Integrand size = 24, antiderivative size = 257 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{5/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{7/2}}{3 e (b d-a e) (d+e x)^{3/2}}-\frac {2 (7 b B d-4 A b e-3 a B e) (a+b x)^{5/2}}{3 e^2 (b d-a e) \sqrt {d+e x}}-\frac {5 b (7 b B d-4 A b e-3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 e^4}+\frac {5 b (7 b B d-4 A b e-3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{6 e^3 (b d-a e)}+\frac {5 \sqrt {b} (b d-a e) (7 b B d-4 A b e-3 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 e^{9/2}} \]
-2/3*(-A*e+B*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)/(e*x+d)^(3/2)+5/4*(-a*e+b*d)*(- 4*A*b*e-3*B*a*e+7*B*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/ 2))*b^(1/2)/e^(9/2)-2/3*(-4*A*b*e-3*B*a*e+7*B*b*d)*(b*x+a)^(5/2)/e^2/(-a*e +b*d)/(e*x+d)^(1/2)+5/6*b*(-4*A*b*e-3*B*a*e+7*B*b*d)*(b*x+a)^(3/2)*(e*x+d) ^(1/2)/e^3/(-a*e+b*d)-5/4*b*(-4*A*b*e-3*B*a*e+7*B*b*d)*(b*x+a)^(1/2)*(e*x+ d)^(1/2)/e^4
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.17 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.44 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{5/2}} \, dx=\frac {2 (a+b x)^{7/2} \left (7 B d-7 A e-\frac {(7 b B d-4 A b e-3 a B e) \left (\frac {b (d+e x)}{b d-a e}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {7}{2},\frac {9}{2},\frac {e (a+b x)}{-b d+a e}\right )}{b}\right )}{21 e (-b d+a e) (d+e x)^{3/2}} \]
(2*(a + b*x)^(7/2)*(7*B*d - 7*A*e - ((7*b*B*d - 4*A*b*e - 3*a*B*e)*((b*(d + e*x))/(b*d - a*e))^(3/2)*Hypergeometric2F1[3/2, 7/2, 9/2, (e*(a + b*x))/ (-(b*d) + a*e)])/b))/(21*e*(-(b*d) + a*e)*(d + e*x)^(3/2))
Time = 0.28 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.86, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {87, 57, 60, 60, 66, 221}
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.
\(\displaystyle \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {(-3 a B e-4 A b e+7 b B d) \int \frac {(a+b x)^{5/2}}{(d+e x)^{3/2}}dx}{3 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}\) |
\(\Big \downarrow \) 57 |
\(\displaystyle \frac {(-3 a B e-4 A b e+7 b B d) \left (\frac {5 b \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}}dx}{e}-\frac {2 (a+b x)^{5/2}}{e \sqrt {d+e x}}\right )}{3 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(-3 a B e-4 A b e+7 b B d) \left (\frac {5 b \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}}dx}{4 e}\right )}{e}-\frac {2 (a+b x)^{5/2}}{e \sqrt {d+e x}}\right )}{3 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(-3 a B e-4 A b e+7 b B d) \left (\frac {5 b \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}}dx}{2 e}\right )}{4 e}\right )}{e}-\frac {2 (a+b x)^{5/2}}{e \sqrt {d+e x}}\right )}{3 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}\) |
\(\Big \downarrow \) 66 |
\(\displaystyle \frac {(-3 a B e-4 A b e+7 b B d) \left (\frac {5 b \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \int \frac {1}{b-\frac {e (a+b x)}{d+e x}}d\frac {\sqrt {a+b x}}{\sqrt {d+e x}}}{e}\right )}{4 e}\right )}{e}-\frac {2 (a+b x)^{5/2}}{e \sqrt {d+e x}}\right )}{3 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {(-3 a B e-4 A b e+7 b B d) \left (\frac {5 b \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} e^{3/2}}\right )}{4 e}\right )}{e}-\frac {2 (a+b x)^{5/2}}{e \sqrt {d+e x}}\right )}{3 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}\) |
(-2*(B*d - A*e)*(a + b*x)^(7/2))/(3*e*(b*d - a*e)*(d + e*x)^(3/2)) + ((7*b *B*d - 4*A*b*e - 3*a*B*e)*((-2*(a + b*x)^(5/2))/(e*Sqrt[d + e*x]) + (5*b*( ((a + b*x)^(3/2)*Sqrt[d + e*x])/(2*e) - (3*(b*d - a*e)*((Sqrt[a + b*x]*Sqr t[d + e*x])/e - ((b*d - a*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt [d + e*x])])/(Sqrt[b]*e^(3/2))))/(4*e)))/e))/(3*e*(b*d - a*e))
3.23.28.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c , d, m, n, x]
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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(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] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Leaf count of result is larger than twice the leaf count of optimal. \(1249\) vs. \(2(219)=438\).
Time = 1.06 (sec) , antiderivative size = 1250, normalized size of antiderivative = 4.86
1/24*(b*x+a)^(1/2)*(316*B*a*b*d*e^2*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+ 120*A*b^2*d^2*e*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-120*A*ln(1/2*(2*b*e*x+ 2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^3*d^2*e^2*x+ 60*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^ (1/2))*a*b^2*d^2*e^2+45*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^ (1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b*d^2*e^2+12*B*b^2*e^3*x^3*(b*e)^(1/2)*((b *x+a)*(e*x+d))^(1/2)+160*A*b^2*d*e^2*x*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2) +105*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e )^(1/2))*b^3*d^4-80*A*a*b*d*e^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+120*A* ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2) )*a*b^2*d*e^3*x+90*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2) +a*e+b*d)/(b*e)^(1/2))*a^2*b*d*e^3*x-60*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+ d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^3*d^3*e+60*A*ln(1/2*(2*b*e*x +2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^2*e^4*x^2 -60*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e) ^(1/2))*b^3*d*e^3*x^2+45*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e) ^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b*e^4*x^2+230*B*a*b*d^2*e*((b*x+a)*(e*x+d ))^(1/2)*(b*e)^(1/2)-16*A*a^2*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-210* B*b^2*d^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-150*B*ln(1/2*(2*b*e*x+2*((b* x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^2*d*e^3*x^2-3...
Time = 1.40 (sec) , antiderivative size = 855, normalized size of antiderivative = 3.33 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{5/2}} \, dx=\left [\frac {15 \, {\left (7 \, B b^{2} d^{4} - 2 \, {\left (5 \, B a b + 2 \, A b^{2}\right )} d^{3} e + {\left (3 \, B a^{2} + 4 \, A a b\right )} d^{2} e^{2} + {\left (7 \, B b^{2} d^{2} e^{2} - 2 \, {\left (5 \, B a b + 2 \, A b^{2}\right )} d e^{3} + {\left (3 \, B a^{2} + 4 \, A a b\right )} e^{4}\right )} x^{2} + 2 \, {\left (7 \, B b^{2} d^{3} e - 2 \, {\left (5 \, B a b + 2 \, A b^{2}\right )} d^{2} e^{2} + {\left (3 \, B a^{2} + 4 \, A a b\right )} d e^{3}\right )} x\right )} \sqrt {\frac {b}{e}} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b e^{2} x + b d e + a e^{2}\right )} \sqrt {b x + a} \sqrt {e x + d} \sqrt {\frac {b}{e}} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \, {\left (6 \, B b^{2} e^{3} x^{3} - 105 \, B b^{2} d^{3} - 8 \, A a^{2} e^{3} + 5 \, {\left (23 \, B a b + 12 \, A b^{2}\right )} d^{2} e - 8 \, {\left (2 \, B a^{2} + 5 \, A a b\right )} d e^{2} - 3 \, {\left (7 \, B b^{2} d e^{2} - {\left (9 \, B a b + 4 \, A b^{2}\right )} e^{3}\right )} x^{2} - 2 \, {\left (70 \, B b^{2} d^{2} e - {\left (79 \, B a b + 40 \, A b^{2}\right )} d e^{2} + 4 \, {\left (3 \, B a^{2} + 7 \, A a b\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{48 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}}, -\frac {15 \, {\left (7 \, B b^{2} d^{4} - 2 \, {\left (5 \, B a b + 2 \, A b^{2}\right )} d^{3} e + {\left (3 \, B a^{2} + 4 \, A a b\right )} d^{2} e^{2} + {\left (7 \, B b^{2} d^{2} e^{2} - 2 \, {\left (5 \, B a b + 2 \, A b^{2}\right )} d e^{3} + {\left (3 \, B a^{2} + 4 \, A a b\right )} e^{4}\right )} x^{2} + 2 \, {\left (7 \, B b^{2} d^{3} e - 2 \, {\left (5 \, B a b + 2 \, A b^{2}\right )} d^{2} e^{2} + {\left (3 \, B a^{2} + 4 \, A a b\right )} d e^{3}\right )} x\right )} \sqrt {-\frac {b}{e}} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {b x + a} \sqrt {e x + d} \sqrt {-\frac {b}{e}}}{2 \, {\left (b^{2} e x^{2} + a b d + {\left (b^{2} d + a b e\right )} x\right )}}\right ) - 2 \, {\left (6 \, B b^{2} e^{3} x^{3} - 105 \, B b^{2} d^{3} - 8 \, A a^{2} e^{3} + 5 \, {\left (23 \, B a b + 12 \, A b^{2}\right )} d^{2} e - 8 \, {\left (2 \, B a^{2} + 5 \, A a b\right )} d e^{2} - 3 \, {\left (7 \, B b^{2} d e^{2} - {\left (9 \, B a b + 4 \, A b^{2}\right )} e^{3}\right )} x^{2} - 2 \, {\left (70 \, B b^{2} d^{2} e - {\left (79 \, B a b + 40 \, A b^{2}\right )} d e^{2} + 4 \, {\left (3 \, B a^{2} + 7 \, A a b\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{24 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}}\right ] \]
[1/48*(15*(7*B*b^2*d^4 - 2*(5*B*a*b + 2*A*b^2)*d^3*e + (3*B*a^2 + 4*A*a*b) *d^2*e^2 + (7*B*b^2*d^2*e^2 - 2*(5*B*a*b + 2*A*b^2)*d*e^3 + (3*B*a^2 + 4*A *a*b)*e^4)*x^2 + 2*(7*B*b^2*d^3*e - 2*(5*B*a*b + 2*A*b^2)*d^2*e^2 + (3*B*a ^2 + 4*A*a*b)*d*e^3)*x)*sqrt(b/e)*log(8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 + 4*(2*b*e^2*x + b*d*e + a*e^2)*sqrt(b*x + a)*sqrt(e*x + d)*sqrt (b/e) + 8*(b^2*d*e + a*b*e^2)*x) + 4*(6*B*b^2*e^3*x^3 - 105*B*b^2*d^3 - 8* A*a^2*e^3 + 5*(23*B*a*b + 12*A*b^2)*d^2*e - 8*(2*B*a^2 + 5*A*a*b)*d*e^2 - 3*(7*B*b^2*d*e^2 - (9*B*a*b + 4*A*b^2)*e^3)*x^2 - 2*(70*B*b^2*d^2*e - (79* B*a*b + 40*A*b^2)*d*e^2 + 4*(3*B*a^2 + 7*A*a*b)*e^3)*x)*sqrt(b*x + a)*sqrt (e*x + d))/(e^6*x^2 + 2*d*e^5*x + d^2*e^4), -1/24*(15*(7*B*b^2*d^4 - 2*(5* B*a*b + 2*A*b^2)*d^3*e + (3*B*a^2 + 4*A*a*b)*d^2*e^2 + (7*B*b^2*d^2*e^2 - 2*(5*B*a*b + 2*A*b^2)*d*e^3 + (3*B*a^2 + 4*A*a*b)*e^4)*x^2 + 2*(7*B*b^2*d^ 3*e - 2*(5*B*a*b + 2*A*b^2)*d^2*e^2 + (3*B*a^2 + 4*A*a*b)*d*e^3)*x)*sqrt(- b/e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(b*x + a)*sqrt(e*x + d)*sqrt(-b/ e)/(b^2*e*x^2 + a*b*d + (b^2*d + a*b*e)*x)) - 2*(6*B*b^2*e^3*x^3 - 105*B*b ^2*d^3 - 8*A*a^2*e^3 + 5*(23*B*a*b + 12*A*b^2)*d^2*e - 8*(2*B*a^2 + 5*A*a* b)*d*e^2 - 3*(7*B*b^2*d*e^2 - (9*B*a*b + 4*A*b^2)*e^3)*x^2 - 2*(70*B*b^2*d ^2*e - (79*B*a*b + 40*A*b^2)*d*e^2 + 4*(3*B*a^2 + 7*A*a*b)*e^3)*x)*sqrt(b* x + a)*sqrt(e*x + d))/(e^6*x^2 + 2*d*e^5*x + d^2*e^4)]
\[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{5/2}} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e*(a*e-b*d)>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 564 vs. \(2 (219) = 438\).
Time = 0.41 (sec) , antiderivative size = 564, normalized size of antiderivative = 2.19 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{5/2}} \, dx=\frac {{\left ({\left (3 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (B b^{5} d e^{6} {\left | b \right |} - B a b^{4} e^{7} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{4} d e^{7} - a b^{3} e^{8}} - \frac {7 \, B b^{6} d^{2} e^{5} {\left | b \right |} - 10 \, B a b^{5} d e^{6} {\left | b \right |} - 4 \, A b^{6} d e^{6} {\left | b \right |} + 3 \, B a^{2} b^{4} e^{7} {\left | b \right |} + 4 \, A a b^{5} e^{7} {\left | b \right |}}{b^{4} d e^{7} - a b^{3} e^{8}}\right )} - \frac {20 \, {\left (7 \, B b^{7} d^{3} e^{4} {\left | b \right |} - 17 \, B a b^{6} d^{2} e^{5} {\left | b \right |} - 4 \, A b^{7} d^{2} e^{5} {\left | b \right |} + 13 \, B a^{2} b^{5} d e^{6} {\left | b \right |} + 8 \, A a b^{6} d e^{6} {\left | b \right |} - 3 \, B a^{3} b^{4} e^{7} {\left | b \right |} - 4 \, A a^{2} b^{5} e^{7} {\left | b \right |}\right )}}{b^{4} d e^{7} - a b^{3} e^{8}}\right )} {\left (b x + a\right )} - \frac {15 \, {\left (7 \, B b^{8} d^{4} e^{3} {\left | b \right |} - 24 \, B a b^{7} d^{3} e^{4} {\left | b \right |} - 4 \, A b^{8} d^{3} e^{4} {\left | b \right |} + 30 \, B a^{2} b^{6} d^{2} e^{5} {\left | b \right |} + 12 \, A a b^{7} d^{2} e^{5} {\left | b \right |} - 16 \, B a^{3} b^{5} d e^{6} {\left | b \right |} - 12 \, A a^{2} b^{6} d e^{6} {\left | b \right |} + 3 \, B a^{4} b^{4} e^{7} {\left | b \right |} + 4 \, A a^{3} b^{5} e^{7} {\left | b \right |}\right )}}{b^{4} d e^{7} - a b^{3} e^{8}}\right )} \sqrt {b x + a}}{12 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {3}{2}}} - \frac {5 \, {\left (7 \, B b^{2} d^{2} {\left | b \right |} - 10 \, B a b d e {\left | b \right |} - 4 \, A b^{2} d e {\left | b \right |} + 3 \, B a^{2} e^{2} {\left | b \right |} + 4 \, A a b e^{2} {\left | b \right |}\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{4 \, \sqrt {b e} e^{4}} \]
1/12*((3*(b*x + a)*(2*(B*b^5*d*e^6*abs(b) - B*a*b^4*e^7*abs(b))*(b*x + a)/ (b^4*d*e^7 - a*b^3*e^8) - (7*B*b^6*d^2*e^5*abs(b) - 10*B*a*b^5*d*e^6*abs(b ) - 4*A*b^6*d*e^6*abs(b) + 3*B*a^2*b^4*e^7*abs(b) + 4*A*a*b^5*e^7*abs(b))/ (b^4*d*e^7 - a*b^3*e^8)) - 20*(7*B*b^7*d^3*e^4*abs(b) - 17*B*a*b^6*d^2*e^5 *abs(b) - 4*A*b^7*d^2*e^5*abs(b) + 13*B*a^2*b^5*d*e^6*abs(b) + 8*A*a*b^6*d *e^6*abs(b) - 3*B*a^3*b^4*e^7*abs(b) - 4*A*a^2*b^5*e^7*abs(b))/(b^4*d*e^7 - a*b^3*e^8))*(b*x + a) - 15*(7*B*b^8*d^4*e^3*abs(b) - 24*B*a*b^7*d^3*e^4* abs(b) - 4*A*b^8*d^3*e^4*abs(b) + 30*B*a^2*b^6*d^2*e^5*abs(b) + 12*A*a*b^7 *d^2*e^5*abs(b) - 16*B*a^3*b^5*d*e^6*abs(b) - 12*A*a^2*b^6*d*e^6*abs(b) + 3*B*a^4*b^4*e^7*abs(b) + 4*A*a^3*b^5*e^7*abs(b))/(b^4*d*e^7 - a*b^3*e^8))* sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(3/2) - 5/4*(7*B*b^2*d^2*abs (b) - 10*B*a*b*d*e*abs(b) - 4*A*b^2*d*e*abs(b) + 3*B*a^2*e^2*abs(b) + 4*A* a*b*e^2*abs(b))*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)* b*e - a*b*e)))/(sqrt(b*e)*e^4)
Timed out. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{5/2}} \,d x \]