\(\int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx\) [402]

Optimal result

Integrand size = 22, antiderivative size = 287 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {5 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^5}+\frac {5 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^4 (b c-a d)}+\frac {4 a (3 b c-5 a d) (c+d x)^{5/2}}{3 b^3 (b c-a d) \sqrt {a+b x}}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b^3}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {5 (b c-a d) \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{11/2} \sqrt {d}} \] Output:

5/8*(21*a^2*d^2-14*a*b*c*d+b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^5+5/12*( 
21*a^2*d^2-14*a*b*c*d+b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(3/2)/b^4/(-a*d+b*c)+ 
4/3*a*(-5*a*d+3*b*c)*(d*x+c)^(5/2)/b^3/(-a*d+b*c)/(b*x+a)^(1/2)+1/3*(b*x+a 
)^(1/2)*(d*x+c)^(5/2)/b^3-2/3*a^2*(d*x+c)^(7/2)/b^2/(-a*d+b*c)/(b*x+a)^(3/ 
2)+5/8*(-a*d+b*c)*(21*a^2*d^2-14*a*b*c*d+b^2*c^2)*arctanh(d^(1/2)*(b*x+a)^ 
(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(11/2)/d^(1/2)
 

Mathematica [A] (verified)

Time = 10.49 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.76 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {\sqrt {c+d x} \left (\frac {315 a^4 d^2+420 a^3 b d (-c+d x)-6 a b^3 x \left (-27 c^2+16 c d x+3 d^2 x^2\right )+b^4 x^2 \left (33 c^2+26 c d x+8 d^2 x^2\right )+a^2 b^2 \left (113 c^2-574 c d x+63 d^2 x^2\right )}{(a+b x)^{3/2}}+\frac {15 \sqrt {b c-a d} \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {d} \sqrt {\frac {b (c+d x)}{b c-a d}}}\right )}{24 b^5} \] Input:

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

Output:

(Sqrt[c + d*x]*((315*a^4*d^2 + 420*a^3*b*d*(-c + d*x) - 6*a*b^3*x*(-27*c^2 
 + 16*c*d*x + 3*d^2*x^2) + b^4*x^2*(33*c^2 + 26*c*d*x + 8*d^2*x^2) + a^2*b 
^2*(113*c^2 - 574*c*d*x + 63*d^2*x^2))/(a + b*x)^(3/2) + (15*Sqrt[b*c - a* 
d]*(b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqr 
t[b*c - a*d]])/(Sqrt[d]*Sqrt[(b*(c + d*x))/(b*c - a*d)])))/(24*b^5)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {100, 27, 87, 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.

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

\(\Big \downarrow \) 100

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 87

\(\displaystyle -\frac {-\frac {3 \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \int \frac {(c+d x)^{5/2}}{\sqrt {a+b x}}dx}{b c-a d}-\frac {4 a (c+d x)^{7/2} (3 b c-5 a d)}{\sqrt {a+b x} (b c-a d)}}{3 b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {-\frac {3 \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \left (\frac {5 (b c-a d) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}}dx}{6 b}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}\right )}{b c-a d}-\frac {4 a (c+d x)^{7/2} (3 b c-5 a d)}{\sqrt {a+b x} (b c-a d)}}{3 b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {-\frac {3 \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}}dx}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}\right )}{b c-a d}-\frac {4 a (c+d x)^{7/2} (3 b c-5 a d)}{\sqrt {a+b x} (b c-a d)}}{3 b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {-\frac {3 \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}\right )}{b c-a d}-\frac {4 a (c+d x)^{7/2} (3 b c-5 a d)}{\sqrt {a+b x} (b c-a d)}}{3 b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 66

\(\displaystyle -\frac {-\frac {3 \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}\right )}{b c-a d}-\frac {4 a (c+d x)^{7/2} (3 b c-5 a d)}{\sqrt {a+b x} (b c-a d)}}{3 b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {-\frac {3 \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} \sqrt {d}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}\right )}{b c-a d}-\frac {4 a (c+d x)^{7/2} (3 b c-5 a d)}{\sqrt {a+b x} (b c-a d)}}{3 b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (a+b x)^{3/2} (b c-a d)}\)

Input:

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

Output:

(-2*a^2*(c + d*x)^(7/2))/(3*b^2*(b*c - a*d)*(a + b*x)^(3/2)) - ((-4*a*(3*b 
*c - 5*a*d)*(c + d*x)^(7/2))/((b*c - a*d)*Sqrt[a + b*x]) - (3*(b^2*c^2 - 1 
4*a*b*c*d + 21*a^2*d^2)*((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)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[ 
b]*Sqrt[c + d*x])])/(b^(3/2)*Sqrt[d])))/(4*b)))/(6*b)))/(b*c - a*d))/(3*b^ 
2*(b*c - a*d))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

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. \(1001\) vs. \(2(243)=486\).

Time = 0.26 (sec) , antiderivative size = 1002, normalized size of antiderivative = 3.49

method	result	size
default	\(\text {Expression too large to display}\)	\(1002\)

Input:

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

Output:

-1/48*(d*x+c)^(1/2)*(-16*b^4*d^2*x^4*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+3 
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^3*b^2*d^3*x^2-525*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^3*c*d^2*x^2+225*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^4*c^2*d*x^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))*b^ 
5*c^3*x^2+36*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a*b^3*d^2*x^3-52*((b*x+a) 
*(d*x+c))^(1/2)*(d*b)^(1/2)*b^4*c*d*x^3+630*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*b*d^3*x-1050*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^2*c 
*d^2*x+450*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^3*c^2*d*x-30*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^4*c^3*x-126*((b*x+a)*(d*x+c))^(1/2)* 
(d*b)^(1/2)*a^2*b^2*d^2*x^2+192*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a*b^3* 
c*d*x^2-66*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*b^4*c^2*x^2+315*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^5*d^3-5 
25*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*b*c*d^2+225*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^2*c^2*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))*a^2*b^3*c^3-840*((b*x+a)*(d*x...
 

Fricas [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 804, normalized size of antiderivative = 2.80 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/96*(15*(a^2*b^3*c^3 - 15*a^3*b^2*c^2*d + 35*a^4*b*c*d^2 - 21*a^5*d^3 + 
 (b^5*c^3 - 15*a*b^4*c^2*d + 35*a^2*b^3*c*d^2 - 21*a^3*b^2*d^3)*x^2 + 2*(a 
*b^4*c^3 - 15*a^2*b^3*c^2*d + 35*a^3*b^2*c*d^2 - 21*a^4*b*d^3)*x)*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*( 
8*b^5*d^3*x^4 + 113*a^2*b^3*c^2*d - 420*a^3*b^2*c*d^2 + 315*a^4*b*d^3 + 2* 
(13*b^5*c*d^2 - 9*a*b^4*d^3)*x^3 + 3*(11*b^5*c^2*d - 32*a*b^4*c*d^2 + 21*a 
^2*b^3*d^3)*x^2 + 2*(81*a*b^4*c^2*d - 287*a^2*b^3*c*d^2 + 210*a^3*b^2*d^3) 
*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^8*d*x^2 + 2*a*b^7*d*x + a^2*b^6*d), -1 
/48*(15*(a^2*b^3*c^3 - 15*a^3*b^2*c^2*d + 35*a^4*b*c*d^2 - 21*a^5*d^3 + (b 
^5*c^3 - 15*a*b^4*c^2*d + 35*a^2*b^3*c*d^2 - 21*a^3*b^2*d^3)*x^2 + 2*(a*b^ 
4*c^3 - 15*a^2*b^3*c^2*d + 35*a^3*b^2*c*d^2 - 21*a^4*b*d^3)*x)*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*(8*b^5*d^3*x^4 + 113*a^ 
2*b^3*c^2*d - 420*a^3*b^2*c*d^2 + 315*a^4*b*d^3 + 2*(13*b^5*c*d^2 - 9*a*b^ 
4*d^3)*x^3 + 3*(11*b^5*c^2*d - 32*a*b^4*c*d^2 + 21*a^2*b^3*d^3)*x^2 + 2*(8 
1*a*b^4*c^2*d - 287*a^2*b^3*c*d^2 + 210*a^3*b^2*d^3)*x)*sqrt(b*x + a)*sqrt 
(d*x + c))/(b^8*d*x^2 + 2*a*b^7*d*x + a^2*b^6*d)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^2*(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. 848 vs. \(2 (243) = 486\).

Time = 0.38 (sec) , antiderivative size = 848, normalized size of antiderivative = 2.95 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

1/24*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b* 
x + a)*d^2*abs(b)/b^7 + (13*b^21*c*d^5*abs(b) - 25*a*b^20*d^6*abs(b))/(b^2 
7*d^4)) + 3*(11*b^22*c^2*d^4*abs(b) - 58*a*b^21*c*d^5*abs(b) + 55*a^2*b^20 
*d^6*abs(b))/(b^27*d^4)) - 5/16*(b^3*c^3*abs(b) - 15*a*b^2*c^2*d*abs(b) + 
35*a^2*b*c*d^2*abs(b) - 21*a^3*d^3*abs(b))*log((sqrt(b*d)*sqrt(b*x + a) - 
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(b*d)*b^6) + 4/3*(6*a*b^7*c^5 
*d*abs(b) - 37*a^2*b^6*c^4*d^2*abs(b) + 88*a^3*b^5*c^3*d^3*abs(b) - 102*a^ 
4*b^4*c^2*d^4*abs(b) + 58*a^5*b^3*c*d^5*abs(b) - 13*a^6*b^2*d^6*abs(b) - 1 
2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^5* 
c^4*d*abs(b) + 60*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - 
a*b*d))^2*a^2*b^4*c^3*d^2*abs(b) - 108*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2 
*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^3*c^2*d^3*abs(b) + 84*(sqrt(b*d)*sqrt 
(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^2*c*d^4*abs(b) - 
24*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b 
*d^5*abs(b) + 6*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a* 
b*d))^4*a*b^3*c^3*d*abs(b) - 27*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b 
*x + a)*b*d - a*b*d))^4*a^2*b^2*c^2*d^2*abs(b) + 36*(sqrt(b*d)*sqrt(b*x + 
a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b*c*d^3*abs(b) - 15*(sqrt( 
b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^4*d^4*abs(b) 
)/((b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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