\(\int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx\) [332]

Optimal result

Integrand size = 22, antiderivative size = 314 \[ \int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx=\frac {(b c-a d)^2 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^5 d^2}+\frac {(b c-a d) \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{240 b^3 d^2}-\frac {(3 b c+17 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 b^2 d^2}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b^2 d}+\frac {(b c-a d)^3 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{11/2} d^{5/2}} \] Output:

1/128*(-a*d+b*c)^2*(63*a^2*d^2+14*a*b*c*d+3*b^2*c^2)*(b*x+a)^(1/2)*(d*x+c) 
^(1/2)/b^5/d^2+1/192*(-a*d+b*c)*(63*a^2*d^2+14*a*b*c*d+3*b^2*c^2)*(b*x+a)^ 
(1/2)*(d*x+c)^(3/2)/b^4/d^2+1/240*(63*a^2*d^2+14*a*b*c*d+3*b^2*c^2)*(b*x+a 
)^(1/2)*(d*x+c)^(5/2)/b^3/d^2-1/40*(17*a*d+3*b*c)*(b*x+a)^(1/2)*(d*x+c)^(7 
/2)/b^2/d^2+1/5*(b*x+a)^(3/2)*(d*x+c)^(7/2)/b^2/d+1/128*(-a*d+b*c)^3*(63*a 
^2*d^2+14*a*b*c*d+3*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^(5/2)
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.78 \[ \int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (945 a^4 d^4-210 a^3 b d^3 (11 c+3 d x)+2 a^2 b^2 d^2 \left (782 c^2+749 c d x+252 d^2 x^2\right )-2 a b^3 d \left (45 c^3+481 c^2 d x+592 c d^2 x^2+216 d^3 x^3\right )+b^4 \left (-45 c^4+30 c^3 d x+744 c^2 d^2 x^2+1008 c d^3 x^3+384 d^4 x^4\right )\right )}{1920 b^5 d^2}+\frac {(b c-a d)^3 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{11/2} d^{5/2}} \] Input:

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

Output:

(Sqrt[a + b*x]*Sqrt[c + d*x]*(945*a^4*d^4 - 210*a^3*b*d^3*(11*c + 3*d*x) + 
 2*a^2*b^2*d^2*(782*c^2 + 749*c*d*x + 252*d^2*x^2) - 2*a*b^3*d*(45*c^3 + 4 
81*c^2*d*x + 592*c*d^2*x^2 + 216*d^3*x^3) + b^4*(-45*c^4 + 30*c^3*d*x + 74 
4*c^2*d^2*x^2 + 1008*c*d^3*x^3 + 384*d^4*x^4)))/(1920*b^5*d^2) + ((b*c - a 
*d)^3*(3*b^2*c^2 + 14*a*b*c*d + 63*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x] 
)/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(11/2)*d^(5/2))
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 264, 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.

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

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(x*Sqrt[a + b*x]*(c + d*x)^(7/2))/(5*b*d) - ((3*(b*c + 3*a*d)*Sqrt[a + b*x 
]*(c + d*x)^(7/2))/(4*b*d) - ((3*b^2*c^2 + 14*a*b*c*d + 63*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)*S 
qrt[d])))/(4*b)))/(6*b)))/(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 = 0.24 (sec) , antiderivative size = 788, normalized size of antiderivative = 2.51

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

Input:

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

Output:

-1/3840*(d*x+c)^(1/2)*(b*x+a)^(1/2)*(-768*b^4*d^4*x^4*((b*x+a)*(d*x+c))^(1 
/2)*(d*b)^(1/2)+864*a*b^3*d^4*x^3*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-2016 
*b^4*c*d^3*x^3*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-1008*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)-1488*b^4*c^2*d^2*x^2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+ 
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))*a^5*d^5-2625*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+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^3*b^2*c^2*d^3-450*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^3* 
d^2-75*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*b^4*c^4*d-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^5*c^5+1260*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2) 
*a^3*b*d^4*x-2996*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^2*b^2*c*d^3*x+1924 
*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a*b^3*c^2*d^2*x-60*((b*x+a)*(d*x+c))^ 
(1/2)*(d*b)^(1/2)*b^4*c^3*d*x-1890*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^4 
*d^4+4620*((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+180*((b*x+a)*(d*x+c))^(1/2)*(d*b) 
^(1/2)*a*b^3*c^3*d+90*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*b^4*c^4)/b^5/d^2 
/((b*x+a)*(d*x+c))^(1/2)/(d*b)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 706, normalized size of antiderivative = 2.25 \[ \int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx=\left [-\frac {15 \, {\left (3 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 150 \, a^{3} b^{2} c^{2} d^{3} + 175 \, a^{4} b c d^{4} - 63 \, a^{5} d^{5}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (384 \, b^{5} d^{5} x^{4} - 45 \, b^{5} c^{4} d - 90 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 2310 \, a^{3} b^{2} c d^{4} + 945 \, a^{4} b d^{5} + 144 \, {\left (7 \, b^{5} c d^{4} - 3 \, a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (93 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 63 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (15 \, b^{5} c^{3} d^{2} - 481 \, a b^{4} c^{2} d^{3} + 749 \, a^{2} b^{3} c d^{4} - 315 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, b^{6} d^{3}}, -\frac {15 \, {\left (3 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 150 \, a^{3} b^{2} c^{2} d^{3} + 175 \, a^{4} b c d^{4} - 63 \, a^{5} d^{5}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (384 \, b^{5} d^{5} x^{4} - 45 \, b^{5} c^{4} d - 90 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 2310 \, a^{3} b^{2} c d^{4} + 945 \, a^{4} b d^{5} + 144 \, {\left (7 \, b^{5} c d^{4} - 3 \, a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (93 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 63 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (15 \, b^{5} c^{3} d^{2} - 481 \, a b^{4} c^{2} d^{3} + 749 \, a^{2} b^{3} c d^{4} - 315 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, b^{6} d^{3}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^2*(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. 831 vs. \(2 (270) = 540\).

Time = 0.22 (sec) , antiderivative size = 831, normalized size of antiderivative = 2.65 \[ \int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx =\text {Too large to display} \] Input:

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

Output:

1/1920*(20*(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))*c*d*abs(b)/b^2 + 80*(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(-sqrt(b*d)*sqrt 
(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*d^2))*c^2*abs 
(b)/b^4 + (sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(6*(8*b*x + 8*a + (b* 
c*d^7 - 41*a*d^8)/d^8)*(b*x + a) - (7*b^2*c^2*d^6 + 26*a*b*c*d^7 - 513*a^2 
*d^8)/d^8)*(b*x + a) + 5*(7*b^3*c^3*d^5 + 19*a*b^2*c^2*d^6 + 37*a^2*b*c*d^ 
7 - 447*a^3*d^8)/d^8)*(b*x + a) - 15*(7*b^4*c^4*d^4 + 12*a*b^3*c^3*d^5 + 1 
8*a^2*b^2*c^2*d^6 + 28*a^3*b*c*d^7 - 193*a^4*d^8)/d^8)*sqrt(b*x + a) - 15* 
(7*b^6*c^5 + 5*a*b^5*c^4*d + 6*a^2*b^4*c^3*d^2 + 10*a^3*b^3*c^2*d^3 + 35*a 
^4*b^2*c*d^4 - 63*a^5*b*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)*d^4))*d^2*abs(b)/b^6)/b
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 651, normalized size of antiderivative = 2.07 \[ \int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx=\frac {945 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{4} b \,d^{5}-2310 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{3} b^{2} c \,d^{4}-630 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{3} b^{2} d^{5} x +1564 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{3} c^{2} d^{3}+1498 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{3} c \,d^{4} x +504 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{3} d^{5} x^{2}-90 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{4} c^{3} d^{2}-962 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{4} c^{2} d^{3} x -1184 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{4} c \,d^{4} x^{2}-432 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{4} d^{5} x^{3}-45 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} c^{4} d +30 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} c^{3} d^{2} x +744 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} c^{2} d^{3} x^{2}+1008 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} c \,d^{4} x^{3}+384 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} d^{5} x^{4}-945 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{5} d^{5}+2625 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{4} b c \,d^{4}-2250 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{3} b^{2} c^{2} d^{3}+450 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} b^{3} c^{3} d^{2}+75 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a \,b^{4} c^{4} d +45 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{5} c^{5}}{1920 b^{6} d^{3}} \] Input:

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

Output:

(945*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b*d**5 - 2310*sqrt(c + d*x)*sqrt(a + 
 b*x)*a**3*b**2*c*d**4 - 630*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 + 1498*sqrt(c + d*x 
)*sqrt(a + b*x)*a**2*b**3*c*d**4*x + 504*sqrt(c + d*x)*sqrt(a + b*x)*a**2* 
b**3*d**5*x**2 - 90*sqrt(c + d*x)*sqrt(a + b*x)*a*b**4*c**3*d**2 - 962*sqr 
t(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 - 432*sqrt(c + d*x)*sqrt(a + b*x)*a*b**4*d**5*x**3 
 - 45*sqrt(c + d*x)*sqrt(a + b*x)*b**5*c**4*d + 30*sqrt(c + d*x)*sqrt(a + 
b*x)*b**5*c**3*d**2*x + 744*sqrt(c + d*x)*sqrt(a + b*x)*b**5*c**2*d**3*x** 
2 + 1008*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 - 945*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 + 2625*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 - 2250*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**2*c**2*d**3 + 450*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 + 75*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*s 
qrt(c + d*x))/sqrt(a*d - b*c))*a*b**4*c**4*d + 45*sqrt(d)*sqrt(b)*log((sqr 
t(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b**5*c**5)/(1 
920*b**6*d**3)