\(\int \frac {x (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx\) [403]

Optimal result

Integrand size = 20, antiderivative size = 183 \[ \int \frac {x (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {5 d (3 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^4}-\frac {2 (3 b c-8 a d) (c+d x)^{3/2}}{3 b^3 \sqrt {a+b x}}+\frac {d \sqrt {a+b x} (c+d x)^{3/2}}{2 b^3}+\frac {2 a (c+d x)^{5/2}}{3 b^2 (a+b x)^{3/2}}+\frac {5 \sqrt {d} (3 b c-7 a d) (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{9/2}} \] Output:

5/4*d*(-7*a*d+3*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^4-2/3*(-8*a*d+3*b*c)*(d 
*x+c)^(3/2)/b^3/(b*x+a)^(1/2)+1/2*d*(b*x+a)^(1/2)*(d*x+c)^(3/2)/b^3+2/3*a* 
(d*x+c)^(5/2)/b^2/(b*x+a)^(3/2)+5/4*d^(1/2)*(-7*a*d+3*b*c)*(-a*d+b*c)*arct 
anh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(9/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.12 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.67 \[ \int \frac {x (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {2 \sqrt {c+d x} \left (a b^3 (c+d x)^3-\frac {(3 b c-7 a d) (b c-a d)^2 (a+b x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {1}{2},\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )}{\sqrt {\frac {b (c+d x)}{b c-a d}}}\right )}{3 b^4 (b c-a d) (a+b x)^{3/2}} \] Input:

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

Output:

(2*Sqrt[c + d*x]*(a*b^3*(c + d*x)^3 - ((3*b*c - 7*a*d)*(b*c - a*d)^2*(a + 
b*x)*Hypergeometric2F1[-5/2, -1/2, 1/2, (d*(a + b*x))/(-(b*c) + a*d)])/Sqr 
t[(b*(c + d*x))/(b*c - a*d)]))/(3*b^4*(b*c - a*d)*(a + b*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {87, 57, 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.

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

\(\Big \downarrow \) 87

\(\displaystyle \frac {(3 b c-7 a d) \int \frac {(c+d x)^{5/2}}{(a+b x)^{3/2}}dx}{3 b (b c-a d)}+\frac {2 a (c+d x)^{7/2}}{3 b (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(2*a*(c + d*x)^(7/2))/(3*b*(b*c - a*d)*(a + b*x)^(3/2)) + ((3*b*c - 7*a*d) 
*((-2*(c + d*x)^(5/2))/(b*Sqrt[a + b*x]) + (5*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)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(b^(3/2)*Sqr 
t[d])))/(4*b)))/b))/(3*b*(b*c - a*d))
 

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. \(749\) vs. \(2(145)=290\).

Time = 0.25 (sec) , antiderivative size = 750, normalized size of antiderivative = 4.10

method	result	size
default	\(\frac {\sqrt {x d +c}\, \left (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^{2} b^{2} d^{3} x^{2}-150 \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 \,d^{2} x^{2}+45 \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^{2} d \,x^{2}+12 b^{3} d^{2} x^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+210 \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 \,d^{3} x -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^{2} b^{2} c \,d^{2} x +90 \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^{2} d x -42 a \,b^{2} d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+54 b^{3} c d \,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^{3}-150 \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^{2}+45 \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 -280 a^{2} b \,d^{2} x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+316 a \,b^{2} c d x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-48 b^{3} c^{2} x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-210 a^{3} d^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+230 a^{2} b c d \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-32 a \,b^{2} c^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\right )}{24 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \left (b x +a \right )^{\frac {3}{2}} b^{4}}\)	\(750\)

Input:

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

Output:

1/24*(d*x+c)^(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^2*b^2*d^3*x^2-150*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*d^2*x^2+45*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 
^2*d*x^2+12*b^3*d^2*x^3*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+210*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*d^ 
3*x-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^2*b^2*c*d^2*x+90*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^2*d*x-42*a*b^2*d^2*x^2*((b*x+a)*(d* 
x+c))^(1/2)*(d*b)^(1/2)+54*b^3*c*d*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^3-150*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^2+45*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-280*a^2*b*d^2*x*((b*x+ 
a)*(d*x+c))^(1/2)*(d*b)^(1/2)+316*a*b^2*c*d*x*((b*x+a)*(d*x+c))^(1/2)*(d*b 
)^(1/2)-48*b^3*c^2*x*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-210*a^3*d^2*((b*x 
+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+230*a^2*b*c*d*((b*x+a)*(d*x+c))^(1/2)*(d*b) 
^(1/2)-32*a*b^2*c^2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2))/((b*x+a)*(d*x+c)) 
^(1/2)/(d*b)^(1/2)/(b*x+a)^(3/2)/b^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (145) = 290\).

Time = 0.32 (sec) , antiderivative size = 619, normalized size of antiderivative = 3.38 \[ \int \frac {x (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\left [\frac {15 \, {\left (3 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 7 \, a^{4} d^{2} + {\left (3 \, b^{4} c^{2} - 10 \, a b^{3} c d + 7 \, a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (3 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 7 \, a^{3} b d^{2}\right )} x\right )} \sqrt {\frac {d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (6 \, b^{3} d^{2} x^{3} - 16 \, a b^{2} c^{2} + 115 \, a^{2} b c d - 105 \, a^{3} d^{2} + 3 \, {\left (9 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{2} - 2 \, {\left (12 \, b^{3} c^{2} - 79 \, a b^{2} c d + 70 \, a^{2} b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, -\frac {15 \, {\left (3 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 7 \, a^{4} d^{2} + {\left (3 \, b^{4} c^{2} - 10 \, a b^{3} c d + 7 \, a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (3 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 7 \, a^{3} b d^{2}\right )} x\right )} \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) - 2 \, {\left (6 \, b^{3} d^{2} x^{3} - 16 \, a b^{2} c^{2} + 115 \, a^{2} b c d - 105 \, a^{3} d^{2} + 3 \, {\left (9 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{2} - 2 \, {\left (12 \, b^{3} c^{2} - 79 \, a b^{2} c d + 70 \, a^{2} b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \] Input:

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

Output:

[1/48*(15*(3*a^2*b^2*c^2 - 10*a^3*b*c*d + 7*a^4*d^2 + (3*b^4*c^2 - 10*a*b^ 
3*c*d + 7*a^2*b^2*d^2)*x^2 + 2*(3*a*b^3*c^2 - 10*a^2*b^2*c*d + 7*a^3*b*d^2 
)*x)*sqrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b^ 
2*d*x + b^2*c + a*b*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d 
+ a*b*d^2)*x) + 4*(6*b^3*d^2*x^3 - 16*a*b^2*c^2 + 115*a^2*b*c*d - 105*a^3* 
d^2 + 3*(9*b^3*c*d - 7*a*b^2*d^2)*x^2 - 2*(12*b^3*c^2 - 79*a*b^2*c*d + 70* 
a^2*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4) 
, -1/24*(15*(3*a^2*b^2*c^2 - 10*a^3*b*c*d + 7*a^4*d^2 + (3*b^4*c^2 - 10*a* 
b^3*c*d + 7*a^2*b^2*d^2)*x^2 + 2*(3*a*b^3*c^2 - 10*a^2*b^2*c*d + 7*a^3*b*d 
^2)*x)*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(d*x 
+ c)*sqrt(-d/b)/(b*d^2*x^2 + a*c*d + (b*c*d + a*d^2)*x)) - 2*(6*b^3*d^2*x^ 
3 - 16*a*b^2*c^2 + 115*a^2*b*c*d - 105*a^3*d^2 + 3*(9*b^3*c*d - 7*a*b^2*d^ 
2)*x^2 - 2*(12*b^3*c^2 - 79*a*b^2*c*d + 70*a^2*b*d^2)*x)*sqrt(b*x + a)*sqr 
t(d*x + c))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x*(d*x+c)^(5/2)/(b*x+a)^(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(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. 840 vs. \(2 (145) = 290\).

Time = 0.33 (sec) , antiderivative size = 840, normalized size of antiderivative = 4.59 \[ \int \frac {x (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*d^2*abs 
(b)/b^6 + (9*b^12*c*d^3*abs(b) - 13*a*b^11*d^4*abs(b))/(b^17*d^2)) - 5/8*( 
3*sqrt(b*d)*b^2*c^2*abs(b) - 10*sqrt(b*d)*a*b*c*d*abs(b) + 7*sqrt(b*d)*a^2 
*d^2*abs(b))*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a 
*b*d))^2)/b^6 - 4/3*(3*sqrt(b*d)*b^7*c^5*abs(b) - 22*sqrt(b*d)*a*b^6*c^4*d 
*abs(b) + 58*sqrt(b*d)*a^2*b^5*c^3*d^2*abs(b) - 72*sqrt(b*d)*a^3*b^4*c^2*d 
^3*abs(b) + 43*sqrt(b*d)*a^4*b^3*c*d^4*abs(b) - 10*sqrt(b*d)*a^5*b^2*d^5*a 
bs(b) - 6*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d 
- a*b*d))^2*b^5*c^4*abs(b) + 36*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt( 
b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^4*c^3*d*abs(b) - 72*sqrt(b*d)*(sqrt( 
b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^3*c^2*d^ 
2*abs(b) + 60*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)* 
b*d - a*b*d))^2*a^3*b^2*c*d^3*abs(b) - 18*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + 
a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b*d^4*abs(b) + 3*sqrt(b*d) 
*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^3*c^3 
*abs(b) - 18*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b 
*d - a*b*d))^4*a*b^2*c^2*d*abs(b) + 27*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) 
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b*c*d^2*abs(b) - 12*sqrt(b*d) 
*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*d^3 
*abs(b))/((b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 1.27 (sec) , antiderivative size = 584, normalized size of antiderivative = 3.19 \[ \int \frac {x (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {840 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{3} d^{2}-1200 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} b c d +840 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} b \,d^{2} x +360 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a \,b^{2} c^{2}-1200 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a \,b^{2} c d x +360 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{3} c^{2} x +175 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{3} d^{2}-270 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{2} b c d +175 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{2} b \,d^{2} x +95 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a \,b^{2} c^{2}-270 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a \,b^{2} c d x +95 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, b^{3} c^{2} x -840 \sqrt {d x +c}\, a^{3} b \,d^{2}+920 \sqrt {d x +c}\, a^{2} b^{2} c d -1120 \sqrt {d x +c}\, a^{2} b^{2} d^{2} x -128 \sqrt {d x +c}\, a \,b^{3} c^{2}+1264 \sqrt {d x +c}\, a \,b^{3} c d x -168 \sqrt {d x +c}\, a \,b^{3} d^{2} x^{2}-192 \sqrt {d x +c}\, b^{4} c^{2} x +216 \sqrt {d x +c}\, b^{4} c d \,x^{2}+48 \sqrt {d x +c}\, b^{4} d^{2} x^{3}}{96 \sqrt {b x +a}\, b^{5} \left (b x +a \right )} \] Input:

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

Output:

(840*sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sq 
rt(c + d*x))/sqrt(a*d - b*c))*a**3*d**2 - 1200*sqrt(d)*sqrt(b)*sqrt(a + b* 
x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a* 
*2*b*c*d + 840*sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + 
sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*b*d**2*x + 360*sqrt(d)*sqrt(b 
)*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a 
*d - b*c))*a*b**2*c**2 - 1200*sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)*s 
qrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a*b**2*c*d*x + 360* 
sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c 
+ d*x))/sqrt(a*d - b*c))*b**3*c**2*x + 175*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a 
**3*d**2 - 270*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**2*b*c*d + 175*sqrt(d)*sqrt 
(b)*sqrt(a + b*x)*a**2*b*d**2*x + 95*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a*b**2* 
c**2 - 270*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a*b**2*c*d*x + 95*sqrt(d)*sqrt(b) 
*sqrt(a + b*x)*b**3*c**2*x - 840*sqrt(c + d*x)*a**3*b*d**2 + 920*sqrt(c + 
d*x)*a**2*b**2*c*d - 1120*sqrt(c + d*x)*a**2*b**2*d**2*x - 128*sqrt(c + d* 
x)*a*b**3*c**2 + 1264*sqrt(c + d*x)*a*b**3*c*d*x - 168*sqrt(c + d*x)*a*b** 
3*d**2*x**2 - 192*sqrt(c + d*x)*b**4*c**2*x + 216*sqrt(c + d*x)*b**4*c*d*x 
**2 + 48*sqrt(c + d*x)*b**4*d**2*x**3)/(96*sqrt(a + b*x)*b**5*(a + b*x))