Integrand size = 24, antiderivative size = 257 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {5 e (3 b B d+4 A b e-7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 b^4}+\frac {5 e (3 b B d+4 A b e-7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{6 b^3 (b d-a e)}-\frac {2 (3 b B d+4 A b e-7 a B e) (d+e x)^{5/2}}{3 b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {5 \sqrt {e} (b d-a e) (3 b B d+4 A b e-7 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 b^{9/2}} \]
-2/3*(A*b-B*a)*(e*x+d)^(7/2)/b/(-a*e+b*d)/(b*x+a)^(3/2)+5/4*(-a*e+b*d)*(4* A*b*e-7*B*a*e+3*B*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2) )*e^(1/2)/b^(9/2)-2/3*(4*A*b*e-7*B*a*e+3*B*b*d)*(e*x+d)^(5/2)/b^2/(-a*e+b* d)/(b*x+a)^(1/2)+5/6*e*(4*A*b*e-7*B*a*e+3*B*b*d)*(e*x+d)^(3/2)*(b*x+a)^(1/ 2)/b^3/(-a*e+b*d)+5/4*e*(4*A*b*e-7*B*a*e+3*B*b*d)*(b*x+a)^(1/2)*(e*x+d)^(1 /2)/b^4
Time = 1.44 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.91 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {\sqrt {d+e x} \left (B \left (-105 a^3 e^2+5 a^2 b e (23 d-28 e x)+a b^2 \left (-16 d^2+158 d e x-21 e^2 x^2\right )+3 b^3 x \left (-8 d^2+9 d e x+2 e^2 x^2\right )\right )+4 A b \left (15 a^2 e^2-10 a b e (d-2 e x)+b^2 \left (-2 d^2-14 d e x+3 e^2 x^2\right )\right )\right )}{12 b^4 (a+b x)^{3/2}}+\frac {5 \sqrt {e} (-b d+a e) (3 b B d+4 A b e-7 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \left (\sqrt {a-\frac {b d}{e}}-\sqrt {a+b x}\right )}\right )}{2 b^{9/2}} \]
(Sqrt[d + e*x]*(B*(-105*a^3*e^2 + 5*a^2*b*e*(23*d - 28*e*x) + a*b^2*(-16*d ^2 + 158*d*e*x - 21*e^2*x^2) + 3*b^3*x*(-8*d^2 + 9*d*e*x + 2*e^2*x^2)) + 4 *A*b*(15*a^2*e^2 - 10*a*b*e*(d - 2*e*x) + b^2*(-2*d^2 - 14*d*e*x + 3*e^2*x ^2))))/(12*b^4*(a + b*x)^(3/2)) + (5*Sqrt[e]*(-(b*d) + a*e)*(3*b*B*d + 4*A *b*e - 7*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*(Sqrt[a - (b*d)/e ] - Sqrt[a + b*x]))])/(2*b^(9/2))
Time = 0.28 (sec) , antiderivative size = 220, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-7 a B e+4 A b e+3 b B d) \int \frac {(d+e x)^{5/2}}{(a+b x)^{3/2}}dx}{3 b (b d-a e)}-\frac {2 (d+e x)^{7/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {(-7 a B e+4 A b e+3 b B d) \left (\frac {5 e \int \frac {(d+e x)^{3/2}}{\sqrt {a+b x}}dx}{b}-\frac {2 (d+e x)^{5/2}}{b \sqrt {a+b x}}\right )}{3 b (b d-a e)}-\frac {2 (d+e x)^{7/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(-7 a B e+4 A b e+3 b B d) \left (\frac {5 e \left (\frac {3 (b d-a e) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x}}dx}{4 b}+\frac {\sqrt {a+b x} (d+e x)^{3/2}}{2 b}\right )}{b}-\frac {2 (d+e x)^{5/2}}{b \sqrt {a+b x}}\right )}{3 b (b d-a e)}-\frac {2 (d+e x)^{7/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(-7 a B e+4 A b e+3 b B d) \left (\frac {5 e \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}}dx}{2 b}+\frac {\sqrt {a+b x} \sqrt {d+e x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (d+e x)^{3/2}}{2 b}\right )}{b}-\frac {2 (d+e x)^{5/2}}{b \sqrt {a+b x}}\right )}{3 b (b d-a e)}-\frac {2 (d+e x)^{7/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {(-7 a B e+4 A b e+3 b B d) \left (\frac {5 e \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \int \frac {1}{b-\frac {e (a+b x)}{d+e x}}d\frac {\sqrt {a+b x}}{\sqrt {d+e x}}}{b}+\frac {\sqrt {a+b x} \sqrt {d+e x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (d+e x)^{3/2}}{2 b}\right )}{b}-\frac {2 (d+e x)^{5/2}}{b \sqrt {a+b x}}\right )}{3 b (b d-a e)}-\frac {2 (d+e x)^{7/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(-7 a B e+4 A b e+3 b B d) \left (\frac {5 e \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{3/2} \sqrt {e}}+\frac {\sqrt {a+b x} \sqrt {d+e x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (d+e x)^{3/2}}{2 b}\right )}{b}-\frac {2 (d+e x)^{5/2}}{b \sqrt {a+b x}}\right )}{3 b (b d-a e)}-\frac {2 (d+e x)^{7/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}\) |
(-2*(A*b - a*B)*(d + e*x)^(7/2))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)) + ((3*b *B*d + 4*A*b*e - 7*a*B*e)*((-2*(d + e*x)^(5/2))/(b*Sqrt[a + b*x]) + (5*e*( (Sqrt[a + b*x]*(d + e*x)^(3/2))/(2*b) + (3*(b*d - a*e)*((Sqrt[a + b*x]*Sqr t[d + e*x])/b + ((b*d - a*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt [d + e*x])])/(b^(3/2)*Sqrt[e])))/(4*b)))/b))/(3*b*(b*d - a*e))
3.23.54.3.1 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 = 1.10 (sec) , antiderivative size = 1250, normalized size of antiderivative = 4.86
-1/24*(e*x+d)^(1/2)*(32*B*a*b^2*d^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+12 0*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^2*b^2*e^3*x-210*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^3*b*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))*a^2*b^2*d*e^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^3*b *d*e^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))*a*b^3*e^3*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^4*d*e^2*x^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))*a^2*b^2*e^3*x^2-4 5*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^4*d^2*e*x^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^3*d*e^2*x^2-316*B*a*b^2*d*e*x*((b*x+a)*(e* x+d))^(1/2)*(b*e)^(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))*a^4*e^3-12*B*b^3*e^2*x^3*((b*x+a)*(e*x+d)) ^(1/2)*(b*e)^(1/2)-24*A*b^3*e^2*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+48 *B*b^3*d^2*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-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^2*d^2*e-120* A*a^2*b*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-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*b^3*d^2*e*x+16...
Time = 1.13 (sec) , antiderivative size = 887, normalized size of antiderivative = 3.45 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{5/2}} \, dx=\left [\frac {15 \, {\left (3 \, B a^{2} b^{2} d^{2} - 2 \, {\left (5 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} d e + {\left (7 \, B a^{4} - 4 \, A a^{3} b\right )} e^{2} + {\left (3 \, B b^{4} d^{2} - 2 \, {\left (5 \, B a b^{3} - 2 \, A b^{4}\right )} d e + {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} e^{2}\right )} x^{2} + 2 \, {\left (3 \, B a b^{3} d^{2} - 2 \, {\left (5 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d e + {\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} e^{2}\right )} x\right )} \sqrt {\frac {e}{b}} \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^{2} e x + b^{2} d + a b e\right )} \sqrt {b x + a} \sqrt {e x + d} \sqrt {\frac {e}{b}} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \, {\left (6 \, B b^{3} e^{2} x^{3} - 8 \, {\left (2 \, B a b^{2} + A b^{3}\right )} d^{2} + 5 \, {\left (23 \, B a^{2} b - 8 \, A a b^{2}\right )} d e - 15 \, {\left (7 \, B a^{3} - 4 \, A a^{2} b\right )} e^{2} + 3 \, {\left (9 \, B b^{3} d e - {\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} e^{2}\right )} x^{2} - 2 \, {\left (12 \, B b^{3} d^{2} - {\left (79 \, B a b^{2} - 28 \, A b^{3}\right )} d e + 10 \, {\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{48 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, -\frac {15 \, {\left (3 \, B a^{2} b^{2} d^{2} - 2 \, {\left (5 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} d e + {\left (7 \, B a^{4} - 4 \, A a^{3} b\right )} e^{2} + {\left (3 \, B b^{4} d^{2} - 2 \, {\left (5 \, B a b^{3} - 2 \, A b^{4}\right )} d e + {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} e^{2}\right )} x^{2} + 2 \, {\left (3 \, B a b^{3} d^{2} - 2 \, {\left (5 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d e + {\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} e^{2}\right )} x\right )} \sqrt {-\frac {e}{b}} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {b x + a} \sqrt {e x + d} \sqrt {-\frac {e}{b}}}{2 \, {\left (b e^{2} x^{2} + a d e + {\left (b d e + a e^{2}\right )} x\right )}}\right ) - 2 \, {\left (6 \, B b^{3} e^{2} x^{3} - 8 \, {\left (2 \, B a b^{2} + A b^{3}\right )} d^{2} + 5 \, {\left (23 \, B a^{2} b - 8 \, A a b^{2}\right )} d e - 15 \, {\left (7 \, B a^{3} - 4 \, A a^{2} b\right )} e^{2} + 3 \, {\left (9 \, B b^{3} d e - {\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} e^{2}\right )} x^{2} - 2 \, {\left (12 \, B b^{3} d^{2} - {\left (79 \, B a b^{2} - 28 \, A b^{3}\right )} d e + 10 \, {\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{24 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \]
[1/48*(15*(3*B*a^2*b^2*d^2 - 2*(5*B*a^3*b - 2*A*a^2*b^2)*d*e + (7*B*a^4 - 4*A*a^3*b)*e^2 + (3*B*b^4*d^2 - 2*(5*B*a*b^3 - 2*A*b^4)*d*e + (7*B*a^2*b^2 - 4*A*a*b^3)*e^2)*x^2 + 2*(3*B*a*b^3*d^2 - 2*(5*B*a^2*b^2 - 2*A*a*b^3)*d* e + (7*B*a^3*b - 4*A*a^2*b^2)*e^2)*x)*sqrt(e/b)*log(8*b^2*e^2*x^2 + b^2*d^ 2 + 6*a*b*d*e + a^2*e^2 + 4*(2*b^2*e*x + b^2*d + a*b*e)*sqrt(b*x + a)*sqrt (e*x + d)*sqrt(e/b) + 8*(b^2*d*e + a*b*e^2)*x) + 4*(6*B*b^3*e^2*x^3 - 8*(2 *B*a*b^2 + A*b^3)*d^2 + 5*(23*B*a^2*b - 8*A*a*b^2)*d*e - 15*(7*B*a^3 - 4*A *a^2*b)*e^2 + 3*(9*B*b^3*d*e - (7*B*a*b^2 - 4*A*b^3)*e^2)*x^2 - 2*(12*B*b^ 3*d^2 - (79*B*a*b^2 - 28*A*b^3)*d*e + 10*(7*B*a^2*b - 4*A*a*b^2)*e^2)*x)*s qrt(b*x + a)*sqrt(e*x + d))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4), -1/24*(15*(3* B*a^2*b^2*d^2 - 2*(5*B*a^3*b - 2*A*a^2*b^2)*d*e + (7*B*a^4 - 4*A*a^3*b)*e^ 2 + (3*B*b^4*d^2 - 2*(5*B*a*b^3 - 2*A*b^4)*d*e + (7*B*a^2*b^2 - 4*A*a*b^3) *e^2)*x^2 + 2*(3*B*a*b^3*d^2 - 2*(5*B*a^2*b^2 - 2*A*a*b^3)*d*e + (7*B*a^3* b - 4*A*a^2*b^2)*e^2)*x)*sqrt(-e/b)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt( b*x + a)*sqrt(e*x + d)*sqrt(-e/b)/(b*e^2*x^2 + a*d*e + (b*d*e + a*e^2)*x)) - 2*(6*B*b^3*e^2*x^3 - 8*(2*B*a*b^2 + A*b^3)*d^2 + 5*(23*B*a^2*b - 8*A*a* b^2)*d*e - 15*(7*B*a^3 - 4*A*a^2*b)*e^2 + 3*(9*B*b^3*d*e - (7*B*a*b^2 - 4* A*b^3)*e^2)*x^2 - 2*(12*B*b^3*d^2 - (79*B*a*b^2 - 28*A*b^3)*d*e + 10*(7*B* a^2*b - 4*A*a*b^2)*e^2)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^6*x^2 + 2*a*b^5 *x + a^2*b^4)]
\[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{5/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{\frac {5}{2}}}{\left (a + b x\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 1401 vs. \(2 (219) = 438\).
Time = 0.69 (sec) , antiderivative size = 1401, normalized size of antiderivative = 5.45 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{5/2}} \, dx=\text {Too large to display} \]
1/4*sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*B*e^2*a bs(b)/b^6 + (9*B*b^12*d*e^3*abs(b) - 13*B*a*b^11*e^4*abs(b) + 4*A*b^12*e^4 *abs(b))/(b^17*e^2)) - 5/8*(3*sqrt(b*e)*B*b^2*d^2*abs(b) - 10*sqrt(b*e)*B* a*b*d*e*abs(b) + 4*sqrt(b*e)*A*b^2*d*e*abs(b) + 7*sqrt(b*e)*B*a^2*e^2*abs( b) - 4*sqrt(b*e)*A*a*b*e^2*abs(b))*log((sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2 *d + (b*x + a)*b*e - a*b*e))^2)/b^6 - 4/3*(3*sqrt(b*e)*B*b^7*d^5*abs(b) - 22*sqrt(b*e)*B*a*b^6*d^4*e*abs(b) + 7*sqrt(b*e)*A*b^7*d^4*e*abs(b) + 58*sq rt(b*e)*B*a^2*b^5*d^3*e^2*abs(b) - 28*sqrt(b*e)*A*a*b^6*d^3*e^2*abs(b) - 7 2*sqrt(b*e)*B*a^3*b^4*d^2*e^3*abs(b) + 42*sqrt(b*e)*A*a^2*b^5*d^2*e^3*abs( b) + 43*sqrt(b*e)*B*a^4*b^3*d*e^4*abs(b) - 28*sqrt(b*e)*A*a^3*b^4*d*e^4*ab s(b) - 10*sqrt(b*e)*B*a^5*b^2*e^5*abs(b) + 7*sqrt(b*e)*A*a^4*b^3*e^5*abs(b ) - 6*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a* b*e))^2*B*b^5*d^4*abs(b) + 36*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b^ 2*d + (b*x + a)*b*e - a*b*e))^2*B*a*b^4*d^3*e*abs(b) - 12*sqrt(b*e)*(sqrt( b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*b^5*d^3*e*ab s(b) - 72*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a^2*b^3*d^2*e^2*abs(b) + 36*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*a*b^4*d^2*e^2*abs(b) + 60*sq rt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2* B*a^3*b^2*d*e^3*abs(b) - 36*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b...
Timed out. \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{5/2}}{{\left (a+b\,x\right )}^{5/2}} \,d x \]