\(\int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{5/2}} \, dx\) [187]

Optimal result

Integrand size = 24, antiderivative size = 155 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{5/2}} \, dx=\frac {2 (B d-A e) (a+b x)^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (2 b B d-A b e-a B e) \sqrt {a+b x}}{e^3 \sqrt {d+e x}}+\frac {b B \sqrt {a+b x} \sqrt {d+e x}}{e^3}-\frac {\sqrt {b} (5 b B d-2 A b e-3 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{7/2}} \] Output:

2/3*(-A*e+B*d)*(b*x+a)^(3/2)/e^2/(e*x+d)^(3/2)+2*(-A*b*e-B*a*e+2*B*b*d)*(b 
*x+a)^(1/2)/e^3/(e*x+d)^(1/2)+b*B*(b*x+a)^(1/2)*(e*x+d)^(1/2)/e^3-b^(1/2)* 
(-2*A*b*e-3*B*a*e+5*B*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^( 
1/2))/e^(7/2)
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{5/2}} \, dx=\frac {\sqrt {a+b x} \left (-2 A b e (3 d+4 e x)-2 a e (2 B d+A e+3 B e x)+b B \left (15 d^2+20 d e x+3 e^2 x^2\right )\right )}{3 e^3 (d+e x)^{3/2}}+\frac {\sqrt {b} (-5 b B d+2 A b e+3 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{e^{7/2}} \] Input:

Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(5/2),x]
 

Output:

(Sqrt[a + b*x]*(-2*A*b*e*(3*d + 4*e*x) - 2*a*e*(2*B*d + A*e + 3*B*e*x) + b 
*B*(15*d^2 + 20*d*e*x + 3*e^2*x^2)))/(3*e^3*(d + e*x)^(3/2)) + (Sqrt[b]*(- 
5*b*B*d + 2*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt 
[a + b*x])])/e^(7/2)
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {87, 57, 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.

\(\displaystyle \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{5/2}} \, dx\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {(-3 a B e-2 A b e+5 b B d) \int \frac {(a+b x)^{3/2}}{(d+e x)^{3/2}}dx}{3 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 57

\(\displaystyle \frac {(-3 a B e-2 A b e+5 b B d) \left (\frac {3 b \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}}dx}{e}-\frac {2 (a+b x)^{3/2}}{e \sqrt {d+e x}}\right )}{3 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(-3 a B e-2 A b e+5 b B d) \left (\frac {3 b \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}}dx}{2 e}\right )}{e}-\frac {2 (a+b x)^{3/2}}{e \sqrt {d+e x}}\right )}{3 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {(-3 a B e-2 A b e+5 b B d) \left (\frac {3 b \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \int \frac {1}{b-\frac {e (a+b x)}{d+e x}}d\frac {\sqrt {a+b x}}{\sqrt {d+e x}}}{e}\right )}{e}-\frac {2 (a+b x)^{3/2}}{e \sqrt {d+e x}}\right )}{3 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(-3 a B e-2 A b e+5 b B d) \left (\frac {3 b \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} e^{3/2}}\right )}{e}-\frac {2 (a+b x)^{3/2}}{e \sqrt {d+e x}}\right )}{3 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}\)

Input:

Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(5/2),x]
 

Output:

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(3*e*(b*d - a*e)*(d + e*x)^(3/2)) + ((5*b 
*B*d - 2*A*b*e - 3*a*B*e)*((-2*(a + b*x)^(3/2))/(e*Sqrt[d + e*x]) + (3*b*( 
(Sqrt[a + b*x]*Sqrt[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))))/e))/(3*e*(b*d - a*e))
 

Defintions of rubi rules used

rule 57

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]
 

rule 60

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]
 

rule 66

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]
 

rule 87

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]))))
 

rule 221

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(697\) vs. \(2(129)=258\).

Time = 0.27 (sec) , antiderivative size = 698, normalized size of antiderivative = 4.50

method	result	size
default	\(\frac {\sqrt {b x +a}\, \left (6 A \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) b^{2} e^{3} x^{2}+9 B \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) a b \,e^{3} x^{2}-15 B \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) b^{2} d \,e^{2} x^{2}+12 A \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) b^{2} d \,e^{2} x +18 B \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) a b d \,e^{2} x -30 B \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) b^{2} d^{2} e x +6 B b \,e^{2} x^{2} \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+6 A \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) b^{2} d^{2} e -16 A b \,e^{2} x \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+9 B \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) a b \,d^{2} e -15 B \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) b^{2} d^{3}-12 B a \,e^{2} x \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+40 B b d e x \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}-4 A a \,e^{2} \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}-12 A b d e \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}-8 B a d e \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+30 B b \,d^{2} \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\right )}{6 \sqrt {b e}\, \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, e^{3} \left (e x +d \right )^{\frac {3}{2}}}\)	\(698\)

Input:

int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/6*(b*x+a)^(1/2)*(6*A*ln(1/2*(2*b*e*x+2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/ 
2)+a*e+d*b)/(b*e)^(1/2))*b^2*e^3*x^2+9*B*ln(1/2*(2*b*e*x+2*((e*x+d)*(b*x+a 
))^(1/2)*(b*e)^(1/2)+a*e+d*b)/(b*e)^(1/2))*a*b*e^3*x^2-15*B*ln(1/2*(2*b*e* 
x+2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+a*e+d*b)/(b*e)^(1/2))*b^2*d*e^2*x^ 
2+12*A*ln(1/2*(2*b*e*x+2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+a*e+d*b)/(b*e 
)^(1/2))*b^2*d*e^2*x+18*B*ln(1/2*(2*b*e*x+2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^ 
(1/2)+a*e+d*b)/(b*e)^(1/2))*a*b*d*e^2*x-30*B*ln(1/2*(2*b*e*x+2*((e*x+d)*(b 
*x+a))^(1/2)*(b*e)^(1/2)+a*e+d*b)/(b*e)^(1/2))*b^2*d^2*e*x+6*B*b*e^2*x^2*( 
(e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+6*A*ln(1/2*(2*b*e*x+2*((e*x+d)*(b*x+a)) 
^(1/2)*(b*e)^(1/2)+a*e+d*b)/(b*e)^(1/2))*b^2*d^2*e-16*A*b*e^2*x*((e*x+d)*( 
b*x+a))^(1/2)*(b*e)^(1/2)+9*B*ln(1/2*(2*b*e*x+2*((e*x+d)*(b*x+a))^(1/2)*(b 
*e)^(1/2)+a*e+d*b)/(b*e)^(1/2))*a*b*d^2*e-15*B*ln(1/2*(2*b*e*x+2*((e*x+d)* 
(b*x+a))^(1/2)*(b*e)^(1/2)+a*e+d*b)/(b*e)^(1/2))*b^2*d^3-12*B*a*e^2*x*((e* 
x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+40*B*b*d*e*x*((e*x+d)*(b*x+a))^(1/2)*(b*e) 
^(1/2)-4*A*a*e^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-12*A*b*d*e*((e*x+d)*( 
b*x+a))^(1/2)*(b*e)^(1/2)-8*B*a*d*e*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+30 
*B*b*d^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2)/((e*x+d)*(b*x+a) 
)^(1/2)/e^3/(e*x+d)^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.92 (sec) , antiderivative size = 537, normalized size of antiderivative = 3.46 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{5/2}} \, dx=\left [-\frac {3 \, {\left (5 \, B b d^{3} - {\left (3 \, B a + 2 \, A b\right )} d^{2} e + {\left (5 \, B b d e^{2} - {\left (3 \, B a + 2 \, A b\right )} e^{3}\right )} x^{2} + 2 \, {\left (5 \, B b d^{2} e - {\left (3 \, B a + 2 \, A b\right )} d e^{2}\right )} x\right )} \sqrt {\frac {b}{e}} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b e^{2} x + b d e + a e^{2}\right )} \sqrt {b x + a} \sqrt {e x + d} \sqrt {\frac {b}{e}} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) - 4 \, {\left (3 \, B b e^{2} x^{2} + 15 \, B b d^{2} - 2 \, A a e^{2} - 2 \, {\left (2 \, B a + 3 \, A b\right )} d e + 2 \, {\left (10 \, B b d e - {\left (3 \, B a + 4 \, A b\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{12 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}}, \frac {3 \, {\left (5 \, B b d^{3} - {\left (3 \, B a + 2 \, A b\right )} d^{2} e + {\left (5 \, B b d e^{2} - {\left (3 \, B a + 2 \, A b\right )} e^{3}\right )} x^{2} + 2 \, {\left (5 \, B b d^{2} e - {\left (3 \, B a + 2 \, A b\right )} d e^{2}\right )} x\right )} \sqrt {-\frac {b}{e}} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {b x + a} \sqrt {e x + d} \sqrt {-\frac {b}{e}}}{2 \, {\left (b^{2} e x^{2} + a b d + {\left (b^{2} d + a b e\right )} x\right )}}\right ) + 2 \, {\left (3 \, B b e^{2} x^{2} + 15 \, B b d^{2} - 2 \, A a e^{2} - 2 \, {\left (2 \, B a + 3 \, A b\right )} d e + 2 \, {\left (10 \, B b d e - {\left (3 \, B a + 4 \, A b\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{6 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}}\right ] \] Input:

integrate((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(5/2),x, algorithm="fricas")
 

Output:

[-1/12*(3*(5*B*b*d^3 - (3*B*a + 2*A*b)*d^2*e + (5*B*b*d*e^2 - (3*B*a + 2*A 
*b)*e^3)*x^2 + 2*(5*B*b*d^2*e - (3*B*a + 2*A*b)*d*e^2)*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*(3* 
B*b*e^2*x^2 + 15*B*b*d^2 - 2*A*a*e^2 - 2*(2*B*a + 3*A*b)*d*e + 2*(10*B*b*d 
*e - (3*B*a + 4*A*b)*e^2)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(e^5*x^2 + 2*d*e 
^4*x + d^2*e^3), 1/6*(3*(5*B*b*d^3 - (3*B*a + 2*A*b)*d^2*e + (5*B*b*d*e^2 
- (3*B*a + 2*A*b)*e^3)*x^2 + 2*(5*B*b*d^2*e - (3*B*a + 2*A*b)*d*e^2)*x)*sq 
rt(-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*(3*B*b*e^2*x^2 + 15*B* 
b*d^2 - 2*A*a*e^2 - 2*(2*B*a + 3*A*b)*d*e + 2*(10*B*b*d*e - (3*B*a + 4*A*b 
)*e^2)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(e^5*x^2 + 2*d*e^4*x + d^2*e^3)]
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{5/2}} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((b*x+a)**(3/2)*(B*x+A)/(e*x+d)**(5/2),x)
 

Output:

Integral((A + B*x)*(a + b*x)**(3/2)/(d + e*x)**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(5/2),x, algorithm="maxima")
 

Output:

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
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (129) = 258\).

Time = 0.21 (sec) , antiderivative size = 369, normalized size of antiderivative = 2.38 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{5/2}} \, dx=\frac {{\left ({\left (b x + a\right )} {\left (\frac {3 \, {\left (B b^{5} d e^{4} {\left | b \right |} - B a b^{4} e^{5} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{4} d e^{5} - a b^{3} e^{6}} + \frac {4 \, {\left (5 \, B b^{6} d^{2} e^{3} {\left | b \right |} - 8 \, B a b^{5} d e^{4} {\left | b \right |} - 2 \, A b^{6} d e^{4} {\left | b \right |} + 3 \, B a^{2} b^{4} e^{5} {\left | b \right |} + 2 \, A a b^{5} e^{5} {\left | b \right |}\right )}}{b^{4} d e^{5} - a b^{3} e^{6}}\right )} + \frac {3 \, {\left (5 \, B b^{7} d^{3} e^{2} {\left | b \right |} - 13 \, B a b^{6} d^{2} e^{3} {\left | b \right |} - 2 \, A b^{7} d^{2} e^{3} {\left | b \right |} + 11 \, B a^{2} b^{5} d e^{4} {\left | b \right |} + 4 \, A a b^{6} d e^{4} {\left | b \right |} - 3 \, B a^{3} b^{4} e^{5} {\left | b \right |} - 2 \, A a^{2} b^{5} e^{5} {\left | b \right |}\right )}}{b^{4} d e^{5} - a b^{3} e^{6}}\right )} \sqrt {b x + a}}{3 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {3}{2}}} + \frac {{\left (5 \, B b d {\left | b \right |} - 3 \, B a e {\left | b \right |} - 2 \, A b e {\left | b \right |}\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b e} e^{3}} \] Input:

integrate((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(5/2),x, algorithm="giac")
 

Output:

1/3*((b*x + a)*(3*(B*b^5*d*e^4*abs(b) - B*a*b^4*e^5*abs(b))*(b*x + a)/(b^4 
*d*e^5 - a*b^3*e^6) + 4*(5*B*b^6*d^2*e^3*abs(b) - 8*B*a*b^5*d*e^4*abs(b) - 
 2*A*b^6*d*e^4*abs(b) + 3*B*a^2*b^4*e^5*abs(b) + 2*A*a*b^5*e^5*abs(b))/(b^ 
4*d*e^5 - a*b^3*e^6)) + 3*(5*B*b^7*d^3*e^2*abs(b) - 13*B*a*b^6*d^2*e^3*abs 
(b) - 2*A*b^7*d^2*e^3*abs(b) + 11*B*a^2*b^5*d*e^4*abs(b) + 4*A*a*b^6*d*e^4 
*abs(b) - 3*B*a^3*b^4*e^5*abs(b) - 2*A*a^2*b^5*e^5*abs(b))/(b^4*d*e^5 - a* 
b^3*e^6))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(3/2) + (5*B*b*d*a 
bs(b) - 3*B*a*e*abs(b) - 2*A*b*e*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^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^{5/2}} \,d x \] Input:

int(((A + B*x)*(a + b*x)^(3/2))/(d + e*x)^(5/2),x)
 

Output:

int(((A + B*x)*(a + b*x)^(3/2))/(d + e*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 504, normalized size of antiderivative = 3.25 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{5/2}} \, dx=\frac {-4 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{2} e^{3}-20 \sqrt {e x +d}\, \sqrt {b x +a}\, a b d \,e^{2}-28 \sqrt {e x +d}\, \sqrt {b x +a}\, a b \,e^{3} x +30 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{2} d^{2} e +40 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{2} d \,e^{2} x +6 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{2} e^{3} x^{2}+30 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) a b \,d^{2} e +60 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) a b d \,e^{2} x +30 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) a b \,e^{3} x^{2}-30 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) b^{2} d^{3}-60 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) b^{2} d^{2} e x -30 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) b^{2} d \,e^{2} x^{2}+5 \sqrt {e}\, \sqrt {b}\, a b \,d^{2} e +10 \sqrt {e}\, \sqrt {b}\, a b d \,e^{2} x +5 \sqrt {e}\, \sqrt {b}\, a b \,e^{3} x^{2}-5 \sqrt {e}\, \sqrt {b}\, b^{2} d^{3}-10 \sqrt {e}\, \sqrt {b}\, b^{2} d^{2} e x -5 \sqrt {e}\, \sqrt {b}\, b^{2} d \,e^{2} x^{2}}{6 e^{4} \left (e^{2} x^{2}+2 d e x +d^{2}\right )} \] Input:

int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(5/2),x)
 

Output:

( - 4*sqrt(d + e*x)*sqrt(a + b*x)*a**2*e**3 - 20*sqrt(d + e*x)*sqrt(a + b* 
x)*a*b*d*e**2 - 28*sqrt(d + e*x)*sqrt(a + b*x)*a*b*e**3*x + 30*sqrt(d + e* 
x)*sqrt(a + b*x)*b**2*d**2*e + 40*sqrt(d + e*x)*sqrt(a + b*x)*b**2*d*e**2* 
x + 6*sqrt(d + e*x)*sqrt(a + b*x)*b**2*e**3*x**2 + 30*sqrt(e)*sqrt(b)*log( 
(sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e - b*d))*a*b*d**2* 
e + 60*sqrt(e)*sqrt(b)*log((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x)) 
/sqrt(a*e - b*d))*a*b*d*e**2*x + 30*sqrt(e)*sqrt(b)*log((sqrt(e)*sqrt(a + 
b*x) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e - b*d))*a*b*e**3*x**2 - 30*sqrt(e)* 
sqrt(b)*log((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e - b*d 
))*b**2*d**3 - 60*sqrt(e)*sqrt(b)*log((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqr 
t(d + e*x))/sqrt(a*e - b*d))*b**2*d**2*e*x - 30*sqrt(e)*sqrt(b)*log((sqrt( 
e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e - b*d))*b**2*d*e**2*x** 
2 + 5*sqrt(e)*sqrt(b)*a*b*d**2*e + 10*sqrt(e)*sqrt(b)*a*b*d*e**2*x + 5*sqr 
t(e)*sqrt(b)*a*b*e**3*x**2 - 5*sqrt(e)*sqrt(b)*b**2*d**3 - 10*sqrt(e)*sqrt 
(b)*b**2*d**2*e*x - 5*sqrt(e)*sqrt(b)*b**2*d*e**2*x**2)/(6*e**4*(d**2 + 2* 
d*e*x + e**2*x**2))