]

Timed out

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Giac [A]
time = 0.95, size = 600, normalized size = 1.47 \begin {gather*} -\frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 39 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 17 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, b^{6}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 39 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 17 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, b^{6}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 39 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 17 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, b^{6}} + \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 39 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 17 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, b^{6}} + \frac {a b^{3} c^{3} \sqrt {x} - 3 \, a^{2} b^{2} c^{2} d \sqrt {x} + 3 \, a^{3} b c d^{2} \sqrt {x} - a^{4} d^{3} \sqrt {x}}{2 \, {\left (b x^{2} + a\right )} b^{5}} + \frac {2 \, {\left (45 \, b^{24} d^{3} x^{\frac {13}{2}} + 195 \, b^{24} c d^{2} x^{\frac {9}{2}} - 130 \, a b^{23} d^{3} x^{\frac {9}{2}} + 351 \, b^{24} c^{2} d x^{\frac {5}{2}} - 702 \, a b^{23} c d^{2} x^{\frac {5}{2}} + 351 \, a^{2} b^{22} d^{3} x^{\frac {5}{2}} + 585 \, b^{24} c^{3} \sqrt {x} - 3510 \, a b^{23} c^{2} d \sqrt {x} + 5265 \, a^{2} b^{22} c d^{2} \sqrt {x} - 2340 \, a^{3} b^{21} d^{3} \sqrt {x}\right )}}{585 \, b^{26}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(d*x^2+c)^3/(b*x^2+a)^2,x, algorithm="giac")

[Out]

-1/8*sqrt(2)*(5*(a*b^3)^(1/4)*b^3*c^3 - 27*(a*b^3)^(1/4)*a*b^2*c^2*d + 39*(a*b^3)^(1/4)*a^2*b*c*d^2 - 17*(a*b^
3)^(1/4)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/b^6 - 1/8*sqrt(2)*(5*(a*b^
3)^(1/4)*b^3*c^3 - 27*(a*b^3)^(1/4)*a*b^2*c^2*d + 39*(a*b^3)^(1/4)*a^2*b*c*d^2 - 17*(a*b^3)^(1/4)*a^3*d^3)*arc
tan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/b^6 - 1/16*sqrt(2)*(5*(a*b^3)^(1/4)*b^3*c^3 -
27*(a*b^3)^(1/4)*a*b^2*c^2*d + 39*(a*b^3)^(1/4)*a^2*b*c*d^2 - 17*(a*b^3)^(1/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a
/b)^(1/4) + x + sqrt(a/b))/b^6 + 1/16*sqrt(2)*(5*(a*b^3)^(1/4)*b^3*c^3 - 27*(a*b^3)^(1/4)*a*b^2*c^2*d + 39*(a*
b^3)^(1/4)*a^2*b*c*d^2 - 17*(a*b^3)^(1/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^6 + 1/2
*(a*b^3*c^3*sqrt(x) - 3*a^2*b^2*c^2*d*sqrt(x) + 3*a^3*b*c*d^2*sqrt(x) - a^4*d^3*sqrt(x))/((b*x^2 + a)*b^5) + 2
/585*(45*b^24*d^3*x^(13/2) + 195*b^24*c*d^2*x^(9/2) - 130*a*b^23*d^3*x^(9/2) + 351*b^24*c^2*d*x^(5/2) - 702*a*
b^23*c*d^2*x^(5/2) + 351*a^2*b^22*d^3*x^(5/2) + 585*b^24*c^3*sqrt(x) - 3510*a*b^23*c^2*d*sqrt(x) + 5265*a^2*b^
22*c*d^2*sqrt(x) - 2340*a^3*b^21*d^3*sqrt(x))/b^26

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Mupad [B]
time = 0.16, size = 1850, normalized size = 4.52 \begin {gather*} \sqrt {x}\,\left (\frac {2\,c^3}{b^2}-\frac {2\,a\,\left (\frac {6\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {4\,a\,d^3}{b^3}-\frac {6\,c\,d^2}{b^2}\right )}{b}-\frac {2\,a^2\,d^3}{b^4}\right )}{b}+\frac {a^2\,\left (\frac {4\,a\,d^3}{b^3}-\frac {6\,c\,d^2}{b^2}\right )}{b^2}\right )-x^{9/2}\,\left (\frac {4\,a\,d^3}{9\,b^3}-\frac {2\,c\,d^2}{3\,b^2}\right )+x^{5/2}\,\left (\frac {6\,c^2\,d}{5\,b^2}+\frac {2\,a\,\left (\frac {4\,a\,d^3}{b^3}-\frac {6\,c\,d^2}{b^2}\right )}{5\,b}-\frac {2\,a^2\,d^3}{5\,b^4}\right )-\frac {\sqrt {x}\,\left (\frac {a^4\,d^3}{2}-\frac {3\,a^3\,b\,c\,d^2}{2}+\frac {3\,a^2\,b^2\,c^2\,d}{2}-\frac {a\,b^3\,c^3}{2}\right )}{b^6\,x^2+a\,b^5}+\frac {2\,d^3\,x^{13/2}}{13\,b^2}-\frac {{\left (-a\right )}^{1/4}\,\mathrm {atan}\left (\frac {\frac {{\left (-a\right )}^{1/4}\,{\left (a\,d-b\,c\right )}^2\,\left (17\,a\,d-5\,b\,c\right )\,\left (\frac {\sqrt {x}\,\left (289\,a^8\,d^6-1326\,a^7\,b\,c\,d^5+2439\,a^6\,b^2\,c^2\,d^4-2276\,a^5\,b^3\,c^3\,d^3+1119\,a^4\,b^4\,c^4\,d^2-270\,a^3\,b^5\,c^5\,d+25\,a^2\,b^6\,c^6\right )}{b^7}+\frac {{\left (-a\right )}^{1/4}\,{\left (a\,d-b\,c\right )}^2\,\left (17\,a\,d-5\,b\,c\right )\,\left (17\,a^5\,d^3-39\,a^4\,b\,c\,d^2+27\,a^3\,b^2\,c^2\,d-5\,a^2\,b^3\,c^3\right )}{b^{29/4}}\right )\,1{}\mathrm {i}}{8\,b^{21/4}}+\frac {{\left (-a\right )}^{1/4}\,{\left (a\,d-b\,c\right )}^2\,\left (17\,a\,d-5\,b\,c\right )\,\left (\frac {\sqrt {x}\,\left (289\,a^8\,d^6-1326\,a^7\,b\,c\,d^5+2439\,a^6\,b^2\,c^2\,d^4-2276\,a^5\,b^3\,c^3\,d^3+1119\,a^4\,b^4\,c^4\,d^2-270\,a^3\,b^5\,c^5\,d+25\,a^2\,b^6\,c^6\right )}{b^7}-\frac {{\left (-a\right )}^{1/4}\,{\left (a\,d-b\,c\right )}^2\,\left (17\,a\,d-5\,b\,c\right )\,\left (17\,a^5\,d^3-39\,a^4\,b\,c\,d^2+27\,a^3\,b^2\,c^2\,d-5\,a^2\,b^3\,c^3\right )}{b^{29/4}}\right )\,1{}\mathrm {i}}{8\,b^{21/4}}}{\frac {{\left (-a\right )}^{1/4}\,{\left (a\,d-b\,c\right )}^2\,\left (17\,a\,d-5\,b\,c\right )\,\left (\frac {\sqrt {x}\,\left (289\,a^8\,d^6-1326\,a^7\,b\,c\,d^5+2439\,a^6\,b^2\,c^2\,d^4-2276\,a^5\,b^3\,c^3\,d^3+1119\,a^4\,b^4\,c^4\,d^2-270\,a^3\,b^5\,c^5\,d+25\,a^2\,b^6\,c^6\right )}{b^7}+\frac {{\left (-a\right )}^{1/4}\,{\left (a\,d-b\,c\right )}^2\,\left (17\,a\,d-5\,b\,c\right )\,\left (17\,a^5\,d^3-39\,a^4\,b\,c\,d^2+27\,a^3\,b^2\,c^2\,d-5\,a^2\,b^3\,c^3\right )}{b^{29/4}}\right )}{8\,b^{21/4}}-\frac {{\left (-a\right )}^{1/4}\,{\left (a\,d-b\,c\right )}^2\,\left (17\,a\,d-5\,b\,c\right )\,\left (\frac {\sqrt {x}\,\left (289\,a^8\,d^6-1326\,a^7\,b\,c\,d^5+2439\,a^6\,b^2\,c^2\,d^4-2276\,a^5\,b^3\,c^3\,d^3+1119\,a^4\,b^4\,c^4\,d^2-270\,a^3\,b^5\,c^5\,d+25\,a^2\,b^6\,c^6\right )}{b^7}-\frac {{\left (-a\right )}^{1/4}\,{\left (a\,d-b\,c\right )}^2\,\left (17\,a\,d-5\,b\,c\right )\,\left (17\,a^5\,d^3-39\,a^4\,b\,c\,d^2+27\,a^3\,b^2\,c^2\,d-5\,a^2\,b^3\,c^3\right )}{b^{29/4}}\right )}{8\,b^{21/4}}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (17\,a\,d-5\,b\,c\right )\,1{}\mathrm {i}}{4\,b^{21/4}}+\frac {{\left (-a\right )}^{1/4}\,\mathrm {atan}\left (\frac {\frac {{\left (-a\right )}^{1/4}\,{\left (a\,d-b\,c\right )}^2\,\left (17\,a\,d-5\,b\,c\right )\,\left (\frac {\sqrt {x}\,\left (289\,a^8\,d^6-1326\,a^7\,b\,c\,d^5+2439\,a^6\,b^2\,c^2\,d^4-2276\,a^5\,b^3\,c^3\,d^3+1119\,a^4\,b^4\,c^4\,d^2-270\,a^3\,b^5\,c^5\,d+25\,a^2\,b^6\,c^6\right )}{b^7}-\frac {{\left (-a\right )}^{1/4}\,{\left (a\,d-b\,c\right )}^2\,\left (17\,a\,d-5\,b\,c\right )\,\left (17\,a^5\,d^3-39\,a^4\,b\,c\,d^2+27\,a^3\,b^2\,c^2\,d-5\,a^2\,b^3\,c^3\right )\,1{}\mathrm {i}}{b^{29/4}}\right )}{8\,b^{21/4}}+\frac {{\left (-a\right )}^{1/4}\,{\left (a\,d-b\,c\right )}^2\,\left (17\,a\,d-5\,b\,c\right )\,\left (\frac {\sqrt {x}\,\left (289\,a^8\,d^6-1326\,a^7\,b\,c\,d^5+2439\,a^6\,b^2\,c^2\,d^4-2276\,a^5\,b^3\,c^3\,d^3+1119\,a^4\,b^4\,c^4\,d^2-270\,a^3\,b^5\,c^5\,d+25\,a^2\,b^6\,c^6\right )}{b^7}+\frac {{\left (-a\right )}^{1/4}\,{\left (a\,d-b\,c\right )}^2\,\left (17\,a\,d-5\,b\,c\right )\,\left (17\,a^5\,d^3-39\,a^4\,b\,c\,d^2+27\,a^3\,b^2\,c^2\,d-5\,a^2\,b^3\,c^3\right )\,1{}\mathrm {i}}{b^{29/4}}\right )}{8\,b^{21/4}}}{\frac {{\left (-a\right )}^{1/4}\,{\left (a\,d-b\,c\right )}^2\,\left (17\,a\,d-5\,b\,c\right )\,\left (\frac {\sqrt {x}\,\left (289\,a^8\,d^6-1326\,a^7\,b\,c\,d^5+2439\,a^6\,b^2\,c^2\,d^4-2276\,a^5\,b^3\,c^3\,d^3+1119\,a^4\,b^4\,c^4\,d^2-270\,a^3\,b^5\,c^5\,d+25\,a^2\,b^6\,c^6\right )}{b^7}-\frac {{\left (-a\right )}^{1/4}\,{\left (a\,d-b\,c\right )}^2\,\left (17\,a\,d-5\,b\,c\right )\,\left (17\,a^5\,d^3-39\,a^4\,b\,c\,d^2+27\,a^3\,b^2\,c^2\,d-5\,a^2\,b^3\,c^3\right )\,1{}\mathrm {i}}{b^{29/4}}\right )\,1{}\mathrm {i}}{8\,b^{21/4}}-\frac {{\left (-a\right )}^{1/4}\,{\left (a\,d-b\,c\right )}^2\,\left (17\,a\,d-5\,b\,c\right )\,\left (\frac {\sqrt {x}\,\left (289\,a^8\,d^6-1326\,a^7\,b\,c\,d^5+2439\,a^6\,b^2\,c^2\,d^4-2276\,a^5\,b^3\,c^3\,d^3+1119\,a^4\,b^4\,c^4\,d^2-270\,a^3\,b^5\,c^5\,d+25\,a^2\,b^6\,c^6\right )}{b^7}+\frac {{\left (-a\right )}^{1/4}\,{\left (a\,d-b\,c\right )}^2\,\left (17\,a\,d-5\,b\,c\right )\,\left (17\,a^5\,d^3-39\,a^4\,b\,c\,d^2+27\,a^3\,b^2\,c^2\,d-5\,a^2\,b^3\,c^3\right )\,1{}\mathrm {i}}{b^{29/4}}\right )\,1{}\mathrm {i}}{8\,b^{21/4}}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (17\,a\,d-5\,b\,c\right )}{4\,b^{21/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(7/2)*(c + d*x^2)^3)/(a + b*x^2)^2,x)

[Out]

x^(1/2)*((2*c^3)/b^2 - (2*a*((6*c^2*d)/b^2 + (2*a*((4*a*d^3)/b^3 - (6*c*d^2)/b^2))/b - (2*a^2*d^3)/b^4))/b + (
a^2*((4*a*d^3)/b^3 - (6*c*d^2)/b^2))/b^2) - x^(9/2)*((4*a*d^3)/(9*b^3) - (2*c*d^2)/(3*b^2)) + x^(5/2)*((6*c^2*
d)/(5*b^2) + (2*a*((4*a*d^3)/b^3 - (6*c*d^2)/b^2))/(5*b) - (2*a^2*d^3)/(5*b^4)) - (x^(1/2)*((a^4*d^3)/2 - (a*b
^3*c^3)/2 + (3*a^2*b^2*c^2*d)/2 - (3*a^3*b*c*d^2)/2))/(a*b^5 + b^6*x^2) + (2*d^3*x^(13/2))/(13*b^2) - ((-a)^(1
/4)*atan((((-a)^(1/4)*(a*d - b*c)^2*(17*a*d - 5*b*c)*((x^(1/2)*(289*a^8*d^6 + 25*a^2*b^6*c^6 - 270*a^3*b^5*c^5
*d + 1119*a^4*b^4*c^4*d^2 - 2276*a^5*b^3*c^3*d^3 + 2439*a^6*b^2*c^2*d^4 - 1326*a^7*b*c*d^5))/b^7 + ((-a)^(1/4)
*(a*d - b*c)^2*(17*a*d - 5*b*c)*(17*a^5*d^3 - 5*a^2*b^3*c^3 + 27*a^3*b^2*c^2*d - 39*a^4*b*c*d^2))/b^(29/4))*1i
)/(8*b^(21/4)) + ((-a)^(1/4)*(a*d - b*c)^2*(17*a*d - 5*b*c)*((x^(1/2)*(289*a^8*d^6 + 25*a^2*b^6*c^6 - 270*a^3*
b^5*c^5*d + 1119*a^4*b^4*c^4*d^2 - 2276*a^5*b^3*c^3*d^3 + 2439*a^6*b^2*c^2*d^4 - 1326*a^7*b*c*d^5))/b^7 - ((-a
)^(1/4)*(a*d - b*c)^2*(17*a*d - 5*b*c)*(17*a^5*d^3 - 5*a^2*b^3*c^3 + 27*a^3*b^2*c^2*d - 39*a^4*b*c*d^2))/b^(29
/4))*1i)/(8*b^(21/4)))/(((-a)^(1/4)*(a*d - b*c)^2*(17*a*d - 5*b*c)*((x^(1/2)*(289*a^8*d^6 + 25*a^2*b^6*c^6 - 2
70*a^3*b^5*c^5*d + 1119*a^4*b^4*c^4*d^2 - 2276*a^5*b^3*c^3*d^3 + 2439*a^6*b^2*c^2*d^4 - 1326*a^7*b*c*d^5))/b^7
+ ((-a)^(1/4)*(a*d - b*c)^2*(17*a*d - 5*b*c)*(17*a^5*d^3 - 5*a^2*b^3*c^3 + 27*a^3*b^2*c^2*d - 39*a^4*b*c*d^2)
)/b^(29/4)))/(8*b^(21/4)) - ((-a)^(1/4)*(a*d - b*c)^2*(17*a*d - 5*b*c)*((x^(1/2)*(289*a^8*d^6 + 25*a^2*b^6*c^6
- 270*a^3*b^5*c^5*d + 1119*a^4*b^4*c^4*d^2 - 2276*a^5*b^3*c^3*d^3 + 2439*a^6*b^2*c^2*d^4 - 1326*a^7*b*c*d^5))
/b^7 - ((-a)^(1/4)*(a*d - b*c)^2*(17*a*d - 5*b*c)*(17*a^5*d^3 - 5*a^2*b^3*c^3 + 27*a^3*b^2*c^2*d - 39*a^4*b*c*
d^2))/b^(29/4)))/(8*b^(21/4))))*(a*d - b*c)^2*(17*a*d - 5*b*c)*1i)/(4*b^(21/4)) + ((-a)^(1/4)*atan((((-a)^(1/4
)*(a*d - b*c)^2*(17*a*d - 5*b*c)*((x^(1/2)*(289*a^8*d^6 + 25*a^2*b^6*c^6 - 270*a^3*b^5*c^5*d + 1119*a^4*b^4*c^
4*d^2 - 2276*a^5*b^3*c^3*d^3 + 2439*a^6*b^2*c^2*d^4 - 1326*a^7*b*c*d^5))/b^7 - ((-a)^(1/4)*(a*d - b*c)^2*(17*a
*d - 5*b*c)*(17*a^5*d^3 - 5*a^2*b^3*c^3 + 27*a^3*b^2*c^2*d - 39*a^4*b*c*d^2)*1i)/b^(29/4)))/(8*b^(21/4)) + ((-
a)^(1/4)*(a*d - b*c)^2*(17*a*d - 5*b*c)*((x^(1/2)*(289*a^8*d^6 + 25*a^2*b^6*c^6 - 270*a^3*b^5*c^5*d + 1119*a^4
*b^4*c^4*d^2 - 2276*a^5*b^3*c^3*d^3 + 2439*a^6*b^2*c^2*d^4 - 1326*a^7*b*c*d^5))/b^7 + ((-a)^(1/4)*(a*d - b*c)^
2*(17*a*d - 5*b*c)*(17*a^5*d^3 - 5*a^2*b^3*c^3 + 27*a^3*b^2*c^2*d - 39*a^4*b*c*d^2)*1i)/b^(29/4)))/(8*b^(21/4)
))/(((-a)^(1/4)*(a*d - b*c)^2*(17*a*d - 5*b*c)*((x^(1/2)*(289*a^8*d^6 + 25*a^2*b^6*c^6 - 270*a^3*b^5*c^5*d + 1
119*a^4*b^4*c^4*d^2 - 2276*a^5*b^3*c^3*d^3 + 2439*a^6*b^2*c^2*d^4 - 1326*a^7*b*c*d^5))/b^7 - ((-a)^(1/4)*(a*d
- b*c)^2*(17*a*d - 5*b*c)*(17*a^5*d^3 - 5*a^2*b^3*c^3 + 27*a^3*b^2*c^2*d - 39*a^4*b*c*d^2)*1i)/b^(29/4))*1i)/(
8*b^(21/4)) - ((-a)^(1/4)*(a*d - b*c)^2*(17*a*d - 5*b*c)*((x^(1/2)*(289*a^8*d^6 + 25*a^2*b^6*c^6 - 270*a^3*b^5
*c^5*d + 1119*a^4*b^4*c^4*d^2 - 2276*a^5*b^3*c^3*d^3 + 2439*a^6*b^2*c^2*d^4 - 1326*a^7*b*c*d^5))/b^7 + ((-a)^(
1/4)*(a*d - b*c)^2*(17*a*d - 5*b*c)*(17*a^5*d^3 - 5*a^2*b^3*c^3 + 27*a^3*b^2*c^2*d - 39*a^4*b*c*d^2)*1i)/b^(29
/4))*1i)/(8*b^(21/4))))*(a*d - b*c)^2*(17*a*d - 5*b*c))/(4*b^(21/4))

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