3.4.71 \(\int \frac {A+B x^2}{x^{5/2} (a+b x^2)^3} \, dx\)

Optimal. Leaf size=322 \[ \frac {7 (11 A b-3 a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {7 (11 A b-3 a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {7 (11 A b-3 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {7 (11 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {7 (11 A b-3 a B)}{48 a^3 b x^{3/2}}+\frac {11 A b-3 a B}{16 a^2 b x^{3/2} \left (a+b x^2\right )}+\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2} \]

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Rubi [A]  time = 0.24, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {457, 290, 325, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {11 A b-3 a B}{16 a^2 b x^{3/2} \left (a+b x^2\right )}-\frac {7 (11 A b-3 a B)}{48 a^3 b x^{3/2}}+\frac {7 (11 A b-3 a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {7 (11 A b-3 a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {7 (11 A b-3 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {7 (11 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(x^(5/2)*(a + b*x^2)^3),x]

[Out]

(-7*(11*A*b - 3*a*B))/(48*a^3*b*x^(3/2)) + (A*b - a*B)/(4*a*b*x^(3/2)*(a + b*x^2)^2) + (11*A*b - 3*a*B)/(16*a^
2*b*x^(3/2)*(a + b*x^2)) + (7*(11*A*b - 3*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(1
5/4)*b^(1/4)) - (7*(11*A*b - 3*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(15/4)*b^(1/4
)) + (7*(11*A*b - 3*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(15/4)*b^(1
/4)) - (7*(11*A*b - 3*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(15/4)*b^
(1/4))

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 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 290

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

Rule 325

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

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 457

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d
)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b*e*n*(p + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b
*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] &&
LeQ[-1, m, -(n*(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]

Rubi steps

\begin {align*} \int \frac {A+B x^2}{x^{5/2} \left (a+b x^2\right )^3} \, dx &=\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2}+\frac {\left (\frac {11 A b}{2}-\frac {3 a B}{2}\right ) \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2}+\frac {11 A b-3 a B}{16 a^2 b x^{3/2} \left (a+b x^2\right )}+\frac {(7 (11 A b-3 a B)) \int \frac {1}{x^{5/2} \left (a+b x^2\right )} \, dx}{32 a^2 b}\\ &=-\frac {7 (11 A b-3 a B)}{48 a^3 b x^{3/2}}+\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2}+\frac {11 A b-3 a B}{16 a^2 b x^{3/2} \left (a+b x^2\right )}-\frac {(7 (11 A b-3 a B)) \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{32 a^3}\\ &=-\frac {7 (11 A b-3 a B)}{48 a^3 b x^{3/2}}+\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2}+\frac {11 A b-3 a B}{16 a^2 b x^{3/2} \left (a+b x^2\right )}-\frac {(7 (11 A b-3 a B)) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{16 a^3}\\ &=-\frac {7 (11 A b-3 a B)}{48 a^3 b x^{3/2}}+\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2}+\frac {11 A b-3 a B}{16 a^2 b x^{3/2} \left (a+b x^2\right )}-\frac {(7 (11 A b-3 a B)) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^{7/2}}-\frac {(7 (11 A b-3 a B)) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^{7/2}}\\ &=-\frac {7 (11 A b-3 a B)}{48 a^3 b x^{3/2}}+\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2}+\frac {11 A b-3 a B}{16 a^2 b x^{3/2} \left (a+b x^2\right )}-\frac {(7 (11 A b-3 a B)) \operatorname {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 )}{64 a^{7/2} \sqrt {b}}-\frac {(7 (11 A b-3 a B)) \operatorname {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 )}{64 a^{7/2} \sqrt {b}}+\frac {(7 (11 A b-3 a B)) \operatorname {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 )}{64 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {(7 (11 A b-3 a B)) \operatorname {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 )}{64 \sqrt {2} a^{15/4} \sqrt [4]{b}}\\ &=-\frac {7 (11 A b-3 a B)}{48 a^3 b x^{3/2}}+\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2}+\frac {11 A b-3 a B}{16 a^2 b x^{3/2} \left (a+b x^2\right )}+\frac {7 (11 A b-3 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {7 (11 A b-3 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(7 (11 A b-3 a B)) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {(7 (11 A b-3 a B)) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{15/4} \sqrt [4]{b}}\\ &=-\frac {7 (11 A b-3 a B)}{48 a^3 b x^{3/2}}+\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2}+\frac {11 A b-3 a B}{16 a^2 b x^{3/2} \left (a+b x^2\right )}+\frac {7 (11 A b-3 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {7 (11 A b-3 a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {7 (11 A b-3 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {7 (11 A b-3 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{15/4} \sqrt [4]{b}}\\ \end {align*}

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Mathematica [A]  time = 0.41, size = 400, normalized size = 1.24 \begin {gather*} \frac {-\frac {96 a^{7/4} A b \sqrt {x}}{\left (a+b x^2\right )^2}-\frac {360 a^{3/4} A b \sqrt {x}}{a+b x^2}-\frac {256 a^{3/4} A}{x^{3/2}}+\frac {96 a^{11/4} B \sqrt {x}}{\left (a+b x^2\right )^2}+\frac {168 a^{7/4} B \sqrt {x}}{a+b x^2}+231 \sqrt {2} A b^{3/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-231 \sqrt {2} A b^{3/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+\frac {42 \sqrt {2} (11 A b-3 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}-\frac {42 \sqrt {2} (11 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{b}}-\frac {63 \sqrt {2} a B \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt [4]{b}}+\frac {63 \sqrt {2} a B \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt [4]{b}}}{384 a^{15/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(x^(5/2)*(a + b*x^2)^3),x]

[Out]

((-256*a^(3/4)*A)/x^(3/2) - (96*a^(7/4)*A*b*Sqrt[x])/(a + b*x^2)^2 + (96*a^(11/4)*B*Sqrt[x])/(a + b*x^2)^2 - (
360*a^(3/4)*A*b*Sqrt[x])/(a + b*x^2) + (168*a^(7/4)*B*Sqrt[x])/(a + b*x^2) + (42*Sqrt[2]*(11*A*b - 3*a*B)*ArcT
an[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) - (42*Sqrt[2]*(11*A*b - 3*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*
Sqrt[x])/a^(1/4)])/b^(1/4) + 231*Sqrt[2]*A*b^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]
- (63*Sqrt[2]*a*B*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4) - 231*Sqrt[2]*A*b^(3/4)*
Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + (63*Sqrt[2]*a*B*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(
1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4))/(384*a^(15/4))

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IntegrateAlgebraic [A]  time = 0.62, size = 192, normalized size = 0.60 \begin {gather*} -\frac {7 (3 a B-11 A b) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{32 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {7 (3 a B-11 A b) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{32 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {-32 a^2 A+33 a^2 B x^2-121 a A b x^2+21 a b B x^4-77 A b^2 x^4}{48 a^3 x^{3/2} \left (a+b x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x^2)/(x^(5/2)*(a + b*x^2)^3),x]

[Out]

(-32*a^2*A - 121*a*A*b*x^2 + 33*a^2*B*x^2 - 77*A*b^2*x^4 + 21*a*b*B*x^4)/(48*a^3*x^(3/2)*(a + b*x^2)^2) - (7*(
-11*A*b + 3*a*B)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(32*Sqrt[2]*a^(15/4)*b^(1/4)
) + (7*(-11*A*b + 3*a*B)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(32*Sqrt[2]*a^(15/4
)*b^(1/4))

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fricas [B]  time = 1.14, size = 809, normalized size = 2.51 \begin {gather*} -\frac {84 \, {\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )} \left (-\frac {81 \, B^{4} a^{4} - 1188 \, A B^{3} a^{3} b + 6534 \, A^{2} B^{2} a^{2} b^{2} - 15972 \, A^{3} B a b^{3} + 14641 \, A^{4} b^{4}}{a^{15} b}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a^{8} \sqrt {-\frac {81 \, B^{4} a^{4} - 1188 \, A B^{3} a^{3} b + 6534 \, A^{2} B^{2} a^{2} b^{2} - 15972 \, A^{3} B a b^{3} + 14641 \, A^{4} b^{4}}{a^{15} b}} + {\left (9 \, B^{2} a^{2} - 66 \, A B a b + 121 \, A^{2} b^{2}\right )} x} a^{11} b \left (-\frac {81 \, B^{4} a^{4} - 1188 \, A B^{3} a^{3} b + 6534 \, A^{2} B^{2} a^{2} b^{2} - 15972 \, A^{3} B a b^{3} + 14641 \, A^{4} b^{4}}{a^{15} b}\right )^{\frac {3}{4}} + {\left (3 \, B a^{12} b - 11 \, A a^{11} b^{2}\right )} \sqrt {x} \left (-\frac {81 \, B^{4} a^{4} - 1188 \, A B^{3} a^{3} b + 6534 \, A^{2} B^{2} a^{2} b^{2} - 15972 \, A^{3} B a b^{3} + 14641 \, A^{4} b^{4}}{a^{15} b}\right )^{\frac {3}{4}}}{81 \, B^{4} a^{4} - 1188 \, A B^{3} a^{3} b + 6534 \, A^{2} B^{2} a^{2} b^{2} - 15972 \, A^{3} B a b^{3} + 14641 \, A^{4} b^{4}}\right ) + 21 \, {\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )} \left (-\frac {81 \, B^{4} a^{4} - 1188 \, A B^{3} a^{3} b + 6534 \, A^{2} B^{2} a^{2} b^{2} - 15972 \, A^{3} B a b^{3} + 14641 \, A^{4} b^{4}}{a^{15} b}\right )^{\frac {1}{4}} \log \left (7 \, a^{4} \left (-\frac {81 \, B^{4} a^{4} - 1188 \, A B^{3} a^{3} b + 6534 \, A^{2} B^{2} a^{2} b^{2} - 15972 \, A^{3} B a b^{3} + 14641 \, A^{4} b^{4}}{a^{15} b}\right )^{\frac {1}{4}} - 7 \, {\left (3 \, B a - 11 \, A b\right )} \sqrt {x}\right ) - 21 \, {\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )} \left (-\frac {81 \, B^{4} a^{4} - 1188 \, A B^{3} a^{3} b + 6534 \, A^{2} B^{2} a^{2} b^{2} - 15972 \, A^{3} B a b^{3} + 14641 \, A^{4} b^{4}}{a^{15} b}\right )^{\frac {1}{4}} \log \left (-7 \, a^{4} \left (-\frac {81 \, B^{4} a^{4} - 1188 \, A B^{3} a^{3} b + 6534 \, A^{2} B^{2} a^{2} b^{2} - 15972 \, A^{3} B a b^{3} + 14641 \, A^{4} b^{4}}{a^{15} b}\right )^{\frac {1}{4}} - 7 \, {\left (3 \, B a - 11 \, A b\right )} \sqrt {x}\right ) - 4 \, {\left (7 \, {\left (3 \, B a b - 11 \, A b^{2}\right )} x^{4} - 32 \, A a^{2} + 11 \, {\left (3 \, B a^{2} - 11 \, A a b\right )} x^{2}\right )} \sqrt {x}}{192 \, {\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^(5/2)/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

-1/192*(84*(a^3*b^2*x^6 + 2*a^4*b*x^4 + a^5*x^2)*(-(81*B^4*a^4 - 1188*A*B^3*a^3*b + 6534*A^2*B^2*a^2*b^2 - 159
72*A^3*B*a*b^3 + 14641*A^4*b^4)/(a^15*b))^(1/4)*arctan((sqrt(a^8*sqrt(-(81*B^4*a^4 - 1188*A*B^3*a^3*b + 6534*A
^2*B^2*a^2*b^2 - 15972*A^3*B*a*b^3 + 14641*A^4*b^4)/(a^15*b)) + (9*B^2*a^2 - 66*A*B*a*b + 121*A^2*b^2)*x)*a^11
*b*(-(81*B^4*a^4 - 1188*A*B^3*a^3*b + 6534*A^2*B^2*a^2*b^2 - 15972*A^3*B*a*b^3 + 14641*A^4*b^4)/(a^15*b))^(3/4
) + (3*B*a^12*b - 11*A*a^11*b^2)*sqrt(x)*(-(81*B^4*a^4 - 1188*A*B^3*a^3*b + 6534*A^2*B^2*a^2*b^2 - 15972*A^3*B
*a*b^3 + 14641*A^4*b^4)/(a^15*b))^(3/4))/(81*B^4*a^4 - 1188*A*B^3*a^3*b + 6534*A^2*B^2*a^2*b^2 - 15972*A^3*B*a
*b^3 + 14641*A^4*b^4)) + 21*(a^3*b^2*x^6 + 2*a^4*b*x^4 + a^5*x^2)*(-(81*B^4*a^4 - 1188*A*B^3*a^3*b + 6534*A^2*
B^2*a^2*b^2 - 15972*A^3*B*a*b^3 + 14641*A^4*b^4)/(a^15*b))^(1/4)*log(7*a^4*(-(81*B^4*a^4 - 1188*A*B^3*a^3*b +
6534*A^2*B^2*a^2*b^2 - 15972*A^3*B*a*b^3 + 14641*A^4*b^4)/(a^15*b))^(1/4) - 7*(3*B*a - 11*A*b)*sqrt(x)) - 21*(
a^3*b^2*x^6 + 2*a^4*b*x^4 + a^5*x^2)*(-(81*B^4*a^4 - 1188*A*B^3*a^3*b + 6534*A^2*B^2*a^2*b^2 - 15972*A^3*B*a*b
^3 + 14641*A^4*b^4)/(a^15*b))^(1/4)*log(-7*a^4*(-(81*B^4*a^4 - 1188*A*B^3*a^3*b + 6534*A^2*B^2*a^2*b^2 - 15972
*A^3*B*a*b^3 + 14641*A^4*b^4)/(a^15*b))^(1/4) - 7*(3*B*a - 11*A*b)*sqrt(x)) - 4*(7*(3*B*a*b - 11*A*b^2)*x^4 -
32*A*a^2 + 11*(3*B*a^2 - 11*A*a*b)*x^2)*sqrt(x))/(a^3*b^2*x^6 + 2*a^4*b*x^4 + a^5*x^2)

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giac [A]  time = 0.45, size = 304, normalized size = 0.94 \begin {gather*} \frac {7 \, \sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 11 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\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 )}{64 \, a^{4} b} + \frac {7 \, \sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 11 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\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 )}{64 \, a^{4} b} + \frac {7 \, \sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 11 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{4} b} - \frac {7 \, \sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 11 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{4} b} - \frac {2 \, A}{3 \, a^{3} x^{\frac {3}{2}}} + \frac {7 \, B a b x^{\frac {5}{2}} - 15 \, A b^{2} x^{\frac {5}{2}} + 11 \, B a^{2} \sqrt {x} - 19 \, A a b \sqrt {x}}{16 \, {\left (b x^{2} + a\right )}^{2} a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^(5/2)/(b*x^2+a)^3,x, algorithm="giac")

[Out]

7/64*sqrt(2)*(3*(a*b^3)^(1/4)*B*a - 11*(a*b^3)^(1/4)*A*b)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))
/(a/b)^(1/4))/(a^4*b) + 7/64*sqrt(2)*(3*(a*b^3)^(1/4)*B*a - 11*(a*b^3)^(1/4)*A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)
*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^4*b) + 7/128*sqrt(2)*(3*(a*b^3)^(1/4)*B*a - 11*(a*b^3)^(1/4)*A*b)*lo
g(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b) - 7/128*sqrt(2)*(3*(a*b^3)^(1/4)*B*a - 11*(a*b^3)^(1/4)
*A*b)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b) - 2/3*A/(a^3*x^(3/2)) + 1/16*(7*B*a*b*x^(5/2)
- 15*A*b^2*x^(5/2) + 11*B*a^2*sqrt(x) - 19*A*a*b*sqrt(x))/((b*x^2 + a)^2*a^3)

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maple [A]  time = 0.02, size = 357, normalized size = 1.11 \begin {gather*} -\frac {15 A \,b^{2} x^{\frac {5}{2}}}{16 \left (b \,x^{2}+a \right )^{2} a^{3}}+\frac {7 B b \,x^{\frac {5}{2}}}{16 \left (b \,x^{2}+a \right )^{2} a^{2}}-\frac {19 A b \sqrt {x}}{16 \left (b \,x^{2}+a \right )^{2} a^{2}}+\frac {11 B \sqrt {x}}{16 \left (b \,x^{2}+a \right )^{2} a}-\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{64 a^{4}}-\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{64 a^{4}}-\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A b \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{128 a^{4}}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{64 a^{3}}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{64 a^{3}}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{128 a^{3}}-\frac {2 A}{3 a^{3} x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/x^(5/2)/(b*x^2+a)^3,x)

[Out]

-15/16/a^3/(b*x^2+a)^2*x^(5/2)*b^2*A+7/16/a^2/(b*x^2+a)^2*x^(5/2)*b*B-19/16/a^2/(b*x^2+a)^2*A*x^(1/2)*b+11/16/
a/(b*x^2+a)^2*B*x^(1/2)-77/64/a^4*(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*b-77/128/a^4*(a/
b)^(1/4)*2^(1/2)*A*ln((x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))
*b-77/64/a^4*(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*b+21/64/a^3*(a/b)^(1/4)*2^(1/2)*B*arc
tan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+21/128/a^3*(a/b)^(1/4)*2^(1/2)*B*ln((x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1
/2))/(x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))+21/64/a^3*(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4)*x
^(1/2)+1)-2/3*A/a^3/x^(3/2)

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maxima [A]  time = 2.43, size = 285, normalized size = 0.89 \begin {gather*} \frac {7 \, {\left (3 \, B a b - 11 \, A b^{2}\right )} x^{4} - 32 \, A a^{2} + 11 \, {\left (3 \, B a^{2} - 11 \, A a b\right )} x^{2}}{48 \, {\left (a^{3} b^{2} x^{\frac {11}{2}} + 2 \, a^{4} b x^{\frac {7}{2}} + a^{5} x^{\frac {3}{2}}\right )}} + \frac {7 \, {\left (\frac {2 \, \sqrt {2} {\left (3 \, B a - 11 \, A b\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 (3 \, B a - 11 \, A b\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 (3 \, B a - 11 \, A b\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 (3 \, B a - 11 \, A b\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 )}}{128 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^(5/2)/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

1/48*(7*(3*B*a*b - 11*A*b^2)*x^4 - 32*A*a^2 + 11*(3*B*a^2 - 11*A*a*b)*x^2)/(a^3*b^2*x^(11/2) + 2*a^4*b*x^(7/2)
+ a^5*x^(3/2)) + 7/128*(2*sqrt(2)*(3*B*a - 11*A*b)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sq
rt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(3*B*a - 11*A*b)*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
)*(3*B*a - 11*A*b)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(3*B
*a - 11*A*b)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/a^3

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mupad [B]  time = 0.53, size = 888, normalized size = 2.76

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^2)/(x^(5/2)*(a + b*x^2)^3),x)

[Out]

- ((2*A)/(3*a) + (11*x^2*(11*A*b - 3*B*a))/(48*a^2) + (7*b*x^4*(11*A*b - 3*B*a))/(48*a^3))/(a^2*x^(3/2) + b^2*
x^(11/2) + 2*a*b*x^(7/2)) - (atan((((11*A*b - 3*B*a)*(x^(1/2)*(97140736*A^2*a^9*b^5 + 7225344*B^2*a^11*b^3 - 5
2985856*A*B*a^10*b^4) - (7*(11*A*b - 3*B*a)*(80740352*A*a^13*b^4 - 22020096*B*a^14*b^3))/(64*(-a)^(15/4)*b^(1/
4)))*7i)/(64*(-a)^(15/4)*b^(1/4)) + ((11*A*b - 3*B*a)*(x^(1/2)*(97140736*A^2*a^9*b^5 + 7225344*B^2*a^11*b^3 -
52985856*A*B*a^10*b^4) + (7*(11*A*b - 3*B*a)*(80740352*A*a^13*b^4 - 22020096*B*a^14*b^3))/(64*(-a)^(15/4)*b^(1
/4)))*7i)/(64*(-a)^(15/4)*b^(1/4)))/((7*(11*A*b - 3*B*a)*(x^(1/2)*(97140736*A^2*a^9*b^5 + 7225344*B^2*a^11*b^3
- 52985856*A*B*a^10*b^4) - (7*(11*A*b - 3*B*a)*(80740352*A*a^13*b^4 - 22020096*B*a^14*b^3))/(64*(-a)^(15/4)*b
^(1/4))))/(64*(-a)^(15/4)*b^(1/4)) - (7*(11*A*b - 3*B*a)*(x^(1/2)*(97140736*A^2*a^9*b^5 + 7225344*B^2*a^11*b^3
- 52985856*A*B*a^10*b^4) + (7*(11*A*b - 3*B*a)*(80740352*A*a^13*b^4 - 22020096*B*a^14*b^3))/(64*(-a)^(15/4)*b
^(1/4))))/(64*(-a)^(15/4)*b^(1/4))))*(11*A*b - 3*B*a)*7i)/(32*(-a)^(15/4)*b^(1/4)) - (7*atan(((7*(11*A*b - 3*B
*a)*(x^(1/2)*(97140736*A^2*a^9*b^5 + 7225344*B^2*a^11*b^3 - 52985856*A*B*a^10*b^4) - ((11*A*b - 3*B*a)*(807403
52*A*a^13*b^4 - 22020096*B*a^14*b^3)*7i)/(64*(-a)^(15/4)*b^(1/4))))/(64*(-a)^(15/4)*b^(1/4)) + (7*(11*A*b - 3*
B*a)*(x^(1/2)*(97140736*A^2*a^9*b^5 + 7225344*B^2*a^11*b^3 - 52985856*A*B*a^10*b^4) + ((11*A*b - 3*B*a)*(80740
352*A*a^13*b^4 - 22020096*B*a^14*b^3)*7i)/(64*(-a)^(15/4)*b^(1/4))))/(64*(-a)^(15/4)*b^(1/4)))/(((11*A*b - 3*B
*a)*(x^(1/2)*(97140736*A^2*a^9*b^5 + 7225344*B^2*a^11*b^3 - 52985856*A*B*a^10*b^4) - ((11*A*b - 3*B*a)*(807403
52*A*a^13*b^4 - 22020096*B*a^14*b^3)*7i)/(64*(-a)^(15/4)*b^(1/4)))*7i)/(64*(-a)^(15/4)*b^(1/4)) - ((11*A*b - 3
*B*a)*(x^(1/2)*(97140736*A^2*a^9*b^5 + 7225344*B^2*a^11*b^3 - 52985856*A*B*a^10*b^4) + ((11*A*b - 3*B*a)*(8074
0352*A*a^13*b^4 - 22020096*B*a^14*b^3)*7i)/(64*(-a)^(15/4)*b^(1/4)))*7i)/(64*(-a)^(15/4)*b^(1/4))))*(11*A*b -
3*B*a))/(32*(-a)^(15/4)*b^(1/4))

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sympy [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((B*x**2+A)/x**(5/2)/(b*x**2+a)**3,x)

[Out]

Timed out

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