Integrand size = 26, antiderivative size = 241 \[ \int (e x)^{7/2} \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx=\frac {a^3 (10 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{384 b^2}+\frac {a^2 (10 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{192 b e}+\frac {a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac {(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}-\frac {a^4 (10 A b-3 a B) e^{7/2} \text {arctanh}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{384 b^{5/2}} \]
1/144*a*(10*A*b-3*B*a)*(e*x)^(9/2)*(b*x^3+a)^(3/2)/b/e+1/120*(10*A*b-3*B*a )*(e*x)^(9/2)*(b*x^3+a)^(5/2)/b/e+1/15*B*(e*x)^(9/2)*(b*x^3+a)^(7/2)/b/e-1 /384*a^4*(10*A*b-3*B*a)*e^(7/2)*arctanh((e*x)^(3/2)*b^(1/2)/e^(3/2)/(b*x^3 +a)^(1/2))/b^(5/2)+1/384*a^3*(10*A*b-3*B*a)*e^2*(e*x)^(3/2)*(b*x^3+a)^(1/2 )/b^2+1/192*a^2*(10*A*b-3*B*a)*(e*x)^(9/2)*(b*x^3+a)^(1/2)/b/e
Time = 0.74 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.68 \[ \int (e x)^{7/2} \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx=\frac {e^3 \sqrt {e x} \left (\sqrt {b} x^{3/2} \sqrt {a+b x^3} \left (-45 a^4 B+30 a^3 b \left (5 A+B x^3\right )+96 b^4 x^9 \left (5 A+4 B x^3\right )+16 a b^3 x^6 \left (85 A+63 B x^3\right )+4 a^2 b^2 x^3 \left (295 A+186 B x^3\right )\right )+15 a^4 (-10 A b+3 a B) \log \left (\sqrt {b} x^{3/2}+\sqrt {a+b x^3}\right )\right )}{5760 b^{5/2} \sqrt {x}} \]
(e^3*Sqrt[e*x]*(Sqrt[b]*x^(3/2)*Sqrt[a + b*x^3]*(-45*a^4*B + 30*a^3*b*(5*A + B*x^3) + 96*b^4*x^9*(5*A + 4*B*x^3) + 16*a*b^3*x^6*(85*A + 63*B*x^3) + 4*a^2*b^2*x^3*(295*A + 186*B*x^3)) + 15*a^4*(-10*A*b + 3*a*B)*Log[Sqrt[b]* x^(3/2) + Sqrt[a + b*x^3]]))/(5760*b^(5/2)*Sqrt[x])
Time = 0.38 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.89, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {959, 811, 811, 811, 843, 851, 807, 224, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (e x)^{7/2} \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \frac {(10 A b-3 a B) \int (e x)^{7/2} \left (b x^3+a\right )^{5/2}dx}{10 b}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {(10 A b-3 a B) \left (\frac {5}{8} a \int (e x)^{7/2} \left (b x^3+a\right )^{3/2}dx+\frac {(e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 e}\right )}{10 b}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {(10 A b-3 a B) \left (\frac {5}{8} a \left (\frac {1}{2} a \int (e x)^{7/2} \sqrt {b x^3+a}dx+\frac {(e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 e}\right )+\frac {(e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 e}\right )}{10 b}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {(10 A b-3 a B) \left (\frac {5}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \int \frac {(e x)^{7/2}}{\sqrt {b x^3+a}}dx+\frac {(e x)^{9/2} \sqrt {a+b x^3}}{6 e}\right )+\frac {(e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 e}\right )+\frac {(e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 e}\right )}{10 b}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {(10 A b-3 a B) \left (\frac {5}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {e^2 (e x)^{3/2} \sqrt {a+b x^3}}{3 b}-\frac {a e^3 \int \frac {\sqrt {e x}}{\sqrt {b x^3+a}}dx}{2 b}\right )+\frac {(e x)^{9/2} \sqrt {a+b x^3}}{6 e}\right )+\frac {(e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 e}\right )+\frac {(e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 e}\right )}{10 b}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {(10 A b-3 a B) \left (\frac {5}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {e^2 (e x)^{3/2} \sqrt {a+b x^3}}{3 b}-\frac {a e^2 \int \frac {e x}{\sqrt {b x^3+a}}d\sqrt {e x}}{b}\right )+\frac {(e x)^{9/2} \sqrt {a+b x^3}}{6 e}\right )+\frac {(e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 e}\right )+\frac {(e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 e}\right )}{10 b}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {(10 A b-3 a B) \left (\frac {5}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {e^2 (e x)^{3/2} \sqrt {a+b x^3}}{3 b}-\frac {a e^2 \int \frac {1}{\sqrt {a+\frac {b x}{e^2}}}d(e x)^{3/2}}{3 b}\right )+\frac {(e x)^{9/2} \sqrt {a+b x^3}}{6 e}\right )+\frac {(e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 e}\right )+\frac {(e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 e}\right )}{10 b}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {(10 A b-3 a B) \left (\frac {5}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {e^2 (e x)^{3/2} \sqrt {a+b x^3}}{3 b}-\frac {a e^2 \int \frac {1}{1-\frac {b x}{e^2}}d\frac {(e x)^{3/2}}{\sqrt {a+\frac {b x}{e^2}}}}{3 b}\right )+\frac {(e x)^{9/2} \sqrt {a+b x^3}}{6 e}\right )+\frac {(e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 e}\right )+\frac {(e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 e}\right )}{10 b}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {(10 A b-3 a B) \left (\frac {5}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {e^2 (e x)^{3/2} \sqrt {a+b x^3}}{3 b}-\frac {a e^{7/2} \text {arctanh}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+\frac {b x}{e^2}}}\right )}{3 b^{3/2}}\right )+\frac {(e x)^{9/2} \sqrt {a+b x^3}}{6 e}\right )+\frac {(e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 e}\right )+\frac {(e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 e}\right )}{10 b}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}\) |
(B*(e*x)^(9/2)*(a + b*x^3)^(7/2))/(15*b*e) + ((10*A*b - 3*a*B)*(((e*x)^(9/ 2)*(a + b*x^3)^(5/2))/(12*e) + (5*a*(((e*x)^(9/2)*(a + b*x^3)^(3/2))/(9*e) + (a*(((e*x)^(9/2)*Sqrt[a + b*x^3])/(6*e) + (a*((e^2*(e*x)^(3/2)*Sqrt[a + b*x^3])/(3*b) - (a*e^(7/2)*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/2)*Sqrt[a + (b*x)/e^2])])/(3*b^(3/2))))/4))/2))/8))/(10*b)
3.6.35.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 )) Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m , p, x]
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] - Simp[ 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]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[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]
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] - Simp[(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]
Time = 4.98 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.77
| method | result | size |
| risch | \(\frac {x^{2} \left (384 B \,b^{4} x^{12}+480 A \,b^{4} x^{9}+1008 B a \,b^{3} x^{9}+1360 A a \,b^{3} x^{6}+744 B \,a^{2} b^{2} x^{6}+1180 A \,a^{2} b^{2} x^{3}+30 B \,a^{3} b \,x^{3}+150 A \,a^{3} b -45 B \,a^{4}\right ) \sqrt {b \,x^{3}+a}\, e^{4}}{5760 b^{2} \sqrt {e x}}-\frac {a^{4} \left (10 A b -3 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {\left (b \,x^{3}+a \right ) e x}}{x^{2} \sqrt {b e}}\right ) e^{4} \sqrt {\left (b \,x^{3}+a \right ) e x}}{384 b^{2} \sqrt {b e}\, \sqrt {e x}\, \sqrt {b \,x^{3}+a}}\) | \(186\) |
| elliptic | \(\text {Expression too large to display}\) | \(1443\) |
| default | \(\text {Expression too large to display}\) | \(5626\) |
1/5760/b^2*x^2*(384*B*b^4*x^12+480*A*b^4*x^9+1008*B*a*b^3*x^9+1360*A*a*b^3 *x^6+744*B*a^2*b^2*x^6+1180*A*a^2*b^2*x^3+30*B*a^3*b*x^3+150*A*a^3*b-45*B* a^4)*(b*x^3+a)^(1/2)*e^4/(e*x)^(1/2)-1/384*a^4/b^2*(10*A*b-3*B*a)/(b*e)^(1 /2)*arctanh(((b*x^3+a)*e*x)^(1/2)/x^2/(b*e)^(1/2))*e^4*((b*x^3+a)*e*x)^(1/ 2)/(e*x)^(1/2)/(b*x^3+a)^(1/2)
Time = 0.59 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.70 \[ \int (e x)^{7/2} \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx=\left [-\frac {15 \, {\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} e^{3} \sqrt {\frac {e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e + 4 \, {\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt {b x^{3} + a} \sqrt {e x} \sqrt {\frac {e}{b}}\right ) - 4 \, {\left (384 \, B b^{4} e^{3} x^{13} + 48 \, {\left (21 \, B a b^{3} + 10 \, A b^{4}\right )} e^{3} x^{10} + 8 \, {\left (93 \, B a^{2} b^{2} + 170 \, A a b^{3}\right )} e^{3} x^{7} + 10 \, {\left (3 \, B a^{3} b + 118 \, A a^{2} b^{2}\right )} e^{3} x^{4} - 15 \, {\left (3 \, B a^{4} - 10 \, A a^{3} b\right )} e^{3} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{23040 \, b^{2}}, -\frac {15 \, {\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} e^{3} \sqrt {-\frac {e}{b}} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {e x} b x \sqrt {-\frac {e}{b}}}{2 \, b e x^{3} + a e}\right ) - 2 \, {\left (384 \, B b^{4} e^{3} x^{13} + 48 \, {\left (21 \, B a b^{3} + 10 \, A b^{4}\right )} e^{3} x^{10} + 8 \, {\left (93 \, B a^{2} b^{2} + 170 \, A a b^{3}\right )} e^{3} x^{7} + 10 \, {\left (3 \, B a^{3} b + 118 \, A a^{2} b^{2}\right )} e^{3} x^{4} - 15 \, {\left (3 \, B a^{4} - 10 \, A a^{3} b\right )} e^{3} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{11520 \, b^{2}}\right ] \]
[-1/23040*(15*(3*B*a^5 - 10*A*a^4*b)*e^3*sqrt(e/b)*log(-8*b^2*e*x^6 - 8*a* b*e*x^3 - a^2*e + 4*(2*b^2*x^4 + a*b*x)*sqrt(b*x^3 + a)*sqrt(e*x)*sqrt(e/b )) - 4*(384*B*b^4*e^3*x^13 + 48*(21*B*a*b^3 + 10*A*b^4)*e^3*x^10 + 8*(93*B *a^2*b^2 + 170*A*a*b^3)*e^3*x^7 + 10*(3*B*a^3*b + 118*A*a^2*b^2)*e^3*x^4 - 15*(3*B*a^4 - 10*A*a^3*b)*e^3*x)*sqrt(b*x^3 + a)*sqrt(e*x))/b^2, -1/11520 *(15*(3*B*a^5 - 10*A*a^4*b)*e^3*sqrt(-e/b)*arctan(2*sqrt(b*x^3 + a)*sqrt(e *x)*b*x*sqrt(-e/b)/(2*b*e*x^3 + a*e)) - 2*(384*B*b^4*e^3*x^13 + 48*(21*B*a *b^3 + 10*A*b^4)*e^3*x^10 + 8*(93*B*a^2*b^2 + 170*A*a*b^3)*e^3*x^7 + 10*(3 *B*a^3*b + 118*A*a^2*b^2)*e^3*x^4 - 15*(3*B*a^4 - 10*A*a^3*b)*e^3*x)*sqrt( b*x^3 + a)*sqrt(e*x))/b^2]
Leaf count of result is larger than twice the leaf count of optimal. 1028 vs. \(2 (216) = 432\).
Time = 49.13 (sec) , antiderivative size = 1028, normalized size of antiderivative = 4.27 \[ \int (e x)^{7/2} \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx=\text {Too large to display} \]
Piecewise((2*Piecewise((nan, Eq(e**3, 0)), ((A*a**2*e**3*Piecewise((-a**2* e**3*Piecewise((log(2*b*(e*x)**(3/2)/e**3 + 2*sqrt(b/e**3)*sqrt(a + b*x**3 ))/sqrt(b/e**3), Ne(a, 0)), ((e*x)**(3/2)*log((e*x)**(3/2))/sqrt(b*x**3), True))/(8*b) + sqrt(a + b*x**3)*(a*e**3*(e*x)**(3/2)/(8*b) + (e*x)**(9/2)/ 4), Ne(b/e**3, 0)), (sqrt(a)*(e*x)**(9/2)/3, True)) + 2*A*a*b*Piecewise((a **3*e**6*Piecewise((log(2*b*(e*x)**(3/2)/e**3 + 2*sqrt(b/e**3)*sqrt(a + b* x**3))/sqrt(b/e**3), Ne(a, 0)), ((e*x)**(3/2)*log((e*x)**(3/2))/sqrt(b*x** 3), True))/(16*b**2) + sqrt(a + b*x**3)*(-a**2*e**6*(e*x)**(3/2)/(16*b**2) + a*e**3*(e*x)**(9/2)/(24*b) + (e*x)**(15/2)/6), Ne(b/e**3, 0)), (sqrt(a) *(e*x)**(15/2)/5, True)) + A*b**2*Piecewise((-5*a**4*e**9*Piecewise((log(2 *b*(e*x)**(3/2)/e**3 + 2*sqrt(b/e**3)*sqrt(a + b*x**3))/sqrt(b/e**3), Ne(a , 0)), ((e*x)**(3/2)*log((e*x)**(3/2))/sqrt(b*x**3), True))/(128*b**3) + s qrt(a + b*x**3)*(5*a**3*e**9*(e*x)**(3/2)/(128*b**3) - 5*a**2*e**6*(e*x)** (9/2)/(192*b**2) + a*e**3*(e*x)**(15/2)/(48*b) + (e*x)**(21/2)/8), Ne(b/e* *3, 0)), (sqrt(a)*(e*x)**(21/2)/7, True))/e**3 + B*a**2*Piecewise((a**3*e* *6*Piecewise((log(2*b*(e*x)**(3/2)/e**3 + 2*sqrt(b/e**3)*sqrt(a + b*x**3)) /sqrt(b/e**3), Ne(a, 0)), ((e*x)**(3/2)*log((e*x)**(3/2))/sqrt(b*x**3), Tr ue))/(16*b**2) + sqrt(a + b*x**3)*(-a**2*e**6*(e*x)**(3/2)/(16*b**2) + a*e **3*(e*x)**(9/2)/(24*b) + (e*x)**(15/2)/6), Ne(b/e**3, 0)), (sqrt(a)*(e*x) **(15/2)/5, True)) + 2*B*a*b*Piecewise((-5*a**4*e**9*Piecewise((log(2*b...
\[ \int (e x)^{7/2} \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx=\int { {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {7}{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (197) = 394\).
Time = 0.60 (sec) , antiderivative size = 692, normalized size of antiderivative = 2.87 \[ \int (e x)^{7/2} \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx=\frac {1}{12} \, \sqrt {b e^{4} x^{3} + a e^{4}} \sqrt {e x} {\left (\frac {2 \, x^{3}}{e} + \frac {a}{b e}\right )} A a^{2} x {\left | e \right |}^{2} + \frac {1}{72} \, \sqrt {b e^{4} x^{3} + a e^{4}} {\left (2 \, e^{3} x^{3} {\left (\frac {4 \, x^{3}}{e^{4}} + \frac {a}{b e^{4}}\right )} - \frac {3 \, a^{2}}{b^{2} e}\right )} \sqrt {e x} B a^{2} x {\left | e \right |}^{2} + \frac {1}{36} \, \sqrt {b e^{4} x^{3} + a e^{4}} {\left (2 \, e^{3} x^{3} {\left (\frac {4 \, x^{3}}{e^{4}} + \frac {a}{b e^{4}}\right )} - \frac {3 \, a^{2}}{b^{2} e}\right )} \sqrt {e x} A a b x {\left | e \right |}^{2} + \frac {1}{288} \, \sqrt {b e^{4} x^{3} + a e^{4}} {\left (2 \, {\left (4 \, e^{3} x^{3} {\left (\frac {6 \, x^{3}}{e^{7}} + \frac {a}{b e^{7}}\right )} - \frac {5 \, a^{2}}{b^{2} e^{4}}\right )} e^{3} x^{3} + \frac {15 \, a^{3}}{b^{3} e}\right )} \sqrt {e x} B a b x {\left | e \right |}^{2} + \frac {1}{576} \, \sqrt {b e^{4} x^{3} + a e^{4}} {\left (2 \, {\left (4 \, e^{3} x^{3} {\left (\frac {6 \, x^{3}}{e^{7}} + \frac {a}{b e^{7}}\right )} - \frac {5 \, a^{2}}{b^{2} e^{4}}\right )} e^{3} x^{3} + \frac {15 \, a^{3}}{b^{3} e}\right )} \sqrt {e x} A b^{2} x {\left | e \right |}^{2} + \frac {1}{5760} \, \sqrt {b e^{4} x^{3} + a e^{4}} {\left (2 \, {\left (4 \, {\left (6 \, e^{3} x^{3} {\left (\frac {8 \, x^{3}}{e^{10}} + \frac {a}{b e^{10}}\right )} - \frac {7 \, a^{2}}{b^{2} e^{7}}\right )} e^{3} x^{3} + \frac {35 \, a^{3}}{b^{3} e^{4}}\right )} e^{3} x^{3} - \frac {105 \, a^{4}}{b^{4} e}\right )} \sqrt {e x} B b^{2} x {\left | e \right |}^{2} - \frac {{\left (9 \, B^{2} a^{10} e - 60 \, A B a^{9} b e + 100 \, A^{2} a^{8} b^{2} e\right )}^{2} e^{5} \log \left ({\left | -{\left (3 \, \sqrt {e x} B a^{5} e^{2} x - 10 \, \sqrt {e x} A a^{4} b e^{2} x\right )} \sqrt {b e} + \sqrt {9 \, B^{2} a^{11} e^{6} - 60 \, A B a^{10} b e^{6} + 100 \, A^{2} a^{9} b^{2} e^{6} + {\left (3 \, \sqrt {e x} B a^{5} e^{2} x - 10 \, \sqrt {e x} A a^{4} b e^{2} x\right )}^{2} b e} \right |}\right )}{384 \, \sqrt {b e} b^{2} {\left | 9 \, B^{2} a^{10} e - 60 \, A B a^{9} b e + 100 \, A^{2} a^{8} b^{2} e \right |} {\left | -3 \, B a^{5} + 10 \, A a^{4} b \right |} {\left | e \right |}^{2}} \]
1/12*sqrt(b*e^4*x^3 + a*e^4)*sqrt(e*x)*(2*x^3/e + a/(b*e))*A*a^2*x*abs(e)^ 2 + 1/72*sqrt(b*e^4*x^3 + a*e^4)*(2*e^3*x^3*(4*x^3/e^4 + a/(b*e^4)) - 3*a^ 2/(b^2*e))*sqrt(e*x)*B*a^2*x*abs(e)^2 + 1/36*sqrt(b*e^4*x^3 + a*e^4)*(2*e^ 3*x^3*(4*x^3/e^4 + a/(b*e^4)) - 3*a^2/(b^2*e))*sqrt(e*x)*A*a*b*x*abs(e)^2 + 1/288*sqrt(b*e^4*x^3 + a*e^4)*(2*(4*e^3*x^3*(6*x^3/e^7 + a/(b*e^7)) - 5* a^2/(b^2*e^4))*e^3*x^3 + 15*a^3/(b^3*e))*sqrt(e*x)*B*a*b*x*abs(e)^2 + 1/57 6*sqrt(b*e^4*x^3 + a*e^4)*(2*(4*e^3*x^3*(6*x^3/e^7 + a/(b*e^7)) - 5*a^2/(b ^2*e^4))*e^3*x^3 + 15*a^3/(b^3*e))*sqrt(e*x)*A*b^2*x*abs(e)^2 + 1/5760*sqr t(b*e^4*x^3 + a*e^4)*(2*(4*(6*e^3*x^3*(8*x^3/e^10 + a/(b*e^10)) - 7*a^2/(b ^2*e^7))*e^3*x^3 + 35*a^3/(b^3*e^4))*e^3*x^3 - 105*a^4/(b^4*e))*sqrt(e*x)* B*b^2*x*abs(e)^2 - 1/384*(9*B^2*a^10*e - 60*A*B*a^9*b*e + 100*A^2*a^8*b^2* e)^2*e^5*log(abs(-(3*sqrt(e*x)*B*a^5*e^2*x - 10*sqrt(e*x)*A*a^4*b*e^2*x)*s qrt(b*e) + sqrt(9*B^2*a^11*e^6 - 60*A*B*a^10*b*e^6 + 100*A^2*a^9*b^2*e^6 + (3*sqrt(e*x)*B*a^5*e^2*x - 10*sqrt(e*x)*A*a^4*b*e^2*x)^2*b*e)))/(sqrt(b*e )*b^2*abs(9*B^2*a^10*e - 60*A*B*a^9*b*e + 100*A^2*a^8*b^2*e)*abs(-3*B*a^5 + 10*A*a^4*b)*abs(e)^2)
Timed out. \[ \int (e x)^{7/2} \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx=\int \left (B\,x^3+A\right )\,{\left (e\,x\right )}^{7/2}\,{\left (b\,x^3+a\right )}^{5/2} \,d x \]