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

Optimal. Leaf size=154 \[ -\frac {a^{5/2} (7 A b-9 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}}+\frac {a^2 \sqrt {x} (7 A b-9 a B)}{b^5}-\frac {a x^{3/2} (7 A b-9 a B)}{3 b^4}+\frac {x^{5/2} (7 A b-9 a B)}{5 b^3}-\frac {x^{7/2} (7 A b-9 a B)}{7 a b^2}+\frac {x^{9/2} (A b-a B)}{a b (a+b x)} \]

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Rubi [A]  time = 0.08, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {78, 50, 63, 205} \begin {gather*} \frac {a^2 \sqrt {x} (7 A b-9 a B)}{b^5}-\frac {a^{5/2} (7 A b-9 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}}-\frac {x^{7/2} (7 A b-9 a B)}{7 a b^2}+\frac {x^{5/2} (7 A b-9 a B)}{5 b^3}-\frac {a x^{3/2} (7 A b-9 a B)}{3 b^4}+\frac {x^{9/2} (A b-a B)}{a b (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(7*A*b - 9*a*B)*Sqrt[x])/b^5 - (a*(7*A*b - 9*a*B)*x^(3/2))/(3*b^4) + ((7*A*b - 9*a*B)*x^(5/2))/(5*b^3) -
((7*A*b - 9*a*B)*x^(7/2))/(7*a*b^2) + ((A*b - a*B)*x^(9/2))/(a*b*(a + b*x)) - (a^(5/2)*(7*A*b - 9*a*B)*ArcTan[
(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(11/2)

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 78

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

Rule 205

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

Rubi steps

\begin {align*} \int \frac {x^{7/2} (A+B x)}{(a+b x)^2} \, dx &=\frac {(A b-a B) x^{9/2}}{a b (a+b x)}-\frac {\left (\frac {7 A b}{2}-\frac {9 a B}{2}\right ) \int \frac {x^{7/2}}{a+b x} \, dx}{a b}\\ &=-\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}+\frac {(7 A b-9 a B) \int \frac {x^{5/2}}{a+b x} \, dx}{2 b^2}\\ &=\frac {(7 A b-9 a B) x^{5/2}}{5 b^3}-\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}-\frac {(a (7 A b-9 a B)) \int \frac {x^{3/2}}{a+b x} \, dx}{2 b^3}\\ &=-\frac {a (7 A b-9 a B) x^{3/2}}{3 b^4}+\frac {(7 A b-9 a B) x^{5/2}}{5 b^3}-\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}+\frac {\left (a^2 (7 A b-9 a B)\right ) \int \frac {\sqrt {x}}{a+b x} \, dx}{2 b^4}\\ &=\frac {a^2 (7 A b-9 a B) \sqrt {x}}{b^5}-\frac {a (7 A b-9 a B) x^{3/2}}{3 b^4}+\frac {(7 A b-9 a B) x^{5/2}}{5 b^3}-\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}-\frac {\left (a^3 (7 A b-9 a B)\right ) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 b^5}\\ &=\frac {a^2 (7 A b-9 a B) \sqrt {x}}{b^5}-\frac {a (7 A b-9 a B) x^{3/2}}{3 b^4}+\frac {(7 A b-9 a B) x^{5/2}}{5 b^3}-\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}-\frac {\left (a^3 (7 A b-9 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{b^5}\\ &=\frac {a^2 (7 A b-9 a B) \sqrt {x}}{b^5}-\frac {a (7 A b-9 a B) x^{3/2}}{3 b^4}+\frac {(7 A b-9 a B) x^{5/2}}{5 b^3}-\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}-\frac {a^{5/2} (7 A b-9 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 128, normalized size = 0.83 \begin {gather*} \frac {a^{5/2} (9 a B-7 A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}}+\frac {\sqrt {x} \left (-945 a^4 B+105 a^3 b (7 A-6 B x)+14 a^2 b^2 x (35 A+9 B x)-2 a b^3 x^2 (49 A+27 B x)+6 b^4 x^3 (7 A+5 B x)\right )}{105 b^5 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*(-945*a^4*B + 105*a^3*b*(7*A - 6*B*x) + 6*b^4*x^3*(7*A + 5*B*x) + 14*a^2*b^2*x*(35*A + 9*B*x) - 2*a*b
^3*x^2*(49*A + 27*B*x)))/(105*b^5*(a + b*x)) + (a^(5/2)*(-7*A*b + 9*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^
(11/2)

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IntegrateAlgebraic [A]  time = 0.17, size = 143, normalized size = 0.93 \begin {gather*} \frac {\left (9 a^{7/2} B-7 a^{5/2} A b\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}}+\frac {\sqrt {x} \left (-945 a^4 B+735 a^3 A b-630 a^3 b B x+490 a^2 A b^2 x+126 a^2 b^2 B x^2-98 a A b^3 x^2-54 a b^3 B x^3+42 A b^4 x^3+30 b^4 B x^4\right )}{105 b^5 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*(735*a^3*A*b - 945*a^4*B + 490*a^2*A*b^2*x - 630*a^3*b*B*x - 98*a*A*b^3*x^2 + 126*a^2*b^2*B*x^2 + 42*
A*b^4*x^3 - 54*a*b^3*B*x^3 + 30*b^4*B*x^4))/(105*b^5*(a + b*x)) + ((-7*a^(5/2)*A*b + 9*a^(7/2)*B)*ArcTan[(Sqrt
[b]*Sqrt[x])/Sqrt[a]])/b^(11/2)

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fricas [A]  time = 0.99, size = 341, normalized size = 2.21 \begin {gather*} \left [-\frac {105 \, {\left (9 \, B a^{4} - 7 \, A a^{3} b + {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (30 \, B b^{4} x^{4} - 945 \, B a^{4} + 735 \, A a^{3} b - 6 \, {\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} x^{3} + 14 \, {\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{2} - 70 \, {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt {x}}{210 \, {\left (b^{6} x + a b^{5}\right )}}, \frac {105 \, {\left (9 \, B a^{4} - 7 \, A a^{3} b + {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (30 \, B b^{4} x^{4} - 945 \, B a^{4} + 735 \, A a^{3} b - 6 \, {\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} x^{3} + 14 \, {\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{2} - 70 \, {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt {x}}{105 \, {\left (b^{6} x + a b^{5}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/210*(105*(9*B*a^4 - 7*A*a^3*b + (9*B*a^3*b - 7*A*a^2*b^2)*x)*sqrt(-a/b)*log((b*x - 2*b*sqrt(x)*sqrt(-a/b)
- a)/(b*x + a)) - 2*(30*B*b^4*x^4 - 945*B*a^4 + 735*A*a^3*b - 6*(9*B*a*b^3 - 7*A*b^4)*x^3 + 14*(9*B*a^2*b^2 -
7*A*a*b^3)*x^2 - 70*(9*B*a^3*b - 7*A*a^2*b^2)*x)*sqrt(x))/(b^6*x + a*b^5), 1/105*(105*(9*B*a^4 - 7*A*a^3*b + (
9*B*a^3*b - 7*A*a^2*b^2)*x)*sqrt(a/b)*arctan(b*sqrt(x)*sqrt(a/b)/a) + (30*B*b^4*x^4 - 945*B*a^4 + 735*A*a^3*b
- 6*(9*B*a*b^3 - 7*A*b^4)*x^3 + 14*(9*B*a^2*b^2 - 7*A*a*b^3)*x^2 - 70*(9*B*a^3*b - 7*A*a^2*b^2)*x)*sqrt(x))/(b
^6*x + a*b^5)]

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giac [A]  time = 1.32, size = 146, normalized size = 0.95 \begin {gather*} \frac {{\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} - \frac {B a^{4} \sqrt {x} - A a^{3} b \sqrt {x}}{{\left (b x + a\right )} b^{5}} + \frac {2 \, {\left (15 \, B b^{12} x^{\frac {7}{2}} - 42 \, B a b^{11} x^{\frac {5}{2}} + 21 \, A b^{12} x^{\frac {5}{2}} + 105 \, B a^{2} b^{10} x^{\frac {3}{2}} - 70 \, A a b^{11} x^{\frac {3}{2}} - 420 \, B a^{3} b^{9} \sqrt {x} + 315 \, A a^{2} b^{10} \sqrt {x}\right )}}{105 \, b^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(9*B*a^4 - 7*A*a^3*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^5) - (B*a^4*sqrt(x) - A*a^3*b*sqrt(x))/((b*x +
a)*b^5) + 2/105*(15*B*b^12*x^(7/2) - 42*B*a*b^11*x^(5/2) + 21*A*b^12*x^(5/2) + 105*B*a^2*b^10*x^(3/2) - 70*A*a
*b^11*x^(3/2) - 420*B*a^3*b^9*sqrt(x) + 315*A*a^2*b^10*sqrt(x))/b^14

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maple [A]  time = 0.02, size = 163, normalized size = 1.06 \begin {gather*} \frac {2 B \,x^{\frac {7}{2}}}{7 b^{2}}+\frac {2 A \,x^{\frac {5}{2}}}{5 b^{2}}-\frac {4 B a \,x^{\frac {5}{2}}}{5 b^{3}}-\frac {7 A \,a^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{4}}+\frac {9 B \,a^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{5}}+\frac {A \,a^{3} \sqrt {x}}{\left (b x +a \right ) b^{4}}-\frac {4 A a \,x^{\frac {3}{2}}}{3 b^{3}}-\frac {B \,a^{4} \sqrt {x}}{\left (b x +a \right ) b^{5}}+\frac {2 B \,a^{2} x^{\frac {3}{2}}}{b^{4}}+\frac {6 A \,a^{2} \sqrt {x}}{b^{4}}-\frac {8 B \,a^{3} \sqrt {x}}{b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.80, size = 139, normalized size = 0.90 \begin {gather*} -\frac {{\left (B a^{4} - A a^{3} b\right )} \sqrt {x}}{b^{6} x + a b^{5}} + \frac {{\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} + \frac {2 \, {\left (15 \, B b^{3} x^{\frac {7}{2}} - 21 \, {\left (2 \, B a b^{2} - A b^{3}\right )} x^{\frac {5}{2}} + 35 \, {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{\frac {3}{2}} - 105 \, {\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} \sqrt {x}\right )}}{105 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(B*a^4 - A*a^3*b)*sqrt(x)/(b^6*x + a*b^5) + (9*B*a^4 - 7*A*a^3*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^5)
+ 2/105*(15*B*b^3*x^(7/2) - 21*(2*B*a*b^2 - A*b^3)*x^(5/2) + 35*(3*B*a^2*b - 2*A*a*b^2)*x^(3/2) - 105*(4*B*a^
3 - 3*A*a^2*b)*sqrt(x))/b^5

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mupad [B]  time = 0.39, size = 209, normalized size = 1.36 \begin {gather*} \sqrt {x}\,\left (\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {2\,A}{b^2}-\frac {4\,B\,a}{b^3}\right )}{b}+\frac {2\,B\,a^2}{b^4}\right )}{b}-\frac {a^2\,\left (\frac {2\,A}{b^2}-\frac {4\,B\,a}{b^3}\right )}{b^2}\right )+x^{5/2}\,\left (\frac {2\,A}{5\,b^2}-\frac {4\,B\,a}{5\,b^3}\right )-x^{3/2}\,\left (\frac {2\,a\,\left (\frac {2\,A}{b^2}-\frac {4\,B\,a}{b^3}\right )}{3\,b}+\frac {2\,B\,a^2}{3\,b^4}\right )+\frac {2\,B\,x^{7/2}}{7\,b^2}-\frac {\sqrt {x}\,\left (B\,a^4-A\,a^3\,b\right )}{x\,b^6+a\,b^5}+\frac {a^{5/2}\,\mathrm {atan}\left (\frac {a^{5/2}\,\sqrt {b}\,\sqrt {x}\,\left (7\,A\,b-9\,B\,a\right )}{9\,B\,a^4-7\,A\,a^3\,b}\right )\,\left (7\,A\,b-9\,B\,a\right )}{b^{11/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(1/2)*((2*a*((2*a*((2*A)/b^2 - (4*B*a)/b^3))/b + (2*B*a^2)/b^4))/b - (a^2*((2*A)/b^2 - (4*B*a)/b^3))/b^2) +
x^(5/2)*((2*A)/(5*b^2) - (4*B*a)/(5*b^3)) - x^(3/2)*((2*a*((2*A)/b^2 - (4*B*a)/b^3))/(3*b) + (2*B*a^2)/(3*b^4)
) + (2*B*x^(7/2))/(7*b^2) - (x^(1/2)*(B*a^4 - A*a^3*b))/(a*b^5 + b^6*x) + (a^(5/2)*atan((a^(5/2)*b^(1/2)*x^(1/
2)*(7*A*b - 9*B*a))/(9*B*a^4 - 7*A*a^3*b))*(7*A*b - 9*B*a))/b^(11/2)

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sympy [A]  time = 156.28, size = 1197, normalized size = 7.77

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*(2*A*x**(5/2)/5 + 2*B*x**(7/2)/7), Eq(a, 0) & Eq(b, 0)), ((2*A*x**(5/2)/5 + 2*B*x**(7/2)/7)/b**
2, Eq(a, 0)), ((2*A*x**(9/2)/9 + 2*B*x**(11/2)/11)/a**2, Eq(b, 0)), (1470*I*A*a**(7/2)*b**2*sqrt(x)*sqrt(1/b)/
(210*I*a**(3/2)*b**6*sqrt(1/b) + 210*I*sqrt(a)*b**7*x*sqrt(1/b)) + 980*I*A*a**(5/2)*b**3*x**(3/2)*sqrt(1/b)/(2
10*I*a**(3/2)*b**6*sqrt(1/b) + 210*I*sqrt(a)*b**7*x*sqrt(1/b)) - 196*I*A*a**(3/2)*b**4*x**(5/2)*sqrt(1/b)/(210
*I*a**(3/2)*b**6*sqrt(1/b) + 210*I*sqrt(a)*b**7*x*sqrt(1/b)) + 84*I*A*sqrt(a)*b**5*x**(7/2)*sqrt(1/b)/(210*I*a
**(3/2)*b**6*sqrt(1/b) + 210*I*sqrt(a)*b**7*x*sqrt(1/b)) - 735*A*a**4*b*log(-I*sqrt(a)*sqrt(1/b) + sqrt(x))/(2
10*I*a**(3/2)*b**6*sqrt(1/b) + 210*I*sqrt(a)*b**7*x*sqrt(1/b)) + 735*A*a**4*b*log(I*sqrt(a)*sqrt(1/b) + sqrt(x
))/(210*I*a**(3/2)*b**6*sqrt(1/b) + 210*I*sqrt(a)*b**7*x*sqrt(1/b)) - 735*A*a**3*b**2*x*log(-I*sqrt(a)*sqrt(1/
b) + sqrt(x))/(210*I*a**(3/2)*b**6*sqrt(1/b) + 210*I*sqrt(a)*b**7*x*sqrt(1/b)) + 735*A*a**3*b**2*x*log(I*sqrt(
a)*sqrt(1/b) + sqrt(x))/(210*I*a**(3/2)*b**6*sqrt(1/b) + 210*I*sqrt(a)*b**7*x*sqrt(1/b)) - 1890*I*B*a**(9/2)*b
*sqrt(x)*sqrt(1/b)/(210*I*a**(3/2)*b**6*sqrt(1/b) + 210*I*sqrt(a)*b**7*x*sqrt(1/b)) - 1260*I*B*a**(7/2)*b**2*x
**(3/2)*sqrt(1/b)/(210*I*a**(3/2)*b**6*sqrt(1/b) + 210*I*sqrt(a)*b**7*x*sqrt(1/b)) + 252*I*B*a**(5/2)*b**3*x**
(5/2)*sqrt(1/b)/(210*I*a**(3/2)*b**6*sqrt(1/b) + 210*I*sqrt(a)*b**7*x*sqrt(1/b)) - 108*I*B*a**(3/2)*b**4*x**(7
/2)*sqrt(1/b)/(210*I*a**(3/2)*b**6*sqrt(1/b) + 210*I*sqrt(a)*b**7*x*sqrt(1/b)) + 60*I*B*sqrt(a)*b**5*x**(9/2)*
sqrt(1/b)/(210*I*a**(3/2)*b**6*sqrt(1/b) + 210*I*sqrt(a)*b**7*x*sqrt(1/b)) + 945*B*a**5*log(-I*sqrt(a)*sqrt(1/
b) + sqrt(x))/(210*I*a**(3/2)*b**6*sqrt(1/b) + 210*I*sqrt(a)*b**7*x*sqrt(1/b)) - 945*B*a**5*log(I*sqrt(a)*sqrt
(1/b) + sqrt(x))/(210*I*a**(3/2)*b**6*sqrt(1/b) + 210*I*sqrt(a)*b**7*x*sqrt(1/b)) + 945*B*a**4*b*x*log(-I*sqrt
(a)*sqrt(1/b) + sqrt(x))/(210*I*a**(3/2)*b**6*sqrt(1/b) + 210*I*sqrt(a)*b**7*x*sqrt(1/b)) - 945*B*a**4*b*x*log
(I*sqrt(a)*sqrt(1/b) + sqrt(x))/(210*I*a**(3/2)*b**6*sqrt(1/b) + 210*I*sqrt(a)*b**7*x*sqrt(1/b)), True))

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