\(\int \frac {x^5}{\sqrt {c+d x} (a x+b x^2)^{3/2}} \, dx\) [202]

Optimal result

Integrand size = 26, antiderivative size = 406 \[ \int \frac {x^5}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\frac {2 \left (8 b^2 c^2+13 a b c d+24 a^2 d^2\right ) x \sqrt {c+d x}}{15 b^3 d^3 \sqrt {a x+b x^2}}-\frac {4 (2 b c+3 a d) x^2 \sqrt {c+d x}}{15 b^2 d^2 \sqrt {a x+b x^2}}+\frac {2 x^3 \sqrt {c+d x}}{5 b d \sqrt {a x+b x^2}}-\frac {2 \sqrt {a} \left (8 b^3 c^3+9 a b^2 c^2 d+16 a^2 b c d^2-48 a^3 d^3\right ) \sqrt {x} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 b^{7/2} d^3 (b c-a d) \sqrt {\frac {a (c+d x)}{c (a+b x)}} \sqrt {a x+b x^2}}+\frac {2 a^{3/2} \left (4 b^2 c^2+5 a b c d-24 a^2 d^2\right ) \sqrt {x} \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 b^{7/2} d^2 (b c-a d) \sqrt {\frac {a (c+d x)}{c (a+b x)}} \sqrt {a x+b x^2}} \] Output:

2/15*(24*a^2*d^2+13*a*b*c*d+8*b^2*c^2)*x*(d*x+c)^(1/2)/b^3/d^3/(b*x^2+a*x) 
^(1/2)-4/15*(3*a*d+2*b*c)*x^2*(d*x+c)^(1/2)/b^2/d^2/(b*x^2+a*x)^(1/2)+2/5* 
x^3*(d*x+c)^(1/2)/b/d/(b*x^2+a*x)^(1/2)-2/15*a^(1/2)*(-48*a^3*d^3+16*a^2*b 
*c*d^2+9*a*b^2*c^2*d+8*b^3*c^3)*x^(1/2)*(d*x+c)^(1/2)*EllipticE(b^(1/2)*x^ 
(1/2)/a^(1/2)/(1+b*x/a)^(1/2),(1-a*d/b/c)^(1/2))/b^(7/2)/d^3/(-a*d+b*c)/(a 
*(d*x+c)/c/(b*x+a))^(1/2)/(b*x^2+a*x)^(1/2)+2/15*a^(3/2)*(-24*a^2*d^2+5*a* 
b*c*d+4*b^2*c^2)*x^(1/2)*(d*x+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x^(1 
/2)/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(7/2)/d^2/(-a*d+b*c)/(a*(d*x+c)/c/(b*x+a 
))^(1/2)/(b*x^2+a*x)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 20.49 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.94 \[ \int \frac {x^5}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\frac {2 \left (a d x (c+d x) \left (15 a^3 d^2-(b c-a d) (4 b c+9 a d) (a+b x)+3 b d (b c-a d) x (a+b x)\right )+\sqrt {\frac {a}{b}} \left (\sqrt {\frac {a}{b}} \left (8 b^3 c^3+9 a b^2 c^2 d+16 a^2 b c d^2-48 a^3 d^3\right ) (a+b x) (c+d x)+i a d \left (8 b^3 c^3+9 a b^2 c^2 d+16 a^2 b c d^2-48 a^3 d^3\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )-4 i a d \left (b^3 c^3+a b^2 c^2 d+10 a^2 b c d^2-12 a^3 d^3\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )\right )\right )}{15 a b^3 d^3 (b c-a d) \sqrt {x (a+b x)} \sqrt {c+d x}} \] Input:

Integrate[x^5/(Sqrt[c + d*x]*(a*x + b*x^2)^(3/2)),x]
 

Output:

(2*(a*d*x*(c + d*x)*(15*a^3*d^2 - (b*c - a*d)*(4*b*c + 9*a*d)*(a + b*x) + 
3*b*d*(b*c - a*d)*x*(a + b*x)) + Sqrt[a/b]*(Sqrt[a/b]*(8*b^3*c^3 + 9*a*b^2 
*c^2*d + 16*a^2*b*c*d^2 - 48*a^3*d^3)*(a + b*x)*(c + d*x) + I*a*d*(8*b^3*c 
^3 + 9*a*b^2*c^2*d + 16*a^2*b*c*d^2 - 48*a^3*d^3)*Sqrt[1 + a/(b*x)]*Sqrt[1 
 + c/(d*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[a/b]/Sqrt[x]], (b*c)/(a*d)] - 
 (4*I)*a*d*(b^3*c^3 + a*b^2*c^2*d + 10*a^2*b*c*d^2 - 12*a^3*d^3)*Sqrt[1 + 
a/(b*x)]*Sqrt[1 + c/(d*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[a/b]/Sqrt[x]], 
 (b*c)/(a*d)])))/(15*a*b^3*d^3*(b*c - a*d)*Sqrt[x*(a + b*x)]*Sqrt[c + d*x] 
)
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {1261, 109, 27, 171, 27, 171, 27, 176, 122, 120, 127, 126}

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.

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

\(\Big \downarrow \) 1261

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

\(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a x^{5/2} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {2 \int \frac {\sqrt {x} \left (3 a c (b c-6 a d)+\left (4 b^2 c^2+5 a b d c-24 a^2 d^2\right ) x\right )}{2 \sqrt {a+b x} \sqrt {c+d x}}dx}{5 b d}-\frac {2 x^{3/2} \sqrt {a+b x} \sqrt {c+d x} (b c-6 a d)}{5 b d}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a x^{5/2} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {\int \frac {\sqrt {x} \left (3 a c (b c-6 a d)+\left (4 b^2 c^2+5 a b d c-24 a^2 d^2\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x}}dx}{5 b d}-\frac {2 x^{3/2} \sqrt {a+b x} \sqrt {c+d x} (b c-6 a d)}{5 b d}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a x^{5/2} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {\frac {2 \int -\frac {a c \left (4 b^2 c^2+5 a b d c-24 a^2 d^2\right )+\left (8 b^3 c^3+9 a b^2 d c^2+16 a^2 b d^2 c-48 a^3 d^3\right ) x}{2 \sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{3 b d}+\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {c+d x} \left (-\frac {24 a^2 d}{b}+5 a c+\frac {4 b c^2}{d}\right )}{5 b d}-\frac {2 x^{3/2} \sqrt {a+b x} \sqrt {c+d x} (b c-6 a d)}{5 b d}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a x^{5/2} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {c+d x} \left (-\frac {24 a^2 d}{b}+5 a c+\frac {4 b c^2}{d}\right )-\frac {\int \frac {a c \left (4 b^2 c^2+5 a b d c-24 a^2 d^2\right )+\left (8 b^3 c^3+9 a b^2 d c^2+16 a^2 b d^2 c-48 a^3 d^3\right ) x}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{3 b d}}{5 b d}-\frac {2 x^{3/2} \sqrt {a+b x} \sqrt {c+d x} (b c-6 a d)}{5 b d}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a x^{5/2} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {c+d x} \left (-\frac {24 a^2 d}{b}+5 a c+\frac {4 b c^2}{d}\right )-\frac {\frac {\left (-48 a^3 d^3+16 a^2 b c d^2+9 a b^2 c^2 d+8 b^3 c^3\right ) \int \frac {\sqrt {c+d x}}{\sqrt {x} \sqrt {a+b x}}dx}{d}-\frac {c (b c-a d) \left (24 a^2 d^2+13 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}}{3 b d}}{5 b d}-\frac {2 x^{3/2} \sqrt {a+b x} \sqrt {c+d x} (b c-6 a d)}{5 b d}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 122

\(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a x^{5/2} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {c+d x} \left (-\frac {24 a^2 d}{b}+5 a c+\frac {4 b c^2}{d}\right )-\frac {\frac {\sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-48 a^3 d^3+16 a^2 b c d^2+9 a b^2 c^2 d+8 b^3 c^3\right ) \int \frac {\sqrt {\frac {d x}{c}+1}}{\sqrt {x} \sqrt {\frac {b x}{a}+1}}dx}{d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c (b c-a d) \left (24 a^2 d^2+13 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}}{3 b d}}{5 b d}-\frac {2 x^{3/2} \sqrt {a+b x} \sqrt {c+d x} (b c-6 a d)}{5 b d}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 120

\(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a x^{5/2} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {c+d x} \left (-\frac {24 a^2 d}{b}+5 a c+\frac {4 b c^2}{d}\right )-\frac {\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-48 a^3 d^3+16 a^2 b c d^2+9 a b^2 c^2 d+8 b^3 c^3\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c (b c-a d) \left (24 a^2 d^2+13 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}}{3 b d}}{5 b d}-\frac {2 x^{3/2} \sqrt {a+b x} \sqrt {c+d x} (b c-6 a d)}{5 b d}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 127

\(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a x^{5/2} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {c+d x} \left (-\frac {24 a^2 d}{b}+5 a c+\frac {4 b c^2}{d}\right )-\frac {\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-48 a^3 d^3+16 a^2 b c d^2+9 a b^2 c^2 d+8 b^3 c^3\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (b c-a d) \left (24 a^2 d^2+13 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {x} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}}dx}{d \sqrt {a+b x} \sqrt {c+d x}}}{3 b d}}{5 b d}-\frac {2 x^{3/2} \sqrt {a+b x} \sqrt {c+d x} (b c-6 a d)}{5 b d}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 126

\(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a x^{5/2} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {c+d x} \left (-\frac {24 a^2 d}{b}+5 a c+\frac {4 b c^2}{d}\right )-\frac {\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-48 a^3 d^3+16 a^2 b c d^2+9 a b^2 c^2 d+8 b^3 c^3\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {2 \sqrt {-a} c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (b c-a d) \left (24 a^2 d^2+13 a b c d+8 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {c+d x}}}{3 b d}}{5 b d}-\frac {2 x^{3/2} \sqrt {a+b x} \sqrt {c+d x} (b c-6 a d)}{5 b d}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\)

Input:

Int[x^5/(Sqrt[c + d*x]*(a*x + b*x^2)^(3/2)),x]
 

Output:

(x^(3/2)*(a + b*x)^(3/2)*((2*a*x^(5/2)*Sqrt[c + d*x])/(b*(b*c - a*d)*Sqrt[ 
a + b*x]) - ((-2*(b*c - 6*a*d)*x^(3/2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(5*b*d 
) + ((2*(5*a*c + (4*b*c^2)/d - (24*a^2*d)/b)*Sqrt[x]*Sqrt[a + b*x]*Sqrt[c 
+ d*x])/3 - ((2*Sqrt[-a]*(8*b^3*c^3 + 9*a*b^2*c^2*d + 16*a^2*b*c*d^2 - 48* 
a^3*d^3)*Sqrt[1 + (b*x)/a]*Sqrt[c + d*x]*EllipticE[ArcSin[(Sqrt[b]*Sqrt[x] 
)/Sqrt[-a]], (a*d)/(b*c)])/(Sqrt[b]*d*Sqrt[a + b*x]*Sqrt[1 + (d*x)/c]) - ( 
2*Sqrt[-a]*c*(b*c - a*d)*(8*b^2*c^2 + 13*a*b*c*d + 24*a^2*d^2)*Sqrt[1 + (b 
*x)/a]*Sqrt[1 + (d*x)/c]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[-a]], (a* 
d)/(b*c)])/(Sqrt[b]*d*Sqrt[a + b*x]*Sqrt[c + d*x]))/(3*b*d))/(5*b*d))/(b*( 
b*c - a*d))))/(a*x + b*x^2)^(3/2)
 

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 109

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

rule 120

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] 
 :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- 
b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt 
Q[e, 0] &&  !LtQ[-b/d, 0]
 

rule 122

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] 
 :> Simp[Sqrt[e + f*x]*(Sqrt[1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)]) 
)   Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /; FreeQ[{b 
, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])
 

rule 126

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x 
_] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* 
Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & 
& GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
 

rule 127

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x 
_] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x 
]))   Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free 
Q[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])
 

rule 171

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegersQ[2*m, 2*n, 2*p]
 

rule 176

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* 
Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f   Int[Sqrt[e + f*x]/(Sqrt[a + b*x 
]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f   Int[1/(Sqrt[a + b*x]*Sqrt[c 
 + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim 
plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
 

rule 1261

Int[((e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((b_.)*(x_) + (c_.)*(x_)^2) 
^(p_), x_Symbol] :> Simp[(e*x)^m*((b*x + c*x^2)^p/(x^(m + p)*(b + c*x)^p)) 
  Int[x^(m + p)*(f + g*x)^n*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, f, g, m, 
 n}, x] &&  !IGtQ[n, 0]
 

Maple [A] (verified)

Time = 1.75 (sec) , antiderivative size = 589, normalized size of antiderivative = 1.45

method	result	size
elliptic	\(\frac {\sqrt {x \left (b x +a \right ) \left (d x +c \right )}\, \left (-\frac {2 \left (b d \,x^{2}+c b x \right ) a^{3}}{\left (a d -b c \right ) b^{4} \sqrt {\left (x +\frac {a}{b}\right ) \left (b d \,x^{2}+c b x \right )}}+\frac {2 x \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}{5 b^{2} d}+\frac {2 \left (-\frac {a}{b^{2}}-\frac {2 \left (2 a d +2 b c \right )}{5 b^{2} d}\right ) \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}{3 b d}+\frac {2 \left (\frac {c \,a^{3}}{b^{3} \left (a d -b c \right )}-\frac {\left (-\frac {a}{b^{2}}-\frac {2 \left (2 a d +2 b c \right )}{5 b^{2} d}\right ) a c}{3 b d}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{d \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}+\frac {2 \left (\frac {a^{2}}{b^{3}}+\frac {d \,a^{3}}{b^{3} \left (a d -b c \right )}-\frac {3 a c}{5 b^{2} d}-\frac {2 \left (-\frac {a}{b^{2}}-\frac {2 \left (2 a d +2 b c \right )}{5 b^{2} d}\right ) \left (a d +b c \right )}{3 b d}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{d \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}\right )}{\sqrt {x \left (b x +a \right )}\, \sqrt {d x +c}}\)	\(589\)
default	\(-\frac {2 \left (48 \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{4} c \,d^{4}-40 \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{3} b \,c^{2} d^{3}-4 \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} b^{2} c^{3} d^{2}-4 \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a \,b^{3} c^{4} d -48 \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{4} c \,d^{4}+64 \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{3} b \,c^{2} d^{3}-7 \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} b^{2} c^{3} d^{2}-\sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a \,b^{3} c^{4} d -8 \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b^{4} c^{5}-3 a \,b^{3} d^{5} x^{4}+3 b^{4} c \,d^{4} x^{4}+6 a^{2} b^{2} d^{5} x^{3}-5 a \,b^{3} c \,d^{4} x^{3}-b^{4} c^{2} d^{3} x^{3}+24 a^{3} b \,d^{5} x^{2}+a^{2} b^{2} c \,d^{4} x^{2}-6 a \,b^{3} c^{2} d^{3} x^{2}-4 b^{4} c^{3} d^{2} x^{2}+24 a^{3} b c \,d^{4} x -5 a^{2} b^{2} c^{2} d^{3} x -4 a \,b^{3} c^{3} d^{2} x \right ) \sqrt {x \left (b x +a \right )}\, \sqrt {d x +c}}{15 \left (a d -b c \right ) d^{4} b^{4} x \left (b d \,x^{2}+a d x +c b x +a c \right )}\)	\(919\)

Input:

int(x^5/(d*x+c)^(1/2)/(b*x^2+a*x)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

(x*(b*x+a)*(d*x+c))^(1/2)/(x*(b*x+a))^(1/2)/(d*x+c)^(1/2)*(-2*(b*d*x^2+b*c 
*x)/(a*d-b*c)/b^4*a^3/((x+a/b)*(b*d*x^2+b*c*x))^(1/2)+2/5/b^2/d*x*(b*d*x^3 
+a*d*x^2+b*c*x^2+a*c*x)^(1/2)+2/3*(-a/b^2-2/5/b^2/d*(2*a*d+2*b*c))/b/d*(b* 
d*x^3+a*d*x^2+b*c*x^2+a*c*x)^(1/2)+2*(1/b^3*c/(a*d-b*c)*a^3-1/3*(-a/b^2-2/ 
5/b^2/d*(2*a*d+2*b*c))/b/d*a*c)*c/d*((x+c/d)/c*d)^(1/2)*((x+a/b)/(-c/d+a/b 
))^(1/2)*(-1/c*x*d)^(1/2)/(b*d*x^3+a*d*x^2+b*c*x^2+a*c*x)^(1/2)*EllipticF( 
((x+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b))^(1/2))+2*(1/b^3*a^2+d/b^3*a^3/(a*d-b 
*c)-3/5/b^2/d*a*c-2/3*(-a/b^2-2/5/b^2/d*(2*a*d+2*b*c))/b/d*(a*d+b*c))*c/d* 
((x+c/d)/c*d)^(1/2)*((x+a/b)/(-c/d+a/b))^(1/2)*(-1/c*x*d)^(1/2)/(b*d*x^3+a 
*d*x^2+b*c*x^2+a*c*x)^(1/2)*((-c/d+a/b)*EllipticE(((x+c/d)/c*d)^(1/2),(-c/ 
d/(-c/d+a/b))^(1/2))-a/b*EllipticF(((x+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b))^( 
1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.58 \[ \int \frac {x^5}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left ({\left (8 \, a b^{4} c^{4} + 5 \, a^{2} b^{3} c^{3} d + 10 \, a^{3} b^{2} c^{2} d^{2} + 40 \, a^{4} b c d^{3} - 48 \, a^{5} d^{4} + {\left (8 \, b^{5} c^{4} + 5 \, a b^{4} c^{3} d + 10 \, a^{2} b^{3} c^{2} d^{2} + 40 \, a^{3} b^{2} c d^{3} - 48 \, a^{4} b d^{4}\right )} x\right )} \sqrt {b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) + 3 \, {\left (8 \, a b^{4} c^{3} d + 9 \, a^{2} b^{3} c^{2} d^{2} + 16 \, a^{3} b^{2} c d^{3} - 48 \, a^{4} b d^{4} + {\left (8 \, b^{5} c^{3} d + 9 \, a b^{4} c^{2} d^{2} + 16 \, a^{2} b^{3} c d^{3} - 48 \, a^{3} b^{2} d^{4}\right )} x\right )} \sqrt {b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right ) + 3 \, {\left (4 \, a b^{4} c^{2} d^{2} + 5 \, a^{2} b^{3} c d^{3} - 24 \, a^{3} b^{2} d^{4} - 3 \, {\left (b^{5} c d^{3} - a b^{4} d^{4}\right )} x^{2} + 2 \, {\left (2 \, b^{5} c^{2} d^{2} + a b^{4} c d^{3} - 3 \, a^{2} b^{3} d^{4}\right )} x\right )} \sqrt {b x^{2} + a x} \sqrt {d x + c}\right )}}{45 \, {\left (a b^{6} c d^{4} - a^{2} b^{5} d^{5} + {\left (b^{7} c d^{4} - a b^{6} d^{5}\right )} x\right )}} \] Input:

integrate(x^5/(d*x+c)^(1/2)/(b*x^2+a*x)^(3/2),x, algorithm="fricas")
 

Output:

-2/45*((8*a*b^4*c^4 + 5*a^2*b^3*c^3*d + 10*a^3*b^2*c^2*d^2 + 40*a^4*b*c*d^ 
3 - 48*a^5*d^4 + (8*b^5*c^4 + 5*a*b^4*c^3*d + 10*a^2*b^3*c^2*d^2 + 40*a^3* 
b^2*c*d^3 - 48*a^4*b*d^4)*x)*sqrt(b*d)*weierstrassPInverse(4/3*(b^2*c^2 - 
a*b*c*d + a^2*d^2)/(b^2*d^2), -4/27*(2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c 
*d^2 + 2*a^3*d^3)/(b^3*d^3), 1/3*(3*b*d*x + b*c + a*d)/(b*d)) + 3*(8*a*b^4 
*c^3*d + 9*a^2*b^3*c^2*d^2 + 16*a^3*b^2*c*d^3 - 48*a^4*b*d^4 + (8*b^5*c^3* 
d + 9*a*b^4*c^2*d^2 + 16*a^2*b^3*c*d^3 - 48*a^3*b^2*d^4)*x)*sqrt(b*d)*weie 
rstrassZeta(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d^2), -4/27*(2*b^3*c^3 
- 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/(b^3*d^3), weierstrassPInvers 
e(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d^2), -4/27*(2*b^3*c^3 - 3*a*b^2* 
c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/(b^3*d^3), 1/3*(3*b*d*x + b*c + a*d)/(b 
*d))) + 3*(4*a*b^4*c^2*d^2 + 5*a^2*b^3*c*d^3 - 24*a^3*b^2*d^4 - 3*(b^5*c*d 
^3 - a*b^4*d^4)*x^2 + 2*(2*b^5*c^2*d^2 + a*b^4*c*d^3 - 3*a^2*b^3*d^4)*x)*s 
qrt(b*x^2 + a*x)*sqrt(d*x + c))/(a*b^6*c*d^4 - a^2*b^5*d^5 + (b^7*c*d^4 - 
a*b^6*d^5)*x)
 

Sympy [F]

\[ \int \frac {x^5}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {x^{5}}{\left (x \left (a + b x\right )\right )^{\frac {3}{2}} \sqrt {c + d x}}\, dx \] Input:

integrate(x**5/(d*x+c)**(1/2)/(b*x**2+a*x)**(3/2),x)
 

Output:

Integral(x**5/((x*(a + b*x))**(3/2)*sqrt(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {x^5}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\int { \frac {x^{5}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} \sqrt {d x + c}} \,d x } \] Input:

integrate(x^5/(d*x+c)^(1/2)/(b*x^2+a*x)^(3/2),x, algorithm="maxima")
 

Output:

integrate(x^5/((b*x^2 + a*x)^(3/2)*sqrt(d*x + c)), x)
 

Giac [F]

\[ \int \frac {x^5}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\int { \frac {x^{5}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} \sqrt {d x + c}} \,d x } \] Input:

integrate(x^5/(d*x+c)^(1/2)/(b*x^2+a*x)^(3/2),x, algorithm="giac")
 

Output:

integrate(x^5/((b*x^2 + a*x)^(3/2)*sqrt(d*x + c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {x^5}{{\left (b\,x^2+a\,x\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \] Input:

int(x^5/((a*x + b*x^2)^(3/2)*(c + d*x)^(1/2)),x)
 

Output:

int(x^5/((a*x + b*x^2)^(3/2)*(c + d*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^5}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int(x^5/(d*x+c)^(1/2)/(b*x^2+a*x)^(3/2),x)
 

Output:

(18*sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*a*c*d - 12*sqrt(x)*sqrt(c + d*x)*s 
qrt(a + b*x)*a*d**2*x + 12*sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*b*c**2 - 8* 
sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*b*c*d*x + 6*sqrt(x)*sqrt(c + d*x)*sqrt 
(a + b*x)*b*d**2*x**2 - 9*int((sqrt(c + d*x)*sqrt(a + b*x))/(sqrt(x)*a**2* 
c + sqrt(x)*a**2*d*x + 2*sqrt(x)*a*b*c*x + 2*sqrt(x)*a*b*d*x**2 + sqrt(x)* 
b**2*c*x**2 + sqrt(x)*b**2*d*x**3),x)*a**3*c**2*d - 6*int((sqrt(c + d*x)*s 
qrt(a + b*x))/(sqrt(x)*a**2*c + sqrt(x)*a**2*d*x + 2*sqrt(x)*a*b*c*x + 2*s 
qrt(x)*a*b*d*x**2 + sqrt(x)*b**2*c*x**2 + sqrt(x)*b**2*d*x**3),x)*a**2*b*c 
**3 - 9*int((sqrt(c + d*x)*sqrt(a + b*x))/(sqrt(x)*a**2*c + sqrt(x)*a**2*d 
*x + 2*sqrt(x)*a*b*c*x + 2*sqrt(x)*a*b*d*x**2 + sqrt(x)*b**2*c*x**2 + sqrt 
(x)*b**2*d*x**3),x)*a**2*b*c**2*d*x - 6*int((sqrt(c + d*x)*sqrt(a + b*x))/ 
(sqrt(x)*a**2*c + sqrt(x)*a**2*d*x + 2*sqrt(x)*a*b*c*x + 2*sqrt(x)*a*b*d*x 
**2 + sqrt(x)*b**2*c*x**2 + sqrt(x)*b**2*d*x**3),x)*a*b**2*c**3*x + 24*int 
((sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*x)/(a**2*c + a**2*d*x + 2*a*b*c*x + 
2*a*b*d*x**2 + b**2*c*x**2 + b**2*d*x**3),x)*a**3*d**3 + 4*int((sqrt(x)*sq 
rt(c + d*x)*sqrt(a + b*x)*x)/(a**2*c + a**2*d*x + 2*a*b*c*x + 2*a*b*d*x**2 
 + b**2*c*x**2 + b**2*d*x**3),x)*a**2*b*c*d**2 + 24*int((sqrt(x)*sqrt(c + 
d*x)*sqrt(a + b*x)*x)/(a**2*c + a**2*d*x + 2*a*b*c*x + 2*a*b*d*x**2 + b**2 
*c*x**2 + b**2*d*x**3),x)*a**2*b*d**3*x + 2*int((sqrt(x)*sqrt(c + d*x)*sqr 
t(a + b*x)*x)/(a**2*c + a**2*d*x + 2*a*b*c*x + 2*a*b*d*x**2 + b**2*c*x*...