\(\int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^3} \, dx\) [166]

Optimal result

Integrand size = 22, antiderivative size = 152 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^3} \, dx=\frac {2 B e \sqrt {d+e x}}{b^3}-\frac {(4 b B d+3 A b e-7 a B e) \sqrt {d+e x}}{4 b^3 (a+b x)}-\frac {(A b-a B) (d+e x)^{3/2}}{2 b^2 (a+b x)^2}-\frac {3 e (4 b B d+A b e-5 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{7/2} \sqrt {b d-a e}} \] Output:

2*B*e*(e*x+d)^(1/2)/b^3-1/4*(3*A*b*e-7*B*a*e+4*B*b*d)*(e*x+d)^(1/2)/b^3/(b 
*x+a)-1/2*(A*b-B*a)*(e*x+d)^(3/2)/b^2/(b*x+a)^2-3/4*e*(A*b*e-5*B*a*e+4*B*b 
*d)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(7/2)/(-a*e+b*d)^(1/ 
2)
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.91 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^3} \, dx=-\frac {\sqrt {d+e x} \left (A b (2 b d+3 a e+5 b e x)+B \left (-15 a^2 e+a b (2 d-25 e x)+4 b^2 x (d-2 e x)\right )\right )}{4 b^3 (a+b x)^2}+\frac {3 e (4 b B d+A b e-5 a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{4 b^{7/2} \sqrt {-b d+a e}} \] Input:

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

Output:

-1/4*(Sqrt[d + e*x]*(A*b*(2*b*d + 3*a*e + 5*b*e*x) + B*(-15*a^2*e + a*b*(2 
*d - 25*e*x) + 4*b^2*x*(d - 2*e*x))))/(b^3*(a + b*x)^2) + (3*e*(4*b*B*d + 
A*b*e - 5*a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(4*b^ 
(7/2)*Sqrt[-(b*d) + a*e])
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {87, 51, 60, 73, 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.

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

\(\Big \downarrow \) 87

\(\displaystyle \frac {(-5 a B e+A b e+4 b B d) \int \frac {(d+e x)^{3/2}}{(a+b x)^2}dx}{4 b (b d-a e)}-\frac {(d+e x)^{5/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {(-5 a B e+A b e+4 b B d) \left (\frac {3 e \int \frac {\sqrt {d+e x}}{a+b x}dx}{2 b}-\frac {(d+e x)^{3/2}}{b (a+b x)}\right )}{4 b (b d-a e)}-\frac {(d+e x)^{5/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(-5 a B e+A b e+4 b B d) \left (\frac {3 e \left (\frac {(b d-a e) \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b}+\frac {2 \sqrt {d+e x}}{b}\right )}{2 b}-\frac {(d+e x)^{3/2}}{b (a+b x)}\right )}{4 b (b d-a e)}-\frac {(d+e x)^{5/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {(-5 a B e+A b e+4 b B d) \left (\frac {3 e \left (\frac {2 (b d-a e) \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b e}+\frac {2 \sqrt {d+e x}}{b}\right )}{2 b}-\frac {(d+e x)^{3/2}}{b (a+b x)}\right )}{4 b (b d-a e)}-\frac {(d+e x)^{5/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(-5 a B e+A b e+4 b B d) \left (\frac {3 e \left (\frac {2 \sqrt {d+e x}}{b}-\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}\right )}{2 b}-\frac {(d+e x)^{3/2}}{b (a+b x)}\right )}{4 b (b d-a e)}-\frac {(d+e x)^{5/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)}\)

Input:

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

Output:

-1/2*((A*b - a*B)*(d + e*x)^(5/2))/(b*(b*d - a*e)*(a + b*x)^2) + ((4*b*B*d 
 + A*b*e - 5*a*B*e)*(-((d + e*x)^(3/2)/(b*(a + b*x))) + (3*e*((2*Sqrt[d + 
e*x])/b - (2*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a* 
e]])/b^(3/2)))/(2*b)))/(4*b*(b*d - a*e))
 

Defintions of rubi rules used

rule 51

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, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

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 [A] (verified)

Time = 0.50 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(-\frac {3 \left (-\left (b x +a \right )^{2} \left (\left (A e +4 B d \right ) b -5 B a e \right ) e \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -d b \right ) b}}\right )+\sqrt {e x +d}\, \left (\frac {\left (5 \left (-\frac {8 B x}{5}+A \right ) x e +2 d \left (2 B x +A \right )\right ) b^{2}}{3}+a \left (\left (-\frac {25 B x}{3}+A \right ) e +\frac {2 B d}{3}\right ) b -5 B \,a^{2} e \right ) \sqrt {\left (a e -d b \right ) b}\right )}{4 \sqrt {\left (a e -d b \right ) b}\, b^{3} \left (b x +a \right )^{2}}\)	\(143\)
derivativedivides	\(2 e \left (\frac {B \sqrt {e x +d}}{b^{3}}+\frac {\frac {\left (-\frac {5}{8} A \,b^{2} e +\frac {9}{8} B a b e -\frac {1}{2} b^{2} B d \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {3}{8} A a b \,e^{2}+\frac {3}{8} A \,b^{2} d e +\frac {7}{8} B \,a^{2} e^{2}-\frac {11}{8} B a b d e +\frac {1}{2} b^{2} B \,d^{2}\right ) \sqrt {e x +d}}{\left (\left (e x +d \right ) b +a e -d b \right )^{2}}+\frac {3 \left (A b e -5 B a e +4 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -d b \right ) b}}\right )}{8 \sqrt {\left (a e -d b \right ) b}}}{b^{3}}\right )\)	\(171\)
default	\(2 e \left (\frac {B \sqrt {e x +d}}{b^{3}}+\frac {\frac {\left (-\frac {5}{8} A \,b^{2} e +\frac {9}{8} B a b e -\frac {1}{2} b^{2} B d \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {3}{8} A a b \,e^{2}+\frac {3}{8} A \,b^{2} d e +\frac {7}{8} B \,a^{2} e^{2}-\frac {11}{8} B a b d e +\frac {1}{2} b^{2} B \,d^{2}\right ) \sqrt {e x +d}}{\left (\left (e x +d \right ) b +a e -d b \right )^{2}}+\frac {3 \left (A b e -5 B a e +4 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -d b \right ) b}}\right )}{8 \sqrt {\left (a e -d b \right ) b}}}{b^{3}}\right )\)	\(171\)
risch	\(\frac {2 B e \sqrt {e x +d}}{b^{3}}+\frac {2 e \left (\frac {\left (-\frac {5}{8} A \,b^{2} e +\frac {9}{8} B a b e -\frac {1}{2} b^{2} B d \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {3}{8} A a b \,e^{2}+\frac {3}{8} A \,b^{2} d e +\frac {7}{8} B \,a^{2} e^{2}-\frac {11}{8} B a b d e +\frac {1}{2} b^{2} B \,d^{2}\right ) \sqrt {e x +d}}{\left (\left (e x +d \right ) b +a e -d b \right )^{2}}+\frac {3 \left (A b e -5 B a e +4 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -d b \right ) b}}\right )}{8 \sqrt {\left (a e -d b \right ) b}}\right )}{b^{3}}\)	\(172\)

Input:

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

Output:

-3/4*(-(b*x+a)^2*((A*e+4*B*d)*b-5*B*a*e)*e*arctan(b*(e*x+d)^(1/2)/((a*e-b* 
d)*b)^(1/2))+(e*x+d)^(1/2)*(1/3*(5*(-8/5*B*x+A)*x*e+2*d*(2*B*x+A))*b^2+a*( 
(-25/3*B*x+A)*e+2/3*B*d)*b-5*B*a^2*e)*((a*e-b*d)*b)^(1/2))/((a*e-b*d)*b)^( 
1/2)/b^3/(b*x+a)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (130) = 260\).

Time = 0.13 (sec) , antiderivative size = 703, normalized size of antiderivative = 4.62 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^3} \, dx=\left [\frac {3 \, {\left (4 \, B a^{2} b d e - {\left (5 \, B a^{3} - A a^{2} b\right )} e^{2} + {\left (4 \, B b^{3} d e - {\left (5 \, B a b^{2} - A b^{3}\right )} e^{2}\right )} x^{2} + 2 \, {\left (4 \, B a b^{2} d e - {\left (5 \, B a^{2} b - A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (2 \, {\left (B a b^{3} + A b^{4}\right )} d^{2} - {\left (17 \, B a^{2} b^{2} - A a b^{3}\right )} d e + 3 \, {\left (5 \, B a^{3} b - A a^{2} b^{2}\right )} e^{2} - 8 \, {\left (B b^{4} d e - B a b^{3} e^{2}\right )} x^{2} + {\left (4 \, B b^{4} d^{2} - {\left (29 \, B a b^{3} - 5 \, A b^{4}\right )} d e + 5 \, {\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{8 \, {\left (a^{2} b^{5} d - a^{3} b^{4} e + {\left (b^{7} d - a b^{6} e\right )} x^{2} + 2 \, {\left (a b^{6} d - a^{2} b^{5} e\right )} x\right )}}, \frac {3 \, {\left (4 \, B a^{2} b d e - {\left (5 \, B a^{3} - A a^{2} b\right )} e^{2} + {\left (4 \, B b^{3} d e - {\left (5 \, B a b^{2} - A b^{3}\right )} e^{2}\right )} x^{2} + 2 \, {\left (4 \, B a b^{2} d e - {\left (5 \, B a^{2} b - A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) - {\left (2 \, {\left (B a b^{3} + A b^{4}\right )} d^{2} - {\left (17 \, B a^{2} b^{2} - A a b^{3}\right )} d e + 3 \, {\left (5 \, B a^{3} b - A a^{2} b^{2}\right )} e^{2} - 8 \, {\left (B b^{4} d e - B a b^{3} e^{2}\right )} x^{2} + {\left (4 \, B b^{4} d^{2} - {\left (29 \, B a b^{3} - 5 \, A b^{4}\right )} d e + 5 \, {\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{4 \, {\left (a^{2} b^{5} d - a^{3} b^{4} e + {\left (b^{7} d - a b^{6} e\right )} x^{2} + 2 \, {\left (a b^{6} d - a^{2} b^{5} e\right )} x\right )}}\right ] \] Input:

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

Output:

[1/8*(3*(4*B*a^2*b*d*e - (5*B*a^3 - A*a^2*b)*e^2 + (4*B*b^3*d*e - (5*B*a*b 
^2 - A*b^3)*e^2)*x^2 + 2*(4*B*a*b^2*d*e - (5*B*a^2*b - A*a*b^2)*e^2)*x)*sq 
rt(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e* 
x + d))/(b*x + a)) - 2*(2*(B*a*b^3 + A*b^4)*d^2 - (17*B*a^2*b^2 - A*a*b^3) 
*d*e + 3*(5*B*a^3*b - A*a^2*b^2)*e^2 - 8*(B*b^4*d*e - B*a*b^3*e^2)*x^2 + ( 
4*B*b^4*d^2 - (29*B*a*b^3 - 5*A*b^4)*d*e + 5*(5*B*a^2*b^2 - A*a*b^3)*e^2)* 
x)*sqrt(e*x + d))/(a^2*b^5*d - a^3*b^4*e + (b^7*d - a*b^6*e)*x^2 + 2*(a*b^ 
6*d - a^2*b^5*e)*x), 1/4*(3*(4*B*a^2*b*d*e - (5*B*a^3 - A*a^2*b)*e^2 + (4* 
B*b^3*d*e - (5*B*a*b^2 - A*b^3)*e^2)*x^2 + 2*(4*B*a*b^2*d*e - (5*B*a^2*b - 
 A*a*b^2)*e^2)*x)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e* 
x + d)/(b*e*x + b*d)) - (2*(B*a*b^3 + A*b^4)*d^2 - (17*B*a^2*b^2 - A*a*b^3 
)*d*e + 3*(5*B*a^3*b - A*a^2*b^2)*e^2 - 8*(B*b^4*d*e - B*a*b^3*e^2)*x^2 + 
(4*B*b^4*d^2 - (29*B*a*b^3 - 5*A*b^4)*d*e + 5*(5*B*a^2*b^2 - A*a*b^3)*e^2) 
*x)*sqrt(e*x + d))/(a^2*b^5*d - a^3*b^4*e + (b^7*d - a*b^6*e)*x^2 + 2*(a*b 
^6*d - a^2*b^5*e)*x)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((B*x+A)*(e*x+d)^(3/2)/(b*x+a)^3,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(a*e-b*d>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.49 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^3} \, dx=\frac {2 \, \sqrt {e x + d} B e}{b^{3}} + \frac {3 \, {\left (4 \, B b d e - 5 \, B a e^{2} + A b e^{2}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, \sqrt {-b^{2} d + a b e} b^{3}} - \frac {4 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{2} d e - 4 \, \sqrt {e x + d} B b^{2} d^{2} e - 9 \, {\left (e x + d\right )}^{\frac {3}{2}} B a b e^{2} + 5 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{2} e^{2} + 11 \, \sqrt {e x + d} B a b d e^{2} - 3 \, \sqrt {e x + d} A b^{2} d e^{2} - 7 \, \sqrt {e x + d} B a^{2} e^{3} + 3 \, \sqrt {e x + d} A a b e^{3}}{4 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{2} b^{3}} \] Input:

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

Output:

2*sqrt(e*x + d)*B*e/b^3 + 3/4*(4*B*b*d*e - 5*B*a*e^2 + A*b*e^2)*arctan(sqr 
t(e*x + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^3) - 1/4*(4*(e* 
x + d)^(3/2)*B*b^2*d*e - 4*sqrt(e*x + d)*B*b^2*d^2*e - 9*(e*x + d)^(3/2)*B 
*a*b*e^2 + 5*(e*x + d)^(3/2)*A*b^2*e^2 + 11*sqrt(e*x + d)*B*a*b*d*e^2 - 3* 
sqrt(e*x + d)*A*b^2*d*e^2 - 7*sqrt(e*x + d)*B*a^2*e^3 + 3*sqrt(e*x + d)*A* 
a*b*e^3)/(((e*x + d)*b - b*d + a*e)^2*b^3)
 

Mupad [B] (verification not implemented)

Time = 1.12 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.68 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^3} \, dx=\frac {\sqrt {d+e\,x}\,\left (\frac {7\,B\,a^2\,e^3}{4}-\frac {11\,B\,a\,b\,d\,e^2}{4}-\frac {3\,A\,a\,b\,e^3}{4}+B\,b^2\,d^2\,e+\frac {3\,A\,b^2\,d\,e^2}{4}\right )-{\left (d+e\,x\right )}^{3/2}\,\left (\frac {5\,A\,b^2\,e^2}{4}+B\,d\,b^2\,e-\frac {9\,B\,a\,b\,e^2}{4}\right )}{b^5\,{\left (d+e\,x\right )}^2-\left (2\,b^5\,d-2\,a\,b^4\,e\right )\,\left (d+e\,x\right )+b^5\,d^2+a^2\,b^3\,e^2-2\,a\,b^4\,d\,e}+\frac {2\,B\,e\,\sqrt {d+e\,x}}{b^3}+\frac {3\,e\,\mathrm {atan}\left (\frac {\sqrt {b}\,e\,\sqrt {d+e\,x}\,\left (A\,b\,e-5\,B\,a\,e+4\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (A\,b\,e^2-5\,B\,a\,e^2+4\,B\,b\,d\,e\right )}\right )\,\left (A\,b\,e-5\,B\,a\,e+4\,B\,b\,d\right )}{4\,b^{7/2}\,\sqrt {a\,e-b\,d}} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.84 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^3} \, dx=\frac {-3 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a e -3 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) b e x +3 \sqrt {e x +d}\, a b e -\sqrt {e x +d}\, b^{2} d +2 \sqrt {e x +d}\, b^{2} e x}{b^{3} \left (b x +a \right )} \] Input:

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

Output:

( - 3*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b 
*d)))*a*e - 3*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt 
(a*e - b*d)))*b*e*x + 3*sqrt(d + e*x)*a*b*e - sqrt(d + e*x)*b**2*d + 2*sqr 
t(d + e*x)*b**2*e*x)/(b**3*(a + b*x))