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

Optimal. Leaf size=289 \[ \frac {(7 A b-3 a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}}-\frac {(7 A b-3 a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}}+\frac {(7 A b-3 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} \sqrt [4]{b}}-\frac {(7 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{11/4} \sqrt [4]{b}}-\frac {7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac {A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )} \]

[Out]

1/6*(-7*A*b+3*B*a)/a^2/b/x^(3/2)+1/2*(A*b-B*a)/a/b/x^(3/2)/(b*x^2+a)+1/8*(7*A*b-3*B*a)*arctan(1-b^(1/4)*2^(1/2
)*x^(1/2)/a^(1/4))/a^(11/4)/b^(1/4)*2^(1/2)-1/8*(7*A*b-3*B*a)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(11/
4)/b^(1/4)*2^(1/2)+1/16*(7*A*b-3*B*a)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(11/4)/b^(1/4)*2
^(1/2)-1/16*(7*A*b-3*B*a)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(11/4)/b^(1/4)*2^(1/2)

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Rubi [A]  time = 0.21, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {457, 325, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac {7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac {(7 A b-3 a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}}-\frac {(7 A b-3 a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}}+\frac {(7 A b-3 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} \sqrt [4]{b}}-\frac {(7 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{11/4} \sqrt [4]{b}}+\frac {A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(7*A*b - 3*a*B)/(6*a^2*b*x^(3/2)) + (A*b - a*B)/(2*a*b*x^(3/2)*(a + b*x^2)) + ((7*A*b - 3*a*B)*ArcTan[1 - (Sq
rt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(11/4)*b^(1/4)) - ((7*A*b - 3*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*S
qrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(11/4)*b^(1/4)) + ((7*A*b - 3*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x
] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)*b^(1/4)) - ((7*A*b - 3*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x]
+ Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)*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 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 )^2} \, dx &=\frac {A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}+\frac {\left (\frac {7 A b}{2}-\frac {3 a B}{2}\right ) \int \frac {1}{x^{5/2} \left (a+b x^2\right )} \, dx}{2 a b}\\ &=-\frac {7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac {A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {(7 A b-3 a B) \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{4 a^2}\\ &=-\frac {7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac {A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {(7 A b-3 a B) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^2}\\ &=-\frac {7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac {A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {(7 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 )}{4 a^{5/2}}-\frac {(7 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 )}{4 a^{5/2}}\\ &=-\frac {7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac {A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {(7 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 )}{8 a^{5/2} \sqrt {b}}-\frac {(7 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 )}{8 a^{5/2} \sqrt {b}}+\frac {(7 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 )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}}+\frac {(7 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 )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}}\\ &=-\frac {7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac {A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}+\frac {(7 A b-3 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}}-\frac {(7 A b-3 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}}-\frac {(7 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 )}{4 \sqrt {2} a^{11/4} \sqrt [4]{b}}+\frac {(7 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 )}{4 \sqrt {2} a^{11/4} \sqrt [4]{b}}\\ &=-\frac {7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac {A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}+\frac {(7 A b-3 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} \sqrt [4]{b}}-\frac {(7 A b-3 a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} \sqrt [4]{b}}+\frac {(7 A b-3 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}}-\frac {(7 A b-3 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}}\\ \end {align*}

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Mathematica [A]  time = 0.39, size = 355, normalized size = 1.23 \[ \frac {-\frac {24 a^{3/4} A b \sqrt {x}}{a+b x^2}-\frac {32 a^{3/4} A}{x^{3/2}}+\frac {24 a^{7/4} B \sqrt {x}}{a+b x^2}+21 \sqrt {2} A b^{3/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-21 \sqrt {2} A b^{3/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+\frac {6 \sqrt {2} (7 A b-3 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}-\frac {6 \sqrt {2} (7 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{b}}-\frac {9 \sqrt {2} a B \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt [4]{b}}+\frac {9 \sqrt {2} a B \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt [4]{b}}}{48 a^{11/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-32*a^(3/4)*A)/x^(3/2) - (24*a^(3/4)*A*b*Sqrt[x])/(a + b*x^2) + (24*a^(7/4)*B*Sqrt[x])/(a + b*x^2) + (6*Sqrt
[2]*(7*A*b - 3*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) - (6*Sqrt[2]*(7*A*b - 3*a*B)*ArcTan
[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) + 21*Sqrt[2]*A*b^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*
Sqrt[x] + Sqrt[b]*x] - (9*Sqrt[2]*a*B*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4) - 21
*Sqrt[2]*A*b^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + (9*Sqrt[2]*a*B*Log[Sqrt[a] + S
qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4))/(48*a^(11/4))

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fricas [B]  time = 0.53, size = 741, normalized size = 2.56 \[ -\frac {12 \, {\left (a^{2} b x^{4} + a^{3} x^{2}\right )} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a^{6} \sqrt {-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}} + {\left (9 \, B^{2} a^{2} - 42 \, A B a b + 49 \, A^{2} b^{2}\right )} x} a^{8} b \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {3}{4}} + {\left (3 \, B a^{9} b - 7 \, A a^{8} b^{2}\right )} \sqrt {x} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {3}{4}}}{81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}\right ) + 3 \, {\left (a^{2} b x^{4} + a^{3} x^{2}\right )} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} \log \left (a^{3} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} - {\left (3 \, B a - 7 \, A b\right )} \sqrt {x}\right ) - 3 \, {\left (a^{2} b x^{4} + a^{3} x^{2}\right )} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} \log \left (-a^{3} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} - {\left (3 \, B a - 7 \, A b\right )} \sqrt {x}\right ) - 4 \, {\left ({\left (3 \, B a - 7 \, A b\right )} x^{2} - 4 \, A a\right )} \sqrt {x}}{24 \, {\left (a^{2} b x^{4} + a^{3} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(12*(a^2*b*x^4 + a^3*x^2)*(-(81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 + 24
01*A^4*b^4)/(a^11*b))^(1/4)*arctan((sqrt(a^6*sqrt(-(81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116
*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b)) + (9*B^2*a^2 - 42*A*B*a*b + 49*A^2*b^2)*x)*a^8*b*(-(81*B^4*a^4 - 756*A*
B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b))^(3/4) + (3*B*a^9*b - 7*A*a^8*b^2
)*sqrt(x)*(-(81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b))^
(3/4))/(81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 + 2401*A^4*b^4)) + 3*(a^2*b*x^4
+ a^3*x^2)*(-(81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b)
)^(1/4)*log(a^3*(-(81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^1
1*b))^(1/4) - (3*B*a - 7*A*b)*sqrt(x)) - 3*(a^2*b*x^4 + a^3*x^2)*(-(81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A^2*B^
2*a^2*b^2 - 4116*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b))^(1/4)*log(-a^3*(-(81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A
^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b))^(1/4) - (3*B*a - 7*A*b)*sqrt(x)) - 4*((3*B*a - 7*A
*b)*x^2 - 4*A*a)*sqrt(x))/(a^2*b*x^4 + a^3*x^2)

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giac [A]  time = 0.50, size = 283, normalized size = 0.98 \[ \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 7 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{3} b} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 7 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{3} b} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 7 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{3} b} - \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 7 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{3} b} + \frac {B a \sqrt {x} - A b \sqrt {x}}{2 \, {\left (b x^{2} + a\right )} a^{2}} - \frac {2 \, A}{3 \, a^{2} x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(2)*(3*(a*b^3)^(1/4)*B*a - 7*(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^3*b) + 1/8*sqrt(2)*(3*(a*b^3)^(1/4)*B*a - 7*(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^3*b) + 1/16*sqrt(2)*(3*(a*b^3)^(1/4)*B*a - 7*(a*b^3)^(1/4)*A*b)*log(sqrt
(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b) - 1/16*sqrt(2)*(3*(a*b^3)^(1/4)*B*a - 7*(a*b^3)^(1/4)*A*b)*lo
g(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b) + 1/2*(B*a*sqrt(x) - A*b*sqrt(x))/((b*x^2 + a)*a^2) -
2/3*A/(a^2*x^(3/2))

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maple [A]  time = 0.02, size = 317, normalized size = 1.10 \[ -\frac {A b \sqrt {x}}{2 \left (b \,x^{2}+a \right ) a^{2}}+\frac {B \sqrt {x}}{2 \left (b \,x^{2}+a \right ) a}-\frac {7 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 a^{3}}-\frac {7 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 a^{3}}-\frac {7 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A b \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 a^{3}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 a^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 a^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 a^{2}}-\frac {2 A}{3 a^{2} x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/a^2*x^(1/2)/(b*x^2+a)*A*b+1/2/a*x^(1/2)/(b*x^2+a)*B-7/8/a^3*(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1
/4)*x^(1/2)-1)*b-7/16/a^3*(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-7/8/a^3*(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*b+3/8/a^2*
(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+3/16/a^2*(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)))+3/8/a^2*(a/b)^(1/4)*2^(1/2)*B*arctan(
2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-2/3*A/a^2/x^(3/2)

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maxima [A]  time = 2.71, size = 251, normalized size = 0.87 \[ \frac {{\left (3 \, B a - 7 \, A b\right )} x^{2} - 4 \, A a}{6 \, {\left (a^{2} b x^{\frac {7}{2}} + a^{3} x^{\frac {3}{2}}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, B a - 7 \, A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (3 \, B a - 7 \, A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (3 \, B a - 7 \, A b\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (3 \, B a - 7 \, A b\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{16 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*((3*B*a - 7*A*b)*x^2 - 4*A*a)/(a^2*b*x^(7/2) + a^3*x^(3/2)) + 1/16*(2*sqrt(2)*(3*B*a - 7*A*b)*arctan(1/2*s
qrt(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))) +
2*sqrt(2)*(3*B*a - 7*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 - 7*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 - 7*A*b)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqr
t(a))/(a^(3/4)*b^(1/4)))/a^2

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mupad [B]  time = 0.47, size = 859, normalized size = 2.97 \[ -\frac {\frac {2\,A}{3\,a}+\frac {x^2\,\left (7\,A\,b-3\,B\,a\right )}{6\,a^2}}{a\,x^{3/2}+b\,x^{7/2}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,a^6\,b^5-1344\,A\,B\,a^7\,b^4+288\,B^2\,a^8\,b^3\right )-\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (1792\,A\,a^9\,b^4-768\,B\,a^{10}\,b^3\right )}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}+\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,a^6\,b^5-1344\,A\,B\,a^7\,b^4+288\,B^2\,a^8\,b^3\right )+\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (1792\,A\,a^9\,b^4-768\,B\,a^{10}\,b^3\right )}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}}{\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,a^6\,b^5-1344\,A\,B\,a^7\,b^4+288\,B^2\,a^8\,b^3\right )-\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (1792\,A\,a^9\,b^4-768\,B\,a^{10}\,b^3\right )}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}\right )}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}-\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,a^6\,b^5-1344\,A\,B\,a^7\,b^4+288\,B^2\,a^8\,b^3\right )+\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (1792\,A\,a^9\,b^4-768\,B\,a^{10}\,b^3\right )}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}\right )}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}}\right )\,\left (7\,A\,b-3\,B\,a\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{11/4}\,b^{1/4}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,a^6\,b^5-1344\,A\,B\,a^7\,b^4+288\,B^2\,a^8\,b^3\right )-\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (1792\,A\,a^9\,b^4-768\,B\,a^{10}\,b^3\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}\right )}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}+\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,a^6\,b^5-1344\,A\,B\,a^7\,b^4+288\,B^2\,a^8\,b^3\right )+\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (1792\,A\,a^9\,b^4-768\,B\,a^{10}\,b^3\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}\right )}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}}{\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,a^6\,b^5-1344\,A\,B\,a^7\,b^4+288\,B^2\,a^8\,b^3\right )-\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (1792\,A\,a^9\,b^4-768\,B\,a^{10}\,b^3\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}-\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,a^6\,b^5-1344\,A\,B\,a^7\,b^4+288\,B^2\,a^8\,b^3\right )+\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (1792\,A\,a^9\,b^4-768\,B\,a^{10}\,b^3\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}}\right )\,\left (7\,A\,b-3\,B\,a\right )}{4\,{\left (-a\right )}^{11/4}\,b^{1/4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((2*A)/(3*a) + (x^2*(7*A*b - 3*B*a))/(6*a^2))/(a*x^(3/2) + b*x^(7/2)) - (atan((((7*A*b - 3*B*a)*(x^(1/2)*(15
68*A^2*a^6*b^5 + 288*B^2*a^8*b^3 - 1344*A*B*a^7*b^4) - ((7*A*b - 3*B*a)*(1792*A*a^9*b^4 - 768*B*a^10*b^3))/(8*
(-a)^(11/4)*b^(1/4)))*1i)/(8*(-a)^(11/4)*b^(1/4)) + ((7*A*b - 3*B*a)*(x^(1/2)*(1568*A^2*a^6*b^5 + 288*B^2*a^8*
b^3 - 1344*A*B*a^7*b^4) + ((7*A*b - 3*B*a)*(1792*A*a^9*b^4 - 768*B*a^10*b^3))/(8*(-a)^(11/4)*b^(1/4)))*1i)/(8*
(-a)^(11/4)*b^(1/4)))/(((7*A*b - 3*B*a)*(x^(1/2)*(1568*A^2*a^6*b^5 + 288*B^2*a^8*b^3 - 1344*A*B*a^7*b^4) - ((7
*A*b - 3*B*a)*(1792*A*a^9*b^4 - 768*B*a^10*b^3))/(8*(-a)^(11/4)*b^(1/4))))/(8*(-a)^(11/4)*b^(1/4)) - ((7*A*b -
3*B*a)*(x^(1/2)*(1568*A^2*a^6*b^5 + 288*B^2*a^8*b^3 - 1344*A*B*a^7*b^4) + ((7*A*b - 3*B*a)*(1792*A*a^9*b^4 -
768*B*a^10*b^3))/(8*(-a)^(11/4)*b^(1/4))))/(8*(-a)^(11/4)*b^(1/4))))*(7*A*b - 3*B*a)*1i)/(4*(-a)^(11/4)*b^(1/4
)) - (atan((((7*A*b - 3*B*a)*(x^(1/2)*(1568*A^2*a^6*b^5 + 288*B^2*a^8*b^3 - 1344*A*B*a^7*b^4) - ((7*A*b - 3*B*
a)*(1792*A*a^9*b^4 - 768*B*a^10*b^3)*1i)/(8*(-a)^(11/4)*b^(1/4))))/(8*(-a)^(11/4)*b^(1/4)) + ((7*A*b - 3*B*a)*
(x^(1/2)*(1568*A^2*a^6*b^5 + 288*B^2*a^8*b^3 - 1344*A*B*a^7*b^4) + ((7*A*b - 3*B*a)*(1792*A*a^9*b^4 - 768*B*a^
10*b^3)*1i)/(8*(-a)^(11/4)*b^(1/4))))/(8*(-a)^(11/4)*b^(1/4)))/(((7*A*b - 3*B*a)*(x^(1/2)*(1568*A^2*a^6*b^5 +
288*B^2*a^8*b^3 - 1344*A*B*a^7*b^4) - ((7*A*b - 3*B*a)*(1792*A*a^9*b^4 - 768*B*a^10*b^3)*1i)/(8*(-a)^(11/4)*b^
(1/4)))*1i)/(8*(-a)^(11/4)*b^(1/4)) - ((7*A*b - 3*B*a)*(x^(1/2)*(1568*A^2*a^6*b^5 + 288*B^2*a^8*b^3 - 1344*A*B
*a^7*b^4) + ((7*A*b - 3*B*a)*(1792*A*a^9*b^4 - 768*B*a^10*b^3)*1i)/(8*(-a)^(11/4)*b^(1/4)))*1i)/(8*(-a)^(11/4)
*b^(1/4))))*(7*A*b - 3*B*a))/(4*(-a)^(11/4)*b^(1/4))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/x**(5/2)/(b*x**2+a)**2,x)

[Out]

Timed out

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