\(\int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx\) [292]

Optimal result

Integrand size = 22, antiderivative size = 314 \[ \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^2 d^5}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^2 d^4}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{240 b^2 d^3}-\frac {(9 b c+11 a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b^2 d^2}+\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b^2 d}-\frac {(b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{5/2} d^{11/2}} \] Output:

1/128*(-a*d+b*c)^2*(3*a^2*d^2+14*a*b*c*d+63*b^2*c^2)*(b*x+a)^(1/2)*(d*x+c) 
^(1/2)/b^2/d^5-1/192*(-a*d+b*c)*(3*a^2*d^2+14*a*b*c*d+63*b^2*c^2)*(b*x+a)^ 
(3/2)*(d*x+c)^(1/2)/b^2/d^4+1/240*(3*a^2*d^2+14*a*b*c*d+63*b^2*c^2)*(b*x+a 
)^(5/2)*(d*x+c)^(1/2)/b^2/d^3-1/40*(11*a*d+9*b*c)*(b*x+a)^(7/2)*(d*x+c)^(1 
/2)/b^2/d^2+1/5*(b*x+a)^(9/2)*(d*x+c)^(1/2)/b^2/d-1/128*(-a*d+b*c)^3*(3*a^ 
2*d^2+14*a*b*c*d+63*b^2*c^2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c) 
^(1/2))/b^(5/2)/d^(11/2)
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.77 \[ \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-45 a^4 d^4+30 a^3 b d^3 (-3 c+d x)+2 a^2 b^2 d^2 \left (782 c^2-481 c d x+372 d^2 x^2\right )+2 a b^3 d \left (-1155 c^3+749 c^2 d x-592 c d^2 x^2+504 d^3 x^3\right )+b^4 \left (945 c^4-630 c^3 d x+504 c^2 d^2 x^2-432 c d^3 x^3+384 d^4 x^4\right )\right )}{1920 b^2 d^5}-\frac {(b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{128 b^{5/2} d^{11/2}} \] Input:

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

Output:

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-45*a^4*d^4 + 30*a^3*b*d^3*(-3*c + d*x) + 2* 
a^2*b^2*d^2*(782*c^2 - 481*c*d*x + 372*d^2*x^2) + 2*a*b^3*d*(-1155*c^3 + 7 
49*c^2*d*x - 592*c*d^2*x^2 + 504*d^3*x^3) + b^4*(945*c^4 - 630*c^3*d*x + 5 
04*c^2*d^2*x^2 - 432*c*d^3*x^3 + 384*d^4*x^4)))/(1920*b^2*d^5) - ((b*c - a 
*d)^3*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x] 
)/(Sqrt[d]*Sqrt[a + b*x])])/(128*b^(5/2)*d^(11/2))
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.84, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {101, 27, 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.

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

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {\int \frac {(a+b x)^{5/2} (2 a c+3 (3 b c+a d) x)}{\sqrt {c+d x}}dx}{10 b d}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {\frac {3 (a+b x)^{7/2} \sqrt {c+d x} (a d+3 b c)}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}}dx}{8 b d}}{10 b d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {\frac {3 (a+b x)^{7/2} \sqrt {c+d x} (a d+3 b c)}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}}dx}{6 d}\right )}{8 b d}}{10 b d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {\frac {3 (a+b x)^{7/2} \sqrt {c+d x} (a d+3 b c)}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{4 d}\right )}{6 d}\right )}{8 b d}}{10 b d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {\frac {3 (a+b x)^{7/2} \sqrt {c+d x} (a d+3 b c)}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )}{6 d}\right )}{8 b d}}{10 b d}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {\frac {3 (a+b x)^{7/2} \sqrt {c+d x} (a d+3 b c)}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )}{6 d}\right )}{8 b d}}{10 b d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {\frac {3 (a+b x)^{7/2} \sqrt {c+d x} (a d+3 b c)}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 d}\right )}{6 d}\right )}{8 b d}}{10 b d}\)

Input:

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

Output:

(x*(a + b*x)^(7/2)*Sqrt[c + d*x])/(5*b*d) - ((3*(3*b*c + a*d)*(a + b*x)^(7 
/2)*Sqrt[c + d*x])/(4*b*d) - ((63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*(((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)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*d 
^(3/2))))/(4*d)))/(6*d)))/(8*b*d))/(10*b*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 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 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 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. \(787\) vs. \(2(270)=540\).

Time = 0.23 (sec) , antiderivative size = 788, normalized size of antiderivative = 2.51

method	result	size
default	\(\frac {\sqrt {b x +a}\, \sqrt {x d +c}\, \left (768 b^{4} d^{4} x^{4} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+2016 a \,b^{3} d^{4} x^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-864 b^{4} c \,d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+1488 a^{2} b^{2} d^{4} x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-2368 a \,b^{3} c \,d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+1008 b^{4} c^{2} d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+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^{5} d^{5}+75 \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} b c \,d^{4}+450 \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^{2} c^{2} d^{3}-2250 \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^{3} c^{3} d^{2}+2625 \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^{4} c^{4} d -945 \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^{5} c^{5}+60 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{3} b \,d^{4} x -1924 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{2} b^{2} c \,d^{3} x +2996 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a \,b^{3} c^{2} d^{2} x -1260 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, b^{4} c^{3} d x -90 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{4} d^{4}-180 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{3} b c \,d^{3}+3128 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{2} b^{2} c^{2} d^{2}-4620 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a \,b^{3} c^{3} d +1890 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, b^{4} c^{4}\right )}{3840 b^{2} d^{5} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}}\)	\(788\)

Input:

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

Output:

1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(768*b^4*d^4*x^4*((b*x+a)*(d*x+c))^(1/2 
)*(d*b)^(1/2)+2016*a*b^3*d^4*x^3*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-864*b 
^4*c*d^3*x^3*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+1488*a^2*b^2*d^4*x^2*((b* 
x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-2368*a*b^3*c*d^3*x^2*((b*x+a)*(d*x+c))^(1/ 
2)*(d*b)^(1/2)+1008*b^4*c^2*d^2*x^2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+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^5*d^5+75*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^4+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^3*b^2*c^2*d^3-2250*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*d^2+2 
625*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^4*d-945*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^5+60*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^3 
*b*d^4*x-1924*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^2*b^2*c*d^3*x+2996*((b 
*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a*b^3*c^2*d^2*x-1260*((b*x+a)*(d*x+c))^(1 
/2)*(d*b)^(1/2)*b^4*c^3*d*x-90*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^4*d^4 
-180*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^3*b*c*d^3+3128*((b*x+a)*(d*x+c) 
)^(1/2)*(d*b)^(1/2)*a^2*b^2*c^2*d^2-4620*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/ 
2)*a*b^3*c^3*d+1890*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*b^4*c^4)/b^2/d^5/( 
(b*x+a)*(d*x+c))^(1/2)/(d*b)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 706, normalized size of antiderivative = 2.25 \[ \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\left [-\frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (384 \, b^{5} d^{5} x^{4} + 945 \, b^{5} c^{4} d - 2310 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 90 \, a^{3} b^{2} c d^{4} - 45 \, a^{4} b d^{5} - 144 \, {\left (3 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (63 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 93 \, a^{2} b^{3} d^{5}\right )} x^{2} - 2 \, {\left (315 \, b^{5} c^{3} d^{2} - 749 \, a b^{4} c^{2} d^{3} + 481 \, a^{2} b^{3} c d^{4} - 15 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, b^{3} d^{6}}, \frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (384 \, b^{5} d^{5} x^{4} + 945 \, b^{5} c^{4} d - 2310 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 90 \, a^{3} b^{2} c d^{4} - 45 \, a^{4} b d^{5} - 144 \, {\left (3 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (63 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 93 \, a^{2} b^{3} d^{5}\right )} x^{2} - 2 \, {\left (315 \, b^{5} c^{3} d^{2} - 749 \, a b^{4} c^{2} d^{3} + 481 \, a^{2} b^{3} c d^{4} - 15 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, b^{3} d^{6}}\right ] \] Input:

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

Output:

[-1/7680*(15*(63*b^5*c^5 - 175*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3* 
b^2*c^2*d^3 - 5*a^4*b*c*d^4 - 3*a^5*d^5)*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*(384*b^5*d^5*x^4 + 945*b^5 
*c^4*d - 2310*a*b^4*c^3*d^2 + 1564*a^2*b^3*c^2*d^3 - 90*a^3*b^2*c*d^4 - 45 
*a^4*b*d^5 - 144*(3*b^5*c*d^4 - 7*a*b^4*d^5)*x^3 + 8*(63*b^5*c^2*d^3 - 148 
*a*b^4*c*d^4 + 93*a^2*b^3*d^5)*x^2 - 2*(315*b^5*c^3*d^2 - 749*a*b^4*c^2*d^ 
3 + 481*a^2*b^3*c*d^4 - 15*a^3*b^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b 
^3*d^6), 1/3840*(15*(63*b^5*c^5 - 175*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 
30*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 - 3*a^5*d^5)*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*(384*b^5*d^5*x^4 + 945*b^5*c^4*d - 231 
0*a*b^4*c^3*d^2 + 1564*a^2*b^3*c^2*d^3 - 90*a^3*b^2*c*d^4 - 45*a^4*b*d^5 - 
 144*(3*b^5*c*d^4 - 7*a*b^4*d^5)*x^3 + 8*(63*b^5*c^2*d^3 - 148*a*b^4*c*d^4 
 + 93*a^2*b^3*d^5)*x^2 - 2*(315*b^5*c^3*d^2 - 749*a*b^4*c^2*d^3 + 481*a^2* 
b^3*c*d^4 - 15*a^3*b^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^3*d^6)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^2*(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 = 0.16 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.22 \[ \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (4 \, {\left (b x + a\right )} {\left (6 \, {\left (b x + a\right )} {\left (\frac {8 \, {\left (b x + a\right )}}{b^{3} d} - \frac {9 \, b^{7} c d^{7} + 11 \, a b^{6} d^{8}}{b^{9} d^{9}}\right )} + \frac {63 \, b^{8} c^{2} d^{6} + 14 \, a b^{7} c d^{7} + 3 \, a^{2} b^{6} d^{8}}{b^{9} d^{9}}\right )} - \frac {5 \, {\left (63 \, b^{9} c^{3} d^{5} - 49 \, a b^{8} c^{2} d^{6} - 11 \, a^{2} b^{7} c d^{7} - 3 \, a^{3} b^{6} d^{8}\right )}}{b^{9} d^{9}}\right )} {\left (b x + a\right )} + \frac {15 \, {\left (63 \, b^{10} c^{4} d^{4} - 112 \, a b^{9} c^{3} d^{5} + 38 \, a^{2} b^{8} c^{2} d^{6} + 8 \, a^{3} b^{7} c d^{7} + 3 \, a^{4} b^{6} d^{8}\right )}}{b^{9} d^{9}}\right )} \sqrt {b x + a} + \frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{5}}\right )} b}{1920 \, {\left | b \right |}} \] Input:

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

Output:

1/1920*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(b*x + a)*(6*(b*x + a)*( 
8*(b*x + a)/(b^3*d) - (9*b^7*c*d^7 + 11*a*b^6*d^8)/(b^9*d^9)) + (63*b^8*c^ 
2*d^6 + 14*a*b^7*c*d^7 + 3*a^2*b^6*d^8)/(b^9*d^9)) - 5*(63*b^9*c^3*d^5 - 4 
9*a*b^8*c^2*d^6 - 11*a^2*b^7*c*d^7 - 3*a^3*b^6*d^8)/(b^9*d^9))*(b*x + a) + 
 15*(63*b^10*c^4*d^4 - 112*a*b^9*c^3*d^5 + 38*a^2*b^8*c^2*d^6 + 8*a^3*b^7* 
c*d^7 + 3*a^4*b^6*d^8)/(b^9*d^9))*sqrt(b*x + a) + 15*(63*b^5*c^5 - 175*a*b 
^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 - 3*a^ 
5*d^5)*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^5))*b/abs(b)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 651, normalized size of antiderivative = 2.07 \[ \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\frac {-45 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{4} b \,d^{5}-90 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{3} b^{2} c \,d^{4}+30 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{3} b^{2} d^{5} x +1564 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{3} c^{2} d^{3}-962 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{3} c \,d^{4} x +744 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{3} d^{5} x^{2}-2310 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{4} c^{3} d^{2}+1498 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{4} c^{2} d^{3} x -1184 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{4} c \,d^{4} x^{2}+1008 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{4} d^{5} x^{3}+945 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} c^{4} d -630 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} c^{3} d^{2} x +504 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} c^{2} d^{3} x^{2}-432 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} c \,d^{4} x^{3}+384 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} d^{5} x^{4}+45 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{5} d^{5}+75 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{4} b c \,d^{4}+450 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{3} b^{2} c^{2} d^{3}-2250 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} b^{3} c^{3} d^{2}+2625 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a \,b^{4} c^{4} d -945 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{5} c^{5}}{1920 b^{3} d^{6}} \] Input:

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

Output:

( - 45*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b*d**5 - 90*sqrt(c + d*x)*sqrt(a + 
 b*x)*a**3*b**2*c*d**4 + 30*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**2*d**5*x + 
 1564*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**3*c**2*d**3 - 962*sqrt(c + d*x)* 
sqrt(a + b*x)*a**2*b**3*c*d**4*x + 744*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b* 
*3*d**5*x**2 - 2310*sqrt(c + d*x)*sqrt(a + b*x)*a*b**4*c**3*d**2 + 1498*sq 
rt(c + d*x)*sqrt(a + b*x)*a*b**4*c**2*d**3*x - 1184*sqrt(c + d*x)*sqrt(a + 
 b*x)*a*b**4*c*d**4*x**2 + 1008*sqrt(c + d*x)*sqrt(a + b*x)*a*b**4*d**5*x* 
*3 + 945*sqrt(c + d*x)*sqrt(a + b*x)*b**5*c**4*d - 630*sqrt(c + d*x)*sqrt( 
a + b*x)*b**5*c**3*d**2*x + 504*sqrt(c + d*x)*sqrt(a + b*x)*b**5*c**2*d**3 
*x**2 - 432*sqrt(c + d*x)*sqrt(a + b*x)*b**5*c*d**4*x**3 + 384*sqrt(c + d* 
x)*sqrt(a + b*x)*b**5*d**5*x**4 + 45*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + 
 b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**5*d**5 + 75*sqrt(d)*sqr 
t(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))* 
a**4*b*c*d**4 + 450*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*s 
qrt(c + d*x))/sqrt(a*d - b*c))*a**3*b**2*c**2*d**3 - 2250*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**3*c**3*d**2 + 2625*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b) 
*sqrt(c + d*x))/sqrt(a*d - b*c))*a*b**4*c**4*d - 945*sqrt(d)*sqrt(b)*log(( 
sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b**5*c**5) 
/(1920*b**3*d**6)