\(\int \frac {x (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx\) [293]

Optimal result

Integrand size = 20, antiderivative size = 217 \[ \int \frac {x (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=-\frac {5 (b c-a d)^2 (7 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{64 b d^4}+\frac {5 (b c-a d) (7 b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{96 b d^3}-\frac {(7 b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 b d^2}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b d}+\frac {5 (b c-a d)^3 (7 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{9/2}} \] Output:

-5/64*(-a*d+b*c)^2*(a*d+7*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b/d^4+5/96*(-a* 
d+b*c)*(a*d+7*b*c)*(b*x+a)^(3/2)*(d*x+c)^(1/2)/b/d^3-1/24*(a*d+7*b*c)*(b*x 
+a)^(5/2)*(d*x+c)^(1/2)/b/d^2+1/4*(b*x+a)^(7/2)*(d*x+c)^(1/2)/b/d+5/64*(-a 
*d+b*c)^3*(a*d+7*b*c)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2)) 
/b^(3/2)/d^(9/2)
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.81 \[ \int \frac {x (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 a^3 d^3+a^2 b d^2 (-191 c+118 d x)+a b^2 d \left (265 c^2-172 c d x+136 d^2 x^2\right )+b^3 \left (-105 c^3+70 c^2 d x-56 c d^2 x^2+48 d^3 x^3\right )\right )}{192 b d^4}+\frac {5 (b c-a d)^3 (7 b c+a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{64 b^{3/2} d^{9/2}} \] Input:

Integrate[(x*(a + b*x)^(5/2))/Sqrt[c + d*x],x]
 

Output:

(Sqrt[a + b*x]*Sqrt[c + d*x]*(15*a^3*d^3 + a^2*b*d^2*(-191*c + 118*d*x) + 
a*b^2*d*(265*c^2 - 172*c*d*x + 136*d^2*x^2) + b^3*(-105*c^3 + 70*c^2*d*x - 
 56*c*d^2*x^2 + 48*d^3*x^3)))/(192*b*d^4) + (5*(b*c - a*d)^3*(7*b*c + a*d) 
*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(64*b^(3/2)*d^( 
9/2))
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {90, 60, 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.

\(\displaystyle \int \frac {x (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b d}-\frac {(a d+7 b c) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}}dx}{8 b d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b d}-\frac {(a d+7 b c) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}}dx}{6 d}\right )}{8 b d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b d}-\frac {(a d+7 b c) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{4 d}\right )}{6 d}\right )}{8 b d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b d}-\frac {(a d+7 b c) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )}{6 d}\right )}{8 b d}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b d}-\frac {(a d+7 b c) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )}{6 d}\right )}{8 b d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b d}-\frac {(a d+7 b c) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 d}\right )}{6 d}\right )}{8 b d}\)

Input:

Int[(x*(a + b*x)^(5/2))/Sqrt[c + d*x],x]
 

Output:

((a + b*x)^(7/2)*Sqrt[c + d*x])/(4*b*d) - ((7*b*c + a*d)*(((a + b*x)^(5/2) 
*Sqrt[c + d*x])/(3*d) - (5*(b*c - a*d)*(((a + b*x)^(3/2)*Sqrt[c + d*x])/(2 
*d) - (3*(b*c - a*d)*((Sqrt[a + b*x]*Sqrt[c + d*x])/d - ((b*c - a*d)*ArcTa 
nh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*d^(3/2))))/( 
4*d)))/(6*d)))/(8*b*d)
 

Defintions of rubi rules used

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 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

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. \(573\) vs. \(2(179)=358\).

Time = 0.22 (sec) , antiderivative size = 574, normalized size of antiderivative = 2.65

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \sqrt {x d +c}\, \left (-96 b^{3} d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-272 a \,b^{2} d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+112 b^{3} c \,d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{4} d^{4}+60 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{3} b c \,d^{3}-270 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{2} b^{2} c^{2} d^{2}+300 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a \,b^{3} c^{3} d -105 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) b^{4} c^{4}-236 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{2} b \,d^{3} x +344 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a \,b^{2} c \,d^{2} x -140 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, b^{3} c^{2} d x -30 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{3} d^{3}+382 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{2} b c \,d^{2}-530 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a \,b^{2} c^{2} d +210 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, b^{3} c^{3}\right )}{384 b \,d^{4} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}}\)	\(574\)

Input:

int(x*(b*x+a)^(5/2)/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-96*b^3*d^3*x^3*((b*x+a)*(d*x+c))^(1/2 
)*(d*b)^(1/2)-272*a*b^2*d^3*x^2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+112*b^ 
3*c*d^2*x^2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+15*ln(1/2*(2*b*d*x+2*((b*x 
+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^4*d^4+60*ln(1/2*(2* 
b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^3*b*c* 
d^3-270*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d* 
b)^(1/2))*a^2*b^2*c^2*d^2+300*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d 
*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a*b^3*c^3*d-105*ln(1/2*(2*b*d*x+2*((b*x+a) 
*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*b^4*c^4-236*((b*x+a)*(d* 
x+c))^(1/2)*(d*b)^(1/2)*a^2*b*d^3*x+344*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2 
)*a*b^2*c*d^2*x-140*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*b^3*c^2*d*x-30*((b 
*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^3*d^3+382*((b*x+a)*(d*x+c))^(1/2)*(d*b) 
^(1/2)*a^2*b*c*d^2-530*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a*b^2*c^2*d+210 
*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*b^3*c^3)/b/d^4/((b*x+a)*(d*x+c))^(1/2 
)/(d*b)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 546, normalized size of antiderivative = 2.52 \[ \int \frac {x (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\left [-\frac {15 \, {\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (48 \, b^{4} d^{4} x^{3} - 105 \, b^{4} c^{3} d + 265 \, a b^{3} c^{2} d^{2} - 191 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - 17 \, a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 86 \, a b^{3} c d^{3} + 59 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, b^{2} d^{5}}, -\frac {15 \, {\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (48 \, b^{4} d^{4} x^{3} - 105 \, b^{4} c^{3} d + 265 \, a b^{3} c^{2} d^{2} - 191 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - 17 \, a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 86 \, a b^{3} c d^{3} + 59 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, b^{2} d^{5}}\right ] \] Input:

integrate(x*(b*x+a)^(5/2)/(d*x+c)^(1/2),x, algorithm="fricas")
 

Output:

[-1/768*(15*(7*b^4*c^4 - 20*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d 
^3 - a^4*d^4)*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 
- 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c 
*d + a*b*d^2)*x) - 4*(48*b^4*d^4*x^3 - 105*b^4*c^3*d + 265*a*b^3*c^2*d^2 - 
 191*a^2*b^2*c*d^3 + 15*a^3*b*d^4 - 8*(7*b^4*c*d^3 - 17*a*b^3*d^4)*x^2 + 2 
*(35*b^4*c^2*d^2 - 86*a*b^3*c*d^3 + 59*a^2*b^2*d^4)*x)*sqrt(b*x + a)*sqrt( 
d*x + c))/(b^2*d^5), -1/384*(15*(7*b^4*c^4 - 20*a*b^3*c^3*d + 18*a^2*b^2*c 
^2*d^2 - 4*a^3*b*c*d^3 - a^4*d^4)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a 
*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c 
*d + a*b*d^2)*x)) - 2*(48*b^4*d^4*x^3 - 105*b^4*c^3*d + 265*a*b^3*c^2*d^2 
- 191*a^2*b^2*c*d^3 + 15*a^3*b*d^4 - 8*(7*b^4*c*d^3 - 17*a*b^3*d^4)*x^2 + 
2*(35*b^4*c^2*d^2 - 86*a*b^3*c*d^3 + 59*a^2*b^2*d^4)*x)*sqrt(b*x + a)*sqrt 
(d*x + c))/(b^2*d^5)]
 

Sympy [F]

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

integrate(x*(b*x+a)**(5/2)/(d*x+c)**(1/2),x)
 

Output:

Integral(x*(a + b*x)**(5/2)/sqrt(c + d*x), x)
 

Maxima [F(-2)]

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

integrate(x*(b*x+a)^(5/2)/(d*x+c)^(1/2),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(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.34 \[ \int \frac {x (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )}}{b^{2} d} - \frac {7 \, b^{3} c d^{5} + a b^{2} d^{6}}{b^{4} d^{7}}\right )} + \frac {5 \, {\left (7 \, b^{4} c^{2} d^{4} - 6 \, a b^{3} c d^{5} - a^{2} b^{2} d^{6}\right )}}{b^{4} d^{7}}\right )} - \frac {15 \, {\left (7 \, b^{5} c^{3} d^{3} - 13 \, a b^{4} c^{2} d^{4} + 5 \, a^{2} b^{3} c d^{5} + a^{3} b^{2} d^{6}\right )}}{b^{4} d^{7}}\right )} \sqrt {b x + a} - \frac {15 \, {\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b d^{4}}\right )} b}{192 \, {\left | b \right |}} \] Input:

integrate(x*(b*x+a)^(5/2)/(d*x+c)^(1/2),x, algorithm="giac")
 

Output:

1/192*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b 
*x + a)/(b^2*d) - (7*b^3*c*d^5 + a*b^2*d^6)/(b^4*d^7)) + 5*(7*b^4*c^2*d^4 
- 6*a*b^3*c*d^5 - a^2*b^2*d^6)/(b^4*d^7)) - 15*(7*b^5*c^3*d^3 - 13*a*b^4*c 
^2*d^4 + 5*a^2*b^3*c*d^5 + a^3*b^2*d^6)/(b^4*d^7))*sqrt(b*x + a) - 15*(7*b 
^4*c^4 - 20*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - a^4*d^4)*lo 
g(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sq 
rt(b*d)*b*d^4))*b/abs(b)
 

Mupad [F(-1)]

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

int((x*(a + b*x)^(5/2))/(c + d*x)^(1/2),x)
 

Output:

int((x*(a + b*x)^(5/2))/(c + d*x)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 471, normalized size of antiderivative = 2.17 \[ \int \frac {x (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\frac {15 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{3} b \,d^{4}-191 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{2} c \,d^{3}+118 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{2} d^{4} x +265 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{3} c^{2} d^{2}-172 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{3} c \,d^{3} x +136 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{3} d^{4} x^{2}-105 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{4} c^{3} d +70 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{4} c^{2} d^{2} x -56 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{4} c \,d^{3} x^{2}+48 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{4} d^{4} x^{3}-15 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{4} d^{4}-60 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{3} b c \,d^{3}+270 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} b^{2} c^{2} d^{2}-300 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a \,b^{3} c^{3} d +105 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{4} c^{4}}{192 b^{2} d^{5}} \] Input:

int(x*(b*x+a)^(5/2)/(d*x+c)^(1/2),x)
 

Output:

(15*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b*d**4 - 191*sqrt(c + d*x)*sqrt(a + b 
*x)*a**2*b**2*c*d**3 + 118*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**2*d**4*x + 
265*sqrt(c + d*x)*sqrt(a + b*x)*a*b**3*c**2*d**2 - 172*sqrt(c + d*x)*sqrt( 
a + b*x)*a*b**3*c*d**3*x + 136*sqrt(c + d*x)*sqrt(a + b*x)*a*b**3*d**4*x** 
2 - 105*sqrt(c + d*x)*sqrt(a + b*x)*b**4*c**3*d + 70*sqrt(c + d*x)*sqrt(a 
+ b*x)*b**4*c**2*d**2*x - 56*sqrt(c + d*x)*sqrt(a + b*x)*b**4*c*d**3*x**2 
+ 48*sqrt(c + d*x)*sqrt(a + b*x)*b**4*d**4*x**3 - 15*sqrt(d)*sqrt(b)*log(( 
sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**4*d**4 
- 60*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/s 
qrt(a*d - b*c))*a**3*b*c*d**3 + 270*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + 
b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*b**2*c**2*d**2 - 300*s 
qrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a* 
d - b*c))*a*b**3*c**3*d + 105*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + 
 sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b**4*c**4)/(192*b**2*d**5)