3.4.49 \(\int \frac {x^{7/2} (A+B x^2)}{a+b x^2} \, dx\)

Optimal. Leaf size=276 \[ -\frac {a^{5/4} (A b-a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{13/4}}+\frac {a^{5/4} (A b-a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{13/4}}-\frac {a^{5/4} (A b-a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{13/4}}+\frac {a^{5/4} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} b^{13/4}}-\frac {2 a \sqrt {x} (A b-a B)}{b^3}+\frac {2 x^{5/2} (A b-a B)}{5 b^2}+\frac {2 B x^{9/2}}{9 b} \]

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Rubi [A]  time = 0.26, antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {459, 321, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {a^{5/4} (A b-a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{13/4}}+\frac {a^{5/4} (A b-a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{13/4}}-\frac {a^{5/4} (A b-a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{13/4}}+\frac {a^{5/4} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} b^{13/4}}+\frac {2 x^{5/2} (A b-a B)}{5 b^2}-\frac {2 a \sqrt {x} (A b-a B)}{b^3}+\frac {2 B x^{9/2}}{9 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^(7/2)*(A + B*x^2))/(a + b*x^2),x]

[Out]

(-2*a*(A*b - a*B)*Sqrt[x])/b^3 + (2*(A*b - a*B)*x^(5/2))/(5*b^2) + (2*B*x^(9/2))/(9*b) - (a^(5/4)*(A*b - a*B)*
ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(13/4)) + (a^(5/4)*(A*b - a*B)*ArcTan[1 + (Sqrt[2]*b
^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(13/4)) - (a^(5/4)*(A*b - a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt
[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(13/4)) + (a^(5/4)*(A*b - a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/(2*Sqrt[2]*b^(13/4))

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 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && 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 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 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 {x^{7/2} \left (A+B x^2\right )}{a+b x^2} \, dx &=\frac {2 B x^{9/2}}{9 b}-\frac {\left (2 \left (-\frac {9 A b}{2}+\frac {9 a B}{2}\right )\right ) \int \frac {x^{7/2}}{a+b x^2} \, dx}{9 b}\\ &=\frac {2 (A b-a B) x^{5/2}}{5 b^2}+\frac {2 B x^{9/2}}{9 b}-\frac {(a (A b-a B)) \int \frac {x^{3/2}}{a+b x^2} \, dx}{b^2}\\ &=-\frac {2 a (A b-a B) \sqrt {x}}{b^3}+\frac {2 (A b-a B) x^{5/2}}{5 b^2}+\frac {2 B x^{9/2}}{9 b}+\frac {\left (a^2 (A b-a B)\right ) \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{b^3}\\ &=-\frac {2 a (A b-a B) \sqrt {x}}{b^3}+\frac {2 (A b-a B) x^{5/2}}{5 b^2}+\frac {2 B x^{9/2}}{9 b}+\frac {\left (2 a^2 (A b-a B)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=-\frac {2 a (A b-a B) \sqrt {x}}{b^3}+\frac {2 (A b-a B) x^{5/2}}{5 b^2}+\frac {2 B x^{9/2}}{9 b}+\frac {\left (a^{3/2} (A b-a B)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^3}+\frac {\left (a^{3/2} (A b-a B)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=-\frac {2 a (A b-a B) \sqrt {x}}{b^3}+\frac {2 (A b-a B) x^{5/2}}{5 b^2}+\frac {2 B x^{9/2}}{9 b}+\frac {\left (a^{3/2} (A b-a B)\right ) \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 )}{2 b^{7/2}}+\frac {\left (a^{3/2} (A b-a B)\right ) \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 )}{2 b^{7/2}}-\frac {\left (a^{5/4} (A b-a B)\right ) \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 )}{2 \sqrt {2} b^{13/4}}-\frac {\left (a^{5/4} (A b-a B)\right ) \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 )}{2 \sqrt {2} b^{13/4}}\\ &=-\frac {2 a (A b-a B) \sqrt {x}}{b^3}+\frac {2 (A b-a B) x^{5/2}}{5 b^2}+\frac {2 B x^{9/2}}{9 b}-\frac {a^{5/4} (A b-a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{13/4}}+\frac {a^{5/4} (A b-a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{13/4}}+\frac {\left (a^{5/4} (A b-a B)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{13/4}}-\frac {\left (a^{5/4} (A b-a B)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{13/4}}\\ &=-\frac {2 a (A b-a B) \sqrt {x}}{b^3}+\frac {2 (A b-a B) x^{5/2}}{5 b^2}+\frac {2 B x^{9/2}}{9 b}-\frac {a^{5/4} (A b-a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{13/4}}+\frac {a^{5/4} (A b-a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{13/4}}-\frac {a^{5/4} (A b-a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{13/4}}+\frac {a^{5/4} (A b-a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{13/4}}\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 227, normalized size = 0.82 \begin {gather*} \frac {\frac {45 \sqrt {2} a^{5/4} (a B-A b) \left (\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )\right )}{\sqrt [4]{b}}+\frac {90 \sqrt {2} a^{5/4} (a B-A b) \left (\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )-\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )\right )}{\sqrt [4]{b}}+72 b x^{5/2} (A b-a B)+360 a \sqrt {x} (a B-A b)+40 b^2 B x^{9/2}}{180 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^(7/2)*(A + B*x^2))/(a + b*x^2),x]

[Out]

(360*a*(-(A*b) + a*B)*Sqrt[x] + 72*b*(A*b - a*B)*x^(5/2) + 40*b^2*B*x^(9/2) + (90*Sqrt[2]*a^(5/4)*(-(A*b) + a*
B)*(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]))/b^(1/4) +
(45*Sqrt[2]*a^(5/4)*(-(A*b) + a*B)*(Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/b^(1/4))/(180*b^3)

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IntegrateAlgebraic [A]  time = 0.29, size = 181, normalized size = 0.66 \begin {gather*} \frac {\left (a^{9/4} B-a^{5/4} A b\right ) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt {2} b^{13/4}}-\frac {\left (a^{9/4} B-a^{5/4} A b\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} b^{13/4}}+\frac {2 \sqrt {x} \left (45 a^2 B-45 a A b-9 a b B x^2+9 A b^2 x^2+5 b^2 B x^4\right )}{45 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^(7/2)*(A + B*x^2))/(a + b*x^2),x]

[Out]

(2*Sqrt[x]*(-45*a*A*b + 45*a^2*B + 9*A*b^2*x^2 - 9*a*b*B*x^2 + 5*b^2*B*x^4))/(45*b^3) + ((-(a^(5/4)*A*b) + a^(
9/4)*B)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*b^(13/4)) - ((-(a^(5/4)*A*b)
+ a^(9/4)*B)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*b^(13/4))

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fricas [B]  time = 0.99, size = 714, normalized size = 2.59 \begin {gather*} \frac {180 \, b^{3} \left (-\frac {B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {b^{6} \sqrt {-\frac {B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}} + {\left (B^{2} a^{4} - 2 \, A B a^{3} b + A^{2} a^{2} b^{2}\right )} x} b^{10} \left (-\frac {B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}\right )^{\frac {3}{4}} + {\left (B a^{2} b^{10} - A a b^{11}\right )} \sqrt {x} \left (-\frac {B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}\right )^{\frac {3}{4}}}{B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}\right ) + 45 \, b^{3} \left (-\frac {B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}\right )^{\frac {1}{4}} \log \left (b^{3} \left (-\frac {B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}\right )^{\frac {1}{4}} - {\left (B a^{2} - A a b\right )} \sqrt {x}\right ) - 45 \, b^{3} \left (-\frac {B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}\right )^{\frac {1}{4}} \log \left (-b^{3} \left (-\frac {B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}\right )^{\frac {1}{4}} - {\left (B a^{2} - A a b\right )} \sqrt {x}\right ) + 4 \, {\left (5 \, B b^{2} x^{4} + 45 \, B a^{2} - 45 \, A a b - 9 \, {\left (B a b - A b^{2}\right )} x^{2}\right )} \sqrt {x}}{90 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x^2+A)/(b*x^2+a),x, algorithm="fricas")

[Out]

1/90*(180*b^3*(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^4*a^5*b^4)/b^13)^(1/4)*arct
an((sqrt(b^6*sqrt(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^4*a^5*b^4)/b^13) + (B^2*
a^4 - 2*A*B*a^3*b + A^2*a^2*b^2)*x)*b^10*(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^
4*a^5*b^4)/b^13)^(3/4) + (B*a^2*b^10 - A*a*b^11)*sqrt(x)*(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^
3*B*a^6*b^3 + A^4*a^5*b^4)/b^13)^(3/4))/(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^4*a
^5*b^4)) + 45*b^3*(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^4*a^5*b^4)/b^13)^(1/4)*
log(b^3*(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^4*a^5*b^4)/b^13)^(1/4) - (B*a^2 -
A*a*b)*sqrt(x)) - 45*b^3*(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^4*a^5*b^4)/b^13
)^(1/4)*log(-b^3*(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^4*a^5*b^4)/b^13)^(1/4) -
(B*a^2 - A*a*b)*sqrt(x)) + 4*(5*B*b^2*x^4 + 45*B*a^2 - 45*A*a*b - 9*(B*a*b - A*b^2)*x^2)*sqrt(x))/b^3

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giac [A]  time = 0.37, size = 298, normalized size = 1.08 \begin {gather*} -\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a^{2} - \left (a b^{3}\right )^{\frac {1}{4}} A 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 )}{2 \, b^{4}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a^{2} - \left (a b^{3}\right )^{\frac {1}{4}} A 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 )}{2 \, b^{4}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a^{2} - \left (a b^{3}\right )^{\frac {1}{4}} A a b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, b^{4}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a^{2} - \left (a b^{3}\right )^{\frac {1}{4}} A a b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, b^{4}} + \frac {2 \, {\left (5 \, B b^{8} x^{\frac {9}{2}} - 9 \, B a b^{7} x^{\frac {5}{2}} + 9 \, A b^{8} x^{\frac {5}{2}} + 45 \, B a^{2} b^{6} \sqrt {x} - 45 \, A a b^{7} \sqrt {x}\right )}}{45 \, b^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x^2+A)/(b*x^2+a),x, algorithm="giac")

[Out]

-1/2*sqrt(2)*((a*b^3)^(1/4)*B*a^2 - (a*b^3)^(1/4)*A*a*b)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/
(a/b)^(1/4))/b^4 - 1/2*sqrt(2)*((a*b^3)^(1/4)*B*a^2 - (a*b^3)^(1/4)*A*a*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^
(1/4) - 2*sqrt(x))/(a/b)^(1/4))/b^4 - 1/4*sqrt(2)*((a*b^3)^(1/4)*B*a^2 - (a*b^3)^(1/4)*A*a*b)*log(sqrt(2)*sqrt
(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^4 + 1/4*sqrt(2)*((a*b^3)^(1/4)*B*a^2 - (a*b^3)^(1/4)*A*a*b)*log(-sqrt(2)*sq
rt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^4 + 2/45*(5*B*b^8*x^(9/2) - 9*B*a*b^7*x^(5/2) + 9*A*b^8*x^(5/2) + 45*B*a^
2*b^6*sqrt(x) - 45*A*a*b^7*sqrt(x))/b^9

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maple [A]  time = 0.02, size = 330, normalized size = 1.20 \begin {gather*} \frac {2 B \,x^{\frac {9}{2}}}{9 b}+\frac {2 A \,x^{\frac {5}{2}}}{5 b}-\frac {2 B a \,x^{\frac {5}{2}}}{5 b^{2}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A a \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 b^{2}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A a \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 b^{2}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A a \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 )}{4 b^{2}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B \,a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 b^{3}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B \,a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 b^{3}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B \,a^{2} \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 )}{4 b^{3}}-\frac {2 A a \sqrt {x}}{b^{2}}+\frac {2 B \,a^{2} \sqrt {x}}{b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(B*x^2+A)/(b*x^2+a),x)

[Out]

2/9*B*x^(9/2)/b+2/5/b*A*x^(5/2)-2/5/b^2*B*x^(5/2)*a-2/b^2*a*A*x^(1/2)+2/b^3*a^2*B*x^(1/2)+1/2*a/b^2*(a/b)^(1/4
)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+1/2*a/b^2*(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x
^(1/2)-1)+1/4*a/b^2*(a/b)^(1/4)*2^(1/2)*A*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)))-1/2*a^2/b^3*(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-1/2*a^2/b^3*(a
/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)-1/4*a^2/b^3*(a/b)^(1/4)*2^(1/2)*B*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)))

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maxima [A]  time = 2.49, size = 259, normalized size = 0.94 \begin {gather*} -\frac {{\left (\frac {2 \, \sqrt {2} {\left (B a - 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 (B a - 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 (B a - 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 (B a - 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 )} a^{2}}{4 \, b^{3}} + \frac {2 \, {\left (5 \, B b^{2} x^{\frac {9}{2}} - 9 \, {\left (B a b - A b^{2}\right )} x^{\frac {5}{2}} + 45 \, {\left (B a^{2} - A a b\right )} \sqrt {x}\right )}}{45 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x^2+A)/(b*x^2+a),x, algorithm="maxima")

[Out]

-1/4*(2*sqrt(2)*(B*a - 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))) + 2*sqrt(2)*(B*a - 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)*(B*a - 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)*(B*a - 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^2/b^3 + 2/45*(5*B*b^2*x^(9/2) - 9*(B*a*b - A*b^2)*x^(5/
2) + 45*(B*a^2 - A*a*b)*sqrt(x))/b^3

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mupad [B]  time = 0.38, size = 788, normalized size = 2.86 \begin {gather*} x^{5/2}\,\left (\frac {2\,A}{5\,b}-\frac {2\,B\,a}{5\,b^2}\right )+\frac {2\,B\,x^{9/2}}{9\,b}-\frac {{\left (-a\right )}^{5/4}\,\mathrm {atan}\left (\frac {\frac {{\left (-a\right )}^{5/4}\,\left (\frac {16\,\sqrt {x}\,\left (A^2\,a^4\,b^2-2\,A\,B\,a^5\,b+B^2\,a^6\right )}{b^3}-\frac {{\left (-a\right )}^{5/4}\,\left (A\,b-B\,a\right )\,\left (32\,B\,a^4-32\,A\,a^3\,b\right )\,1{}\mathrm {i}}{2\,b^{13/4}}\right )\,\left (A\,b-B\,a\right )}{2\,b^{13/4}}+\frac {{\left (-a\right )}^{5/4}\,\left (\frac {16\,\sqrt {x}\,\left (A^2\,a^4\,b^2-2\,A\,B\,a^5\,b+B^2\,a^6\right )}{b^3}+\frac {{\left (-a\right )}^{5/4}\,\left (A\,b-B\,a\right )\,\left (32\,B\,a^4-32\,A\,a^3\,b\right )\,1{}\mathrm {i}}{2\,b^{13/4}}\right )\,\left (A\,b-B\,a\right )}{2\,b^{13/4}}}{\frac {{\left (-a\right )}^{5/4}\,\left (\frac {16\,\sqrt {x}\,\left (A^2\,a^4\,b^2-2\,A\,B\,a^5\,b+B^2\,a^6\right )}{b^3}-\frac {{\left (-a\right )}^{5/4}\,\left (A\,b-B\,a\right )\,\left (32\,B\,a^4-32\,A\,a^3\,b\right )\,1{}\mathrm {i}}{2\,b^{13/4}}\right )\,\left (A\,b-B\,a\right )\,1{}\mathrm {i}}{2\,b^{13/4}}-\frac {{\left (-a\right )}^{5/4}\,\left (\frac {16\,\sqrt {x}\,\left (A^2\,a^4\,b^2-2\,A\,B\,a^5\,b+B^2\,a^6\right )}{b^3}+\frac {{\left (-a\right )}^{5/4}\,\left (A\,b-B\,a\right )\,\left (32\,B\,a^4-32\,A\,a^3\,b\right )\,1{}\mathrm {i}}{2\,b^{13/4}}\right )\,\left (A\,b-B\,a\right )\,1{}\mathrm {i}}{2\,b^{13/4}}}\right )\,\left (A\,b-B\,a\right )}{b^{13/4}}-\frac {a\,\sqrt {x}\,\left (\frac {2\,A}{b}-\frac {2\,B\,a}{b^2}\right )}{b}-\frac {{\left (-a\right )}^{5/4}\,\mathrm {atan}\left (\frac {\frac {{\left (-a\right )}^{5/4}\,\left (\frac {16\,\sqrt {x}\,\left (A^2\,a^4\,b^2-2\,A\,B\,a^5\,b+B^2\,a^6\right )}{b^3}-\frac {{\left (-a\right )}^{5/4}\,\left (A\,b-B\,a\right )\,\left (32\,B\,a^4-32\,A\,a^3\,b\right )}{2\,b^{13/4}}\right )\,\left (A\,b-B\,a\right )\,1{}\mathrm {i}}{2\,b^{13/4}}+\frac {{\left (-a\right )}^{5/4}\,\left (\frac {16\,\sqrt {x}\,\left (A^2\,a^4\,b^2-2\,A\,B\,a^5\,b+B^2\,a^6\right )}{b^3}+\frac {{\left (-a\right )}^{5/4}\,\left (A\,b-B\,a\right )\,\left (32\,B\,a^4-32\,A\,a^3\,b\right )}{2\,b^{13/4}}\right )\,\left (A\,b-B\,a\right )\,1{}\mathrm {i}}{2\,b^{13/4}}}{\frac {{\left (-a\right )}^{5/4}\,\left (\frac {16\,\sqrt {x}\,\left (A^2\,a^4\,b^2-2\,A\,B\,a^5\,b+B^2\,a^6\right )}{b^3}-\frac {{\left (-a\right )}^{5/4}\,\left (A\,b-B\,a\right )\,\left (32\,B\,a^4-32\,A\,a^3\,b\right )}{2\,b^{13/4}}\right )\,\left (A\,b-B\,a\right )}{2\,b^{13/4}}-\frac {{\left (-a\right )}^{5/4}\,\left (\frac {16\,\sqrt {x}\,\left (A^2\,a^4\,b^2-2\,A\,B\,a^5\,b+B^2\,a^6\right )}{b^3}+\frac {{\left (-a\right )}^{5/4}\,\left (A\,b-B\,a\right )\,\left (32\,B\,a^4-32\,A\,a^3\,b\right )}{2\,b^{13/4}}\right )\,\left (A\,b-B\,a\right )}{2\,b^{13/4}}}\right )\,\left (A\,b-B\,a\right )\,1{}\mathrm {i}}{b^{13/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(7/2)*(A + B*x^2))/(a + b*x^2),x)

[Out]

x^(5/2)*((2*A)/(5*b) - (2*B*a)/(5*b^2)) + (2*B*x^(9/2))/(9*b) - ((-a)^(5/4)*atan((((-a)^(5/4)*((16*x^(1/2)*(B^
2*a^6 + A^2*a^4*b^2 - 2*A*B*a^5*b))/b^3 - ((-a)^(5/4)*(A*b - B*a)*(32*B*a^4 - 32*A*a^3*b))/(2*b^(13/4)))*(A*b
- B*a)*1i)/(2*b^(13/4)) + ((-a)^(5/4)*((16*x^(1/2)*(B^2*a^6 + A^2*a^4*b^2 - 2*A*B*a^5*b))/b^3 + ((-a)^(5/4)*(A
*b - B*a)*(32*B*a^4 - 32*A*a^3*b))/(2*b^(13/4)))*(A*b - B*a)*1i)/(2*b^(13/4)))/(((-a)^(5/4)*((16*x^(1/2)*(B^2*
a^6 + A^2*a^4*b^2 - 2*A*B*a^5*b))/b^3 - ((-a)^(5/4)*(A*b - B*a)*(32*B*a^4 - 32*A*a^3*b))/(2*b^(13/4)))*(A*b -
B*a))/(2*b^(13/4)) - ((-a)^(5/4)*((16*x^(1/2)*(B^2*a^6 + A^2*a^4*b^2 - 2*A*B*a^5*b))/b^3 + ((-a)^(5/4)*(A*b -
B*a)*(32*B*a^4 - 32*A*a^3*b))/(2*b^(13/4)))*(A*b - B*a))/(2*b^(13/4))))*(A*b - B*a)*1i)/b^(13/4) - ((-a)^(5/4)
*atan((((-a)^(5/4)*((16*x^(1/2)*(B^2*a^6 + A^2*a^4*b^2 - 2*A*B*a^5*b))/b^3 - ((-a)^(5/4)*(A*b - B*a)*(32*B*a^4
- 32*A*a^3*b)*1i)/(2*b^(13/4)))*(A*b - B*a))/(2*b^(13/4)) + ((-a)^(5/4)*((16*x^(1/2)*(B^2*a^6 + A^2*a^4*b^2 -
2*A*B*a^5*b))/b^3 + ((-a)^(5/4)*(A*b - B*a)*(32*B*a^4 - 32*A*a^3*b)*1i)/(2*b^(13/4)))*(A*b - B*a))/(2*b^(13/4
)))/(((-a)^(5/4)*((16*x^(1/2)*(B^2*a^6 + A^2*a^4*b^2 - 2*A*B*a^5*b))/b^3 - ((-a)^(5/4)*(A*b - B*a)*(32*B*a^4 -
32*A*a^3*b)*1i)/(2*b^(13/4)))*(A*b - B*a)*1i)/(2*b^(13/4)) - ((-a)^(5/4)*((16*x^(1/2)*(B^2*a^6 + A^2*a^4*b^2
- 2*A*B*a^5*b))/b^3 + ((-a)^(5/4)*(A*b - B*a)*(32*B*a^4 - 32*A*a^3*b)*1i)/(2*b^(13/4)))*(A*b - B*a)*1i)/(2*b^(
13/4))))*(A*b - B*a))/b^(13/4) - (a*x^(1/2)*((2*A)/b - (2*B*a)/b^2))/b

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sympy [A]  time = 127.30, size = 434, normalized size = 1.57 \begin {gather*} \begin {cases} \tilde {\infty } \left (\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {9}{2}}}{9}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 A x^{\frac {9}{2}}}{9} + \frac {2 B x^{\frac {13}{2}}}{13}}{a} & \text {for}\: b = 0 \\\frac {\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {9}{2}}}{9}}{b} & \text {for}\: a = 0 \\- \frac {\sqrt [4]{-1} A a^{\frac {5}{4}} \sqrt [4]{\frac {1}{b}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 b^{2}} + \frac {\sqrt [4]{-1} A a^{\frac {5}{4}} \sqrt [4]{\frac {1}{b}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 b^{2}} - \frac {\sqrt [4]{-1} A a^{\frac {5}{4}} \sqrt [4]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{b^{2}} - \frac {2 A a \sqrt {x}}{b^{2}} + \frac {2 A x^{\frac {5}{2}}}{5 b} + \frac {\sqrt [4]{-1} B a^{\frac {9}{4}} \sqrt [4]{\frac {1}{b}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 b^{3}} - \frac {\sqrt [4]{-1} B a^{\frac {9}{4}} \sqrt [4]{\frac {1}{b}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{2 b^{3}} + \frac {\sqrt [4]{-1} B a^{\frac {9}{4}} \sqrt [4]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{b^{3}} + \frac {2 B a^{2} \sqrt {x}}{b^{3}} - \frac {2 B a x^{\frac {5}{2}}}{5 b^{2}} + \frac {2 B x^{\frac {9}{2}}}{9 b} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(B*x**2+A)/(b*x**2+a),x)

[Out]

Piecewise((zoo*(2*A*x**(5/2)/5 + 2*B*x**(9/2)/9), Eq(a, 0) & Eq(b, 0)), ((2*A*x**(9/2)/9 + 2*B*x**(13/2)/13)/a
, Eq(b, 0)), ((2*A*x**(5/2)/5 + 2*B*x**(9/2)/9)/b, Eq(a, 0)), (-(-1)**(1/4)*A*a**(5/4)*(1/b)**(1/4)*log(-(-1)*
*(1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(2*b**2) + (-1)**(1/4)*A*a**(5/4)*(1/b)**(1/4)*log((-1)**(1/4)*a**(1/4
)*(1/b)**(1/4) + sqrt(x))/(2*b**2) - (-1)**(1/4)*A*a**(5/4)*(1/b)**(1/4)*atan((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1
/b)**(1/4)))/b**2 - 2*A*a*sqrt(x)/b**2 + 2*A*x**(5/2)/(5*b) + (-1)**(1/4)*B*a**(9/4)*(1/b)**(1/4)*log(-(-1)**(
1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(2*b**3) - (-1)**(1/4)*B*a**(9/4)*(1/b)**(1/4)*log((-1)**(1/4)*a**(1/4)*
(1/b)**(1/4) + sqrt(x))/(2*b**3) + (-1)**(1/4)*B*a**(9/4)*(1/b)**(1/4)*atan((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/b
)**(1/4)))/b**3 + 2*B*a**2*sqrt(x)/b**3 - 2*B*a*x**(5/2)/(5*b**2) + 2*B*x**(9/2)/(9*b), True))

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