\(\int \frac {x (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx\) [333]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.82 \[ \int \frac {x (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^3 d^3+5 a^2 b d^2 (53 c+14 d x)-a b^2 d \left (191 c^2+172 c d x+56 d^2 x^2\right )+b^3 \left (15 c^3+118 c^2 d x+136 c d^2 x^2+48 d^3 x^3\right )\right )}{192 b^4 d}-\frac {5 (b c-a d)^3 (b c+7 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{9/2} d^{3/2}} \] Input:

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

Output:

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*a^3*d^3 + 5*a^2*b*d^2*(53*c + 14*d*x) - 
 a*b^2*d*(191*c^2 + 172*c*d*x + 56*d^2*x^2) + b^3*(15*c^3 + 118*c^2*d*x + 
136*c*d^2*x^2 + 48*d^3*x^3)))/(192*b^4*d) - (5*(b*c - a*d)^3*(b*c + 7*a*d) 
*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(9/2)*d^( 
3/2))
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 201, 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.

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(Sqrt[a + b*x]*(c + d*x)^(7/2))/(4*b*d) - ((b*c + 7*a*d)*((Sqrt[a + b*x]*( 
c + d*x)^(5/2))/(3*b) + (5*(b*c - a*d)*((Sqrt[a + b*x]*(c + d*x)^(3/2))/(2 
*b) + (3*(b*c - a*d)*((Sqrt[a + b*x]*Sqrt[c + d*x])/b + ((b*c - a*d)*ArcTa 
nh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(b^(3/2)*Sqrt[d])))/( 
4*b)))/(6*b)))/(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	\(\frac {\sqrt {x d +c}\, \sqrt {b x +a}\, \left (96 b^{3} d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-112 a \,b^{2} d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+272 b^{3} c \,d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+105 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{4} d^{4}-300 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{3} b c \,d^{3}+270 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{2} b^{2} c^{2} d^{2}-60 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a \,b^{3} c^{3} d -15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) b^{4} c^{4}+140 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{2} b \,d^{3} x -344 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a \,b^{2} c \,d^{2} x +236 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, b^{3} c^{2} d x -210 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{3} d^{3}+530 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{2} b c \,d^{2}-382 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a \,b^{2} c^{2} d +30 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, b^{3} c^{3}\right )}{384 b^{4} d \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}}\)	\(574\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 544, normalized size of antiderivative = 2.51 \[ \int \frac {x (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx=\left [-\frac {15 \, {\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 7 \, a^{4} d^{4}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d - 191 \, a b^{3} c^{2} d^{2} + 265 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4} + 8 \, {\left (17 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (59 \, b^{4} c^{2} d^{2} - 86 \, a b^{3} c d^{3} + 35 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, b^{5} d^{2}}, \frac {15 \, {\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 7 \, a^{4} d^{4}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d - 191 \, a b^{3} c^{2} d^{2} + 265 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4} + 8 \, {\left (17 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (59 \, b^{4} c^{2} d^{2} - 86 \, a b^{3} c d^{3} + 35 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, b^{5} d^{2}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x*(d*x+c)^(5/2)/(b*x+a)^(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (179) = 358\).

Time = 0.21 (sec) , antiderivative size = 620, normalized size of antiderivative = 2.86 \[ \int \frac {x (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx=\frac {\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^{3}} + \frac {b^{12} c d^{5} - 25 \, a b^{11} d^{6}}{b^{14} d^{6}}\right )} - \frac {5 \, b^{13} c^{2} d^{4} + 14 \, a b^{12} c d^{5} - 163 \, a^{2} b^{11} d^{6}}{b^{14} d^{6}}\right )} + \frac {3 \, {\left (5 \, b^{14} c^{3} d^{3} + 9 \, a b^{13} c^{2} d^{4} + 15 \, a^{2} b^{12} c d^{5} - 93 \, a^{3} b^{11} d^{6}\right )}}{b^{14} d^{6}}\right )} \sqrt {b x + a} + \frac {3 \, {\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, 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^{2} d^{3}}\right )} d^{2} {\left | b \right |}}{b^{2}} + \frac {48 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, b x + 2 \, a + \frac {b c d - 5 \, a d^{2}}{d^{2}}\right )} \sqrt {b x + a} + \frac {{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\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} d}\right )} c^{2} {\left | b \right |}}{b^{3}} + \frac {16 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (4 \, b x + 4 \, a + \frac {b c d^{3} - 13 \, a d^{4}}{d^{4}}\right )} {\left (b x + a\right )} - \frac {3 \, {\left (b^{2} c^{2} d^{2} + 2 \, a b c d^{3} - 11 \, a^{2} d^{4}\right )}}{d^{4}}\right )} \sqrt {b x + a} - \frac {3 \, {\left (b^{4} c^{3} + a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - 5 \, a^{3} b d^{3}\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} d^{2}}\right )} c d {\left | b \right |}}{b^{4}}}{192 \, b} \] Input:

integrate(x*(d*x+c)^(5/2)/(b*x+a)^(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^3 + (b^12*c*d^5 - 25*a*b^11*d^6)/(b^14*d^6)) - (5*b^13*c^2*d^4 
+ 14*a*b^12*c*d^5 - 163*a^2*b^11*d^6)/(b^14*d^6)) + 3*(5*b^14*c^3*d^3 + 9* 
a*b^13*c^2*d^4 + 15*a^2*b^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))*sqrt(b*x 
 + a) + 3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 
- 35*a^4*d^4)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b* 
d - a*b*d)))/(sqrt(b*d)*b^2*d^3))*d^2*abs(b)/b^2 + 48*(sqrt(b^2*c + (b*x + 
 a)*b*d - a*b*d)*(2*b*x + 2*a + (b*c*d - 5*a*d^2)/d^2)*sqrt(b*x + a) + (b^ 
3*c^2 + 2*a*b^2*c*d - 3*a^2*b*d^2)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt 
(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*d))*c^2*abs(b)/b^3 + 16*(sqrt 
(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*b*x + 4*a + (b*c*d^3 - 13*a*d^4)/d^4 
)*(b*x + a) - 3*(b^2*c^2*d^2 + 2*a*b*c*d^3 - 11*a^2*d^4)/d^4)*sqrt(b*x + a 
) - 3*(b^4*c^3 + a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*log(abs(-sqr 
t(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*d^ 
2))*c*d*abs(b)/b^4)/b
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 471, normalized size of antiderivative = 2.17 \[ \int \frac {x (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx=\frac {-105 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{3} b \,d^{4}+265 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{2} c \,d^{3}+70 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{2} d^{4} x -191 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{3} c^{2} d^{2}-172 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{3} c \,d^{3} x -56 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{3} d^{4} x^{2}+15 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{4} c^{3} d +118 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{4} c^{2} d^{2} x +136 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{4} c \,d^{3} x^{2}+48 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{4} d^{4} x^{3}+105 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{4} d^{4}-300 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{3} b c \,d^{3}+270 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} b^{2} c^{2} d^{2}-60 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a \,b^{3} c^{3} d -15 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{4} c^{4}}{192 b^{5} d^{2}} \] Input:

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

Output:

( - 105*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b*d**4 + 265*sqrt(c + d*x)*sqrt(a 
 + b*x)*a**2*b**2*c*d**3 + 70*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**2*d**4*x 
 - 191*sqrt(c + d*x)*sqrt(a + b*x)*a*b**3*c**2*d**2 - 172*sqrt(c + d*x)*sq 
rt(a + b*x)*a*b**3*c*d**3*x - 56*sqrt(c + d*x)*sqrt(a + b*x)*a*b**3*d**4*x 
**2 + 15*sqrt(c + d*x)*sqrt(a + b*x)*b**4*c**3*d + 118*sqrt(c + d*x)*sqrt( 
a + b*x)*b**4*c**2*d**2*x + 136*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 + 105*sqrt(d)*sqrt(b)*l 
og((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**4*d 
**4 - 300*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d* 
x))/sqrt(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 - 
60*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqr 
t(a*d - b*c))*a*b**3*c**3*d - 15*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**5*d**2)