\(\int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{7/2}} \, dx\) [188]

Optimal result

Integrand size = 24, antiderivative size = 138 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{7/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{5/2}}{5 e (b d-a e) (d+e x)^{5/2}}-\frac {2 B (a+b x)^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac {2 b B \sqrt {a+b x}}{e^3 \sqrt {d+e x}}+\frac {2 b^{3/2} B \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{7/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.24 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{7/2}} \, dx=-\frac {2 (a+b x)^{5/2} \left (-3 B d e^2+3 A e^3-\frac {5 b B d e (d+e x)}{a+b x}+\frac {5 a B e^2 (d+e x)}{a+b x}-\frac {15 b^2 B d (d+e x)^2}{(a+b x)^2}+\frac {15 a b B e (d+e x)^2}{(a+b x)^2}\right )}{15 e^3 (-b d+a e) (d+e x)^{5/2}}+\frac {2 b^{3/2} B \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{e^{7/2}} \] Input:

Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(7/2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {87, 57, 57, 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.

\(\displaystyle \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{7/2}} \, dx\)

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(7/2),x]
 

Output:

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(5*e*(b*d - a*e)*(d + e*x)^(5/2)) + (B*(( 
-2*(a + b*x)^(3/2))/(3*e*(d + e*x)^(3/2)) + (b*((-2*Sqrt[a + b*x])/(e*Sqrt 
[d + e*x]) + (2*Sqrt[b]*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + 
e*x])])/e^(3/2)))/e))/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 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. \(779\) vs. \(2(110)=220\).

Time = 0.27 (sec) , antiderivative size = 780, normalized size of antiderivative = 5.65

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \left (-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 ) a \,b^{2} e^{4} x^{3}+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^{3} d \,e^{3} x^{3}-45 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^{2} d \,e^{3} x^{2}+45 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^{3} d^{2} e^{2} x^{2}+6 A \,b^{2} e^{3} x^{2} \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}-45 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^{2} d^{2} e^{2} x +45 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^{3} d^{3} e x +40 B a b \,e^{3} x^{2} \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}-46 B \,b^{2} d \,e^{2} x^{2} \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+12 A a b \,e^{3} x \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b 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 ) a \,b^{2} d^{3} 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^{3} d^{4}+10 B \,a^{2} e^{3} x \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+48 B a b d \,e^{2} x \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}-70 B \,b^{2} d^{2} e x \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+6 A \,a^{2} e^{3} \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+4 B \,a^{2} d \,e^{2} \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+20 B a b \,d^{2} e \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}-30 B \,b^{2} d^{3} \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\right )}{15 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \left (a e -d b \right ) \sqrt {b e}\, \left (e x +d \right )^{\frac {5}{2}} e^{3}}\)	\(780\)

Input:

int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-1/15*(b*x+a)^(1/2)*(-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))*a*b^2*e^4*x^3+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^3*d*e^3*x^3-45*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 
^2*d*e^3*x^2+45*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^3*d^2*e^2*x^2+6*A*b^2*e^3*x^2*((e*x+d)*(b*x+a))^(1/2 
)*(b*e)^(1/2)-45*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^2*d^2*e^2*x+45*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^3*d^3*e*x+40*B*a*b*e^3*x^2*( 
(e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-46*B*b^2*d*e^2*x^2*((e*x+d)*(b*x+a))^(1 
/2)*(b*e)^(1/2)+12*A*a*b*e^3*x*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-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))* 
a*b^2*d^3*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^3*d^4+10*B*a^2*e^3*x*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1 
/2)+48*B*a*b*d*e^2*x*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-70*B*b^2*d^2*e*x* 
((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+6*A*a^2*e^3*((e*x+d)*(b*x+a))^(1/2)*(b 
*e)^(1/2)+4*B*a^2*d*e^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+20*B*a*b*d^2*e 
*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-30*B*b^2*d^3*((e*x+d)*(b*x+a))^(1/2)* 
(b*e)^(1/2))/((e*x+d)*(b*x+a))^(1/2)/(a*e-b*d)/(b*e)^(1/2)/(e*x+d)^(5/2)/e 
^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 371 vs. \(2 (110) = 220\).

Time = 2.54 (sec) , antiderivative size = 767, normalized size of antiderivative = 5.56 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{7/2}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(7/2),x, algorithm="fricas")
 

Output:

[1/30*(15*(B*b^2*d^4 - B*a*b*d^3*e + (B*b^2*d*e^3 - B*a*b*e^4)*x^3 + 3*(B* 
b^2*d^2*e^2 - B*a*b*d*e^3)*x^2 + 3*(B*b^2*d^3*e - B*a*b*d^2*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*(15*B*b^2*d^3 - 10*B*a*b*d^2*e - 2*B*a^2*d*e^2 - 3*A*a^2*e^3 + (23* 
B*b^2*d*e^2 - (20*B*a*b + 3*A*b^2)*e^3)*x^2 + (35*B*b^2*d^2*e - 24*B*a*b*d 
*e^2 - (5*B*a^2 + 6*A*a*b)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b*d^4*e^3 
 - a*d^3*e^4 + (b*d*e^6 - a*e^7)*x^3 + 3*(b*d^2*e^5 - a*d*e^6)*x^2 + 3*(b* 
d^3*e^4 - a*d^2*e^5)*x), -1/15*(15*(B*b^2*d^4 - B*a*b*d^3*e + (B*b^2*d*e^3 
 - B*a*b*e^4)*x^3 + 3*(B*b^2*d^2*e^2 - B*a*b*d*e^3)*x^2 + 3*(B*b^2*d^3*e - 
 B*a*b*d^2*e^2)*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*( 
15*B*b^2*d^3 - 10*B*a*b*d^2*e - 2*B*a^2*d*e^2 - 3*A*a^2*e^3 + (23*B*b^2*d* 
e^2 - (20*B*a*b + 3*A*b^2)*e^3)*x^2 + (35*B*b^2*d^2*e - 24*B*a*b*d*e^2 - ( 
5*B*a^2 + 6*A*a*b)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b*d^4*e^3 - a*d^3 
*e^4 + (b*d*e^6 - a*e^7)*x^3 + 3*(b*d^2*e^5 - a*d*e^6)*x^2 + 3*(b*d^3*e^4 
- a*d^2*e^5)*x)]
 

Sympy [F]

\[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{7/2}} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((b*x+a)**(3/2)*(B*x+A)/(e*x+d)**(7/2),x)
 

Output:

Integral((A + B*x)*(a + b*x)**(3/2)/(d + e*x)**(7/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{7/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(7/2),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. 398 vs. \(2 (110) = 220\).

Time = 0.26 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.88 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{7/2}} \, dx=-\frac {2 \, B b {\left | b \right |} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b e} e^{3}} - \frac {2 \, {\left ({\left (b x + a\right )} {\left (\frac {{\left (23 \, B b^{7} d^{2} e^{4} {\left | b \right |} - 43 \, B a b^{6} d e^{5} {\left | b \right |} - 3 \, A b^{7} d e^{5} {\left | b \right |} + 20 \, B a^{2} b^{5} e^{6} {\left | b \right |} + 3 \, A a b^{6} e^{6} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{4} d^{2} e^{5} - 2 \, a b^{3} d e^{6} + a^{2} b^{2} e^{7}} + \frac {35 \, {\left (B b^{8} d^{3} e^{3} {\left | b \right |} - 3 \, B a b^{7} d^{2} e^{4} {\left | b \right |} + 3 \, B a^{2} b^{6} d e^{5} {\left | b \right |} - B a^{3} b^{5} e^{6} {\left | b \right |}\right )}}{b^{4} d^{2} e^{5} - 2 \, a b^{3} d e^{6} + a^{2} b^{2} e^{7}}\right )} + \frac {15 \, {\left (B b^{9} d^{4} e^{2} {\left | b \right |} - 4 \, B a b^{8} d^{3} e^{3} {\left | b \right |} + 6 \, B a^{2} b^{7} d^{2} e^{4} {\left | b \right |} - 4 \, B a^{3} b^{6} d e^{5} {\left | b \right |} + B a^{4} b^{5} e^{6} {\left | b \right |}\right )}}{b^{4} d^{2} e^{5} - 2 \, a b^{3} d e^{6} + a^{2} b^{2} e^{7}}\right )} \sqrt {b x + a}}{15 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {5}{2}}} \] Input:

integrate((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(7/2),x, algorithm="giac")
 

Output:

-2*B*b*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) - 2/15*((b*x + a)*((23*B*b^7*d^2*e^4*abs(b) - 
 43*B*a*b^6*d*e^5*abs(b) - 3*A*b^7*d*e^5*abs(b) + 20*B*a^2*b^5*e^6*abs(b) 
+ 3*A*a*b^6*e^6*abs(b))*(b*x + a)/(b^4*d^2*e^5 - 2*a*b^3*d*e^6 + a^2*b^2*e 
^7) + 35*(B*b^8*d^3*e^3*abs(b) - 3*B*a*b^7*d^2*e^4*abs(b) + 3*B*a^2*b^6*d* 
e^5*abs(b) - B*a^3*b^5*e^6*abs(b))/(b^4*d^2*e^5 - 2*a*b^3*d*e^6 + a^2*b^2* 
e^7)) + 15*(B*b^9*d^4*e^2*abs(b) - 4*B*a*b^8*d^3*e^3*abs(b) + 6*B*a^2*b^7* 
d^2*e^4*abs(b) - 4*B*a^3*b^6*d*e^5*abs(b) + B*a^4*b^5*e^6*abs(b))/(b^4*d^2 
*e^5 - 2*a*b^3*d*e^6 + a^2*b^2*e^7))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e 
- a*b*e)^(5/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{7/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \] Input:

int(((A + B*x)*(a + b*x)^(3/2))/(d + e*x)^(7/2),x)
 

Output:

int(((A + B*x)*(a + b*x)^(3/2))/(d + e*x)^(7/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.71 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.92 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{7/2}} \, dx=\frac {-\frac {2 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{2} e^{3}}{5}-\frac {2 \sqrt {e x +d}\, \sqrt {b x +a}\, a b d \,e^{2}}{3}-\frac {22 \sqrt {e x +d}\, \sqrt {b x +a}\, a b \,e^{3} x}{15}-2 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{2} d^{2} e -\frac {14 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{2} d \,e^{2} x}{3}-\frac {46 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{2} e^{3} x^{2}}{15}+2 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) b^{2} d^{3}+6 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) b^{2} d^{2} e x +6 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) b^{2} d \,e^{2} x^{2}+2 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) b^{2} e^{3} x^{3}+\frac {2 \sqrt {e}\, \sqrt {b}\, b^{2} d^{3}}{3}+2 \sqrt {e}\, \sqrt {b}\, b^{2} d^{2} e x +2 \sqrt {e}\, \sqrt {b}\, b^{2} d \,e^{2} x^{2}+\frac {2 \sqrt {e}\, \sqrt {b}\, b^{2} e^{3} x^{3}}{3}}{e^{4} \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right )} \] Input:

int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(7/2),x)
 

Output:

(2*( - 3*sqrt(d + e*x)*sqrt(a + b*x)*a**2*e**3 - 5*sqrt(d + e*x)*sqrt(a + 
b*x)*a*b*d*e**2 - 11*sqrt(d + e*x)*sqrt(a + b*x)*a*b*e**3*x - 15*sqrt(d + 
e*x)*sqrt(a + b*x)*b**2*d**2*e - 35*sqrt(d + e*x)*sqrt(a + b*x)*b**2*d*e** 
2*x - 23*sqrt(d + e*x)*sqrt(a + b*x)*b**2*e**3*x**2 + 15*sqrt(e)*sqrt(b)*l 
og((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e - b*d))*b**2*d 
**3 + 45*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**2*e*x + 45*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 + 15*sq 
rt(e)*sqrt(b)*log((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e 
 - b*d))*b**2*e**3*x**3 + 5*sqrt(e)*sqrt(b)*b**2*d**3 + 15*sqrt(e)*sqrt(b) 
*b**2*d**2*e*x + 15*sqrt(e)*sqrt(b)*b**2*d*e**2*x**2 + 5*sqrt(e)*sqrt(b)*b 
**2*e**3*x**3))/(15*e**4*(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3))