\(\int \frac {x^3 (c+d x)^2}{\sqrt {a x+b x^2}} \, dx\) [134]

Optimal result

Integrand size = 24, antiderivative size = 249 \[ \int \frac {x^3 (c+d x)^2}{\sqrt {a x+b x^2}} \, dx=\frac {a^2 \left (80 b^2 c^2-7 a d (20 b c-9 a d)\right ) \sqrt {a x+b x^2}}{128 b^5}-\frac {a \left (80 b^2 c^2-7 a d (20 b c-9 a d)\right ) x \sqrt {a x+b x^2}}{192 b^4}+\frac {\left (80 c^2-\frac {7 a d (20 b c-9 a d)}{b^2}\right ) x^2 \sqrt {a x+b x^2}}{240 b}+\frac {d (20 b c-9 a d) x^3 \sqrt {a x+b x^2}}{40 b^2}+\frac {d^2 x^4 \sqrt {a x+b x^2}}{5 b}-\frac {a^3 \left (80 b^2 c^2-7 a d (20 b c-9 a d)\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{128 b^{11/2}} \] Output:

1/128*a^2*(80*b^2*c^2-7*a*d*(-9*a*d+20*b*c))*(b*x^2+a*x)^(1/2)/b^5-1/192*a 
*(80*b^2*c^2-7*a*d*(-9*a*d+20*b*c))*x*(b*x^2+a*x)^(1/2)/b^4+1/240*(80*c^2- 
7*a*d*(-9*a*d+20*b*c)/b^2)*x^2*(b*x^2+a*x)^(1/2)/b+1/40*d*(-9*a*d+20*b*c)* 
x^3*(b*x^2+a*x)^(1/2)/b^2+1/5*d^2*x^4*(b*x^2+a*x)^(1/2)/b-1/128*a^3*(80*b^ 
2*c^2-7*a*d*(-9*a*d+20*b*c))*arctanh(b^(1/2)*x/(b*x^2+a*x)^(1/2))/b^(11/2)
 

Mathematica [A] (verified)

Time = 0.87 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.98 \[ \int \frac {x^3 (c+d x)^2}{\sqrt {a x+b x^2}} \, dx=\frac {x (a+b x) \left (1200 a^2 b^2 c^2-2100 a^3 b c d+945 a^4 d^2-800 a b^3 c^2 x+1400 a^2 b^2 c d x-630 a^3 b d^2 x+640 b^4 c^2 x^2-1120 a b^3 c d x^2+504 a^2 b^2 d^2 x^2+960 b^4 c d x^3-432 a b^3 d^2 x^3+384 b^4 d^2 x^4\right )}{1920 b^5 \sqrt {x (a+b x)}}-\frac {a^3 \left (80 b^2 c^2-140 a b c d+63 a^2 d^2\right ) \sqrt {x} \sqrt {a+b x} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )}{64 b^{11/2} \sqrt {x (a+b x)}} \] Input:

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

Output:

(x*(a + b*x)*(1200*a^2*b^2*c^2 - 2100*a^3*b*c*d + 945*a^4*d^2 - 800*a*b^3* 
c^2*x + 1400*a^2*b^2*c*d*x - 630*a^3*b*d^2*x + 640*b^4*c^2*x^2 - 1120*a*b^ 
3*c*d*x^2 + 504*a^2*b^2*d^2*x^2 + 960*b^4*c*d*x^3 - 432*a*b^3*d^2*x^3 + 38 
4*b^4*d^2*x^4))/(1920*b^5*Sqrt[x*(a + b*x)]) - (a^3*(80*b^2*c^2 - 140*a*b* 
c*d + 63*a^2*d^2)*Sqrt[x]*Sqrt[a + b*x]*ArcTanh[(Sqrt[b]*Sqrt[x])/(-Sqrt[a 
] + Sqrt[a + b*x])])/(64*b^(11/2)*Sqrt[x*(a + b*x)])
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.83, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1262, 27, 1221, 1134, 1134, 1160, 1091, 219}

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.

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

\(\Big \downarrow \) 1262

\(\displaystyle \frac {\int \frac {x^3 \left (10 b c^2+d (20 b c-9 a d) x\right )}{2 \sqrt {b x^2+a x}}dx}{5 b}+\frac {d^2 x^4 \sqrt {a x+b x^2}}{5 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {x^3 \left (10 b c^2+d (20 b c-9 a d) x\right )}{\sqrt {b x^2+a x}}dx}{10 b}+\frac {d^2 x^4 \sqrt {a x+b x^2}}{5 b}\)

\(\Big \downarrow \) 1221

\(\displaystyle \frac {\frac {\left (63 a^2 d^2-140 a b c d+80 b^2 c^2\right ) \int \frac {x^3}{\sqrt {b x^2+a x}}dx}{8 b}+\frac {d x^3 \sqrt {a x+b x^2} (20 b c-9 a d)}{4 b}}{10 b}+\frac {d^2 x^4 \sqrt {a x+b x^2}}{5 b}\)

\(\Big \downarrow \) 1134

\(\displaystyle \frac {\frac {\left (63 a^2 d^2-140 a b c d+80 b^2 c^2\right ) \left (\frac {x^2 \sqrt {a x+b x^2}}{3 b}-\frac {5 a \int \frac {x^2}{\sqrt {b x^2+a x}}dx}{6 b}\right )}{8 b}+\frac {d x^3 \sqrt {a x+b x^2} (20 b c-9 a d)}{4 b}}{10 b}+\frac {d^2 x^4 \sqrt {a x+b x^2}}{5 b}\)

\(\Big \downarrow \) 1134

\(\displaystyle \frac {\frac {\left (63 a^2 d^2-140 a b c d+80 b^2 c^2\right ) \left (\frac {x^2 \sqrt {a x+b x^2}}{3 b}-\frac {5 a \left (\frac {x \sqrt {a x+b x^2}}{2 b}-\frac {3 a \int \frac {x}{\sqrt {b x^2+a x}}dx}{4 b}\right )}{6 b}\right )}{8 b}+\frac {d x^3 \sqrt {a x+b x^2} (20 b c-9 a d)}{4 b}}{10 b}+\frac {d^2 x^4 \sqrt {a x+b x^2}}{5 b}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {\frac {\left (63 a^2 d^2-140 a b c d+80 b^2 c^2\right ) \left (\frac {x^2 \sqrt {a x+b x^2}}{3 b}-\frac {5 a \left (\frac {x \sqrt {a x+b x^2}}{2 b}-\frac {3 a \left (\frac {\sqrt {a x+b x^2}}{b}-\frac {a \int \frac {1}{\sqrt {b x^2+a x}}dx}{2 b}\right )}{4 b}\right )}{6 b}\right )}{8 b}+\frac {d x^3 \sqrt {a x+b x^2} (20 b c-9 a d)}{4 b}}{10 b}+\frac {d^2 x^4 \sqrt {a x+b x^2}}{5 b}\)

\(\Big \downarrow \) 1091

\(\displaystyle \frac {\frac {\left (63 a^2 d^2-140 a b c d+80 b^2 c^2\right ) \left (\frac {x^2 \sqrt {a x+b x^2}}{3 b}-\frac {5 a \left (\frac {x \sqrt {a x+b x^2}}{2 b}-\frac {3 a \left (\frac {\sqrt {a x+b x^2}}{b}-\frac {a \int \frac {1}{1-\frac {b x^2}{b x^2+a x}}d\frac {x}{\sqrt {b x^2+a x}}}{b}\right )}{4 b}\right )}{6 b}\right )}{8 b}+\frac {d x^3 \sqrt {a x+b x^2} (20 b c-9 a d)}{4 b}}{10 b}+\frac {d^2 x^4 \sqrt {a x+b x^2}}{5 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\left (\frac {x^2 \sqrt {a x+b x^2}}{3 b}-\frac {5 a \left (\frac {x \sqrt {a x+b x^2}}{2 b}-\frac {3 a \left (\frac {\sqrt {a x+b x^2}}{b}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{b^{3/2}}\right )}{4 b}\right )}{6 b}\right ) \left (63 a^2 d^2-140 a b c d+80 b^2 c^2\right )}{8 b}+\frac {d x^3 \sqrt {a x+b x^2} (20 b c-9 a d)}{4 b}}{10 b}+\frac {d^2 x^4 \sqrt {a x+b x^2}}{5 b}\)

Input:

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

Output:

(d^2*x^4*Sqrt[a*x + b*x^2])/(5*b) + ((d*(20*b*c - 9*a*d)*x^3*Sqrt[a*x + b* 
x^2])/(4*b) + ((80*b^2*c^2 - 140*a*b*c*d + 63*a^2*d^2)*((x^2*Sqrt[a*x + b* 
x^2])/(3*b) - (5*a*((x*Sqrt[a*x + b*x^2])/(2*b) - (3*a*(Sqrt[a*x + b*x^2]/ 
b - (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/b^(3/2)))/(4*b)))/(6*b)))/( 
8*b))/(10*b)
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1091

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[Int[1/(1 
 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
 

rule 1134

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[(m + p)*((2*c*d - b*e)/(c*(m + 2*p + 1)))   Int[(d + e*x)^ 
(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[ 
c*d^2 - b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2 
*p]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

rule 1221

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1 
)/(c*(m + 2*p + 2))), x] + Simp[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c 
*f - b*g))/(c*e*(m + 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] 
 && NeQ[m + 2*p + 2, 0]
 

rule 1262

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

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.64

method	result	size
pseudoelliptic	\(-\frac {63 \left (a^{3} \left (a^{2} d^{2}-\frac {20}{9} a b c d +\frac {80}{63} b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )-\left (\frac {80 \left (\frac {21}{50} d^{2} x^{2}+\frac {7}{6} c d x +c^{2}\right ) a^{2} b^{\frac {5}{2}}}{63}-\frac {160 x a \left (\frac {27}{50} d^{2} x^{2}+\frac {7}{5} c d x +c^{2}\right ) b^{\frac {7}{2}}}{189}+\frac {128 x^{2} \left (\frac {3}{5} d^{2} x^{2}+\frac {3}{2} c d x +c^{2}\right ) b^{\frac {9}{2}}}{189}+d \left (\left (-\frac {2 d x}{3}-\frac {20 c}{9}\right ) b^{\frac {3}{2}}+\sqrt {b}\, a d \right ) a^{3}\right ) \sqrt {x \left (b x +a \right )}\right )}{128 b^{\frac {11}{2}}}\)	\(160\)
risch	\(\frac {\left (384 b^{4} d^{2} x^{4}-432 a \,b^{3} d^{2} x^{3}+960 b^{4} c d \,x^{3}+504 a^{2} b^{2} d^{2} x^{2}-1120 a \,b^{3} c d \,x^{2}+640 c^{2} x^{2} b^{4}-630 a^{3} b \,d^{2} x +1400 a^{2} b^{2} c d x -800 a \,b^{3} c^{2} x +945 a^{4} d^{2}-2100 a^{3} d c b +1200 a^{2} b^{2} c^{2}\right ) x \left (b x +a \right )}{1920 b^{5} \sqrt {x \left (b x +a \right )}}-\frac {a^{3} \left (63 a^{2} d^{2}-140 a b c d +80 b^{2} c^{2}\right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{256 b^{\frac {11}{2}}}\)	\(205\)
default	\(c^{2} \left (\frac {x^{2} \sqrt {b \,x^{2}+a x}}{3 b}-\frac {5 a \left (\frac {x \sqrt {b \,x^{2}+a x}}{2 b}-\frac {3 a \left (\frac {\sqrt {b \,x^{2}+a x}}{b}-\frac {a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )+d^{2} \left (\frac {x^{4} \sqrt {b \,x^{2}+a x}}{5 b}-\frac {9 a \left (\frac {x^{3} \sqrt {b \,x^{2}+a x}}{4 b}-\frac {7 a \left (\frac {x^{2} \sqrt {b \,x^{2}+a x}}{3 b}-\frac {5 a \left (\frac {x \sqrt {b \,x^{2}+a x}}{2 b}-\frac {3 a \left (\frac {\sqrt {b \,x^{2}+a x}}{b}-\frac {a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )}{8 b}\right )}{10 b}\right )+2 c d \left (\frac {x^{3} \sqrt {b \,x^{2}+a x}}{4 b}-\frac {7 a \left (\frac {x^{2} \sqrt {b \,x^{2}+a x}}{3 b}-\frac {5 a \left (\frac {x \sqrt {b \,x^{2}+a x}}{2 b}-\frac {3 a \left (\frac {\sqrt {b \,x^{2}+a x}}{b}-\frac {a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )}{8 b}\right )\)	\(380\)

Input:

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

Output:

-63/128*(a^3*(a^2*d^2-20/9*a*b*c*d+80/63*b^2*c^2)*arctanh((x*(b*x+a))^(1/2 
)/x/b^(1/2))-(80/63*(21/50*d^2*x^2+7/6*c*d*x+c^2)*a^2*b^(5/2)-160/189*x*a* 
(27/50*d^2*x^2+7/5*c*d*x+c^2)*b^(7/2)+128/189*x^2*(3/5*d^2*x^2+3/2*c*d*x+c 
^2)*b^(9/2)+d*((-2/3*d*x-20/9*c)*b^(3/2)+b^(1/2)*a*d)*a^3)*(x*(b*x+a))^(1/ 
2))/b^(11/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.67 \[ \int \frac {x^3 (c+d x)^2}{\sqrt {a x+b x^2}} \, dx=\left [\frac {15 \, {\left (80 \, a^{3} b^{2} c^{2} - 140 \, a^{4} b c d + 63 \, a^{5} d^{2}\right )} \sqrt {b} \log \left (2 \, b x + a - 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + 2 \, {\left (384 \, b^{5} d^{2} x^{4} + 1200 \, a^{2} b^{3} c^{2} - 2100 \, a^{3} b^{2} c d + 945 \, a^{4} b d^{2} + 48 \, {\left (20 \, b^{5} c d - 9 \, a b^{4} d^{2}\right )} x^{3} + 8 \, {\left (80 \, b^{5} c^{2} - 140 \, a b^{4} c d + 63 \, a^{2} b^{3} d^{2}\right )} x^{2} - 10 \, {\left (80 \, a b^{4} c^{2} - 140 \, a^{2} b^{3} c d + 63 \, a^{3} b^{2} d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}}{3840 \, b^{6}}, \frac {15 \, {\left (80 \, a^{3} b^{2} c^{2} - 140 \, a^{4} b c d + 63 \, a^{5} d^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) + {\left (384 \, b^{5} d^{2} x^{4} + 1200 \, a^{2} b^{3} c^{2} - 2100 \, a^{3} b^{2} c d + 945 \, a^{4} b d^{2} + 48 \, {\left (20 \, b^{5} c d - 9 \, a b^{4} d^{2}\right )} x^{3} + 8 \, {\left (80 \, b^{5} c^{2} - 140 \, a b^{4} c d + 63 \, a^{2} b^{3} d^{2}\right )} x^{2} - 10 \, {\left (80 \, a b^{4} c^{2} - 140 \, a^{2} b^{3} c d + 63 \, a^{3} b^{2} d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}}{1920 \, b^{6}}\right ] \] Input:

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

Output:

[1/3840*(15*(80*a^3*b^2*c^2 - 140*a^4*b*c*d + 63*a^5*d^2)*sqrt(b)*log(2*b* 
x + a - 2*sqrt(b*x^2 + a*x)*sqrt(b)) + 2*(384*b^5*d^2*x^4 + 1200*a^2*b^3*c 
^2 - 2100*a^3*b^2*c*d + 945*a^4*b*d^2 + 48*(20*b^5*c*d - 9*a*b^4*d^2)*x^3 
+ 8*(80*b^5*c^2 - 140*a*b^4*c*d + 63*a^2*b^3*d^2)*x^2 - 10*(80*a*b^4*c^2 - 
 140*a^2*b^3*c*d + 63*a^3*b^2*d^2)*x)*sqrt(b*x^2 + a*x))/b^6, 1/1920*(15*( 
80*a^3*b^2*c^2 - 140*a^4*b*c*d + 63*a^5*d^2)*sqrt(-b)*arctan(sqrt(b*x^2 + 
a*x)*sqrt(-b)/(b*x + a)) + (384*b^5*d^2*x^4 + 1200*a^2*b^3*c^2 - 2100*a^3* 
b^2*c*d + 945*a^4*b*d^2 + 48*(20*b^5*c*d - 9*a*b^4*d^2)*x^3 + 8*(80*b^5*c^ 
2 - 140*a*b^4*c*d + 63*a^2*b^3*d^2)*x^2 - 10*(80*a*b^4*c^2 - 140*a^2*b^3*c 
*d + 63*a^3*b^2*d^2)*x)*sqrt(b*x^2 + a*x))/b^6]
 

Sympy [A] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.30 \[ \int \frac {x^3 (c+d x)^2}{\sqrt {a x+b x^2}} \, dx=\begin {cases} - \frac {5 a^{3} \left (- \frac {7 a \left (- \frac {9 a d^{2}}{10 b} + 2 c d\right )}{8 b} + c^{2}\right ) \left (\begin {cases} \frac {\log {\left (a + 2 \sqrt {b} \sqrt {a x + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: \frac {a^{2}}{b} \neq 0 \\\frac {\left (\frac {a}{2 b} + x\right ) \log {\left (\frac {a}{2 b} + x \right )}}{\sqrt {b \left (\frac {a}{2 b} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{16 b^{3}} + \sqrt {a x + b x^{2}} \cdot \left (\frac {5 a^{2} \left (- \frac {7 a \left (- \frac {9 a d^{2}}{10 b} + 2 c d\right )}{8 b} + c^{2}\right )}{8 b^{3}} - \frac {5 a x \left (- \frac {7 a \left (- \frac {9 a d^{2}}{10 b} + 2 c d\right )}{8 b} + c^{2}\right )}{12 b^{2}} + \frac {d^{2} x^{4}}{5 b} + \frac {x^{3} \left (- \frac {9 a d^{2}}{10 b} + 2 c d\right )}{4 b} + \frac {x^{2} \left (- \frac {7 a \left (- \frac {9 a d^{2}}{10 b} + 2 c d\right )}{8 b} + c^{2}\right )}{3 b}\right ) & \text {for}\: b \neq 0 \\\frac {2 \left (\frac {c^{2} \left (a x\right )^{\frac {7}{2}}}{7} + \frac {2 c d \left (a x\right )^{\frac {9}{2}}}{9 a} + \frac {d^{2} \left (a x\right )^{\frac {11}{2}}}{11 a^{2}}\right )}{a^{4}} & \text {for}\: a \neq 0 \\\tilde {\infty } \left (\frac {c^{2} x^{4}}{4} + \frac {2 c d x^{5}}{5} + \frac {d^{2} x^{6}}{6}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-5*a**3*(-7*a*(-9*a*d**2/(10*b) + 2*c*d)/(8*b) + c**2)*Piecewis 
e((log(a + 2*sqrt(b)*sqrt(a*x + b*x**2) + 2*b*x)/sqrt(b), Ne(a**2/b, 0)), 
((a/(2*b) + x)*log(a/(2*b) + x)/sqrt(b*(a/(2*b) + x)**2), True))/(16*b**3) 
 + sqrt(a*x + b*x**2)*(5*a**2*(-7*a*(-9*a*d**2/(10*b) + 2*c*d)/(8*b) + c** 
2)/(8*b**3) - 5*a*x*(-7*a*(-9*a*d**2/(10*b) + 2*c*d)/(8*b) + c**2)/(12*b** 
2) + d**2*x**4/(5*b) + x**3*(-9*a*d**2/(10*b) + 2*c*d)/(4*b) + x**2*(-7*a* 
(-9*a*d**2/(10*b) + 2*c*d)/(8*b) + c**2)/(3*b)), Ne(b, 0)), (2*(c**2*(a*x) 
**(7/2)/7 + 2*c*d*(a*x)**(9/2)/(9*a) + d**2*(a*x)**(11/2)/(11*a**2))/a**4, 
 Ne(a, 0)), (zoo*(c**2*x**4/4 + 2*c*d*x**5/5 + d**2*x**6/6), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.48 \[ \int \frac {x^3 (c+d x)^2}{\sqrt {a x+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a x} d^{2} x^{4}}{5 \, b} + \frac {\sqrt {b x^{2} + a x} c d x^{3}}{2 \, b} - \frac {9 \, \sqrt {b x^{2} + a x} a d^{2} x^{3}}{40 \, b^{2}} + \frac {\sqrt {b x^{2} + a x} c^{2} x^{2}}{3 \, b} - \frac {7 \, \sqrt {b x^{2} + a x} a c d x^{2}}{12 \, b^{2}} + \frac {21 \, \sqrt {b x^{2} + a x} a^{2} d^{2} x^{2}}{80 \, b^{3}} - \frac {5 \, \sqrt {b x^{2} + a x} a c^{2} x}{12 \, b^{2}} + \frac {35 \, \sqrt {b x^{2} + a x} a^{2} c d x}{48 \, b^{3}} - \frac {21 \, \sqrt {b x^{2} + a x} a^{3} d^{2} x}{64 \, b^{4}} - \frac {5 \, a^{3} c^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{16 \, b^{\frac {7}{2}}} + \frac {35 \, a^{4} c d \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{64 \, b^{\frac {9}{2}}} - \frac {63 \, a^{5} d^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{256 \, b^{\frac {11}{2}}} + \frac {5 \, \sqrt {b x^{2} + a x} a^{2} c^{2}}{8 \, b^{3}} - \frac {35 \, \sqrt {b x^{2} + a x} a^{3} c d}{32 \, b^{4}} + \frac {63 \, \sqrt {b x^{2} + a x} a^{4} d^{2}}{128 \, b^{5}} \] Input:

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

Output:

1/5*sqrt(b*x^2 + a*x)*d^2*x^4/b + 1/2*sqrt(b*x^2 + a*x)*c*d*x^3/b - 9/40*s 
qrt(b*x^2 + a*x)*a*d^2*x^3/b^2 + 1/3*sqrt(b*x^2 + a*x)*c^2*x^2/b - 7/12*sq 
rt(b*x^2 + a*x)*a*c*d*x^2/b^2 + 21/80*sqrt(b*x^2 + a*x)*a^2*d^2*x^2/b^3 - 
5/12*sqrt(b*x^2 + a*x)*a*c^2*x/b^2 + 35/48*sqrt(b*x^2 + a*x)*a^2*c*d*x/b^3 
 - 21/64*sqrt(b*x^2 + a*x)*a^3*d^2*x/b^4 - 5/16*a^3*c^2*log(2*b*x + a + 2* 
sqrt(b*x^2 + a*x)*sqrt(b))/b^(7/2) + 35/64*a^4*c*d*log(2*b*x + a + 2*sqrt( 
b*x^2 + a*x)*sqrt(b))/b^(9/2) - 63/256*a^5*d^2*log(2*b*x + a + 2*sqrt(b*x^ 
2 + a*x)*sqrt(b))/b^(11/2) + 5/8*sqrt(b*x^2 + a*x)*a^2*c^2/b^3 - 35/32*sqr 
t(b*x^2 + a*x)*a^3*c*d/b^4 + 63/128*sqrt(b*x^2 + a*x)*a^4*d^2/b^5
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.87 \[ \int \frac {x^3 (c+d x)^2}{\sqrt {a x+b x^2}} \, dx=\frac {1}{1920} \, \sqrt {b x^{2} + a x} {\left (2 \, {\left (4 \, {\left (6 \, {\left (\frac {8 \, d^{2} x}{b} + \frac {20 \, b^{4} c d - 9 \, a b^{3} d^{2}}{b^{5}}\right )} x + \frac {80 \, b^{4} c^{2} - 140 \, a b^{3} c d + 63 \, a^{2} b^{2} d^{2}}{b^{5}}\right )} x - \frac {5 \, {\left (80 \, a b^{3} c^{2} - 140 \, a^{2} b^{2} c d + 63 \, a^{3} b d^{2}\right )}}{b^{5}}\right )} x + \frac {15 \, {\left (80 \, a^{2} b^{2} c^{2} - 140 \, a^{3} b c d + 63 \, a^{4} d^{2}\right )}}{b^{5}}\right )} + \frac {{\left (80 \, a^{3} b^{2} c^{2} - 140 \, a^{4} b c d + 63 \, a^{5} d^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{256 \, b^{\frac {11}{2}}} \] Input:

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

Output:

1/1920*sqrt(b*x^2 + a*x)*(2*(4*(6*(8*d^2*x/b + (20*b^4*c*d - 9*a*b^3*d^2)/ 
b^5)*x + (80*b^4*c^2 - 140*a*b^3*c*d + 63*a^2*b^2*d^2)/b^5)*x - 5*(80*a*b^ 
3*c^2 - 140*a^2*b^2*c*d + 63*a^3*b*d^2)/b^5)*x + 15*(80*a^2*b^2*c^2 - 140* 
a^3*b*c*d + 63*a^4*d^2)/b^5) + 1/256*(80*a^3*b^2*c^2 - 140*a^4*b*c*d + 63* 
a^5*d^2)*log(abs(2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) + a))/b^(11/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.29 \[ \int \frac {x^3 (c+d x)^2}{\sqrt {a x+b x^2}} \, dx=\frac {945 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b \,d^{2}-2100 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{2} c d -630 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{2} d^{2} x +1200 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{3} c^{2}+1400 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{3} c d x +504 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{3} d^{2} x^{2}-800 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{4} c^{2} x -1120 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{4} c d \,x^{2}-432 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{4} d^{2} x^{3}+640 \sqrt {x}\, \sqrt {b x +a}\, b^{5} c^{2} x^{2}+960 \sqrt {x}\, \sqrt {b x +a}\, b^{5} c d \,x^{3}+384 \sqrt {x}\, \sqrt {b x +a}\, b^{5} d^{2} x^{4}-945 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{5} d^{2}+2100 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{4} b c d -1200 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{3} b^{2} c^{2}}{1920 b^{6}} \] Input:

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

Output:

(945*sqrt(x)*sqrt(a + b*x)*a**4*b*d**2 - 2100*sqrt(x)*sqrt(a + b*x)*a**3*b 
**2*c*d - 630*sqrt(x)*sqrt(a + b*x)*a**3*b**2*d**2*x + 1200*sqrt(x)*sqrt(a 
 + b*x)*a**2*b**3*c**2 + 1400*sqrt(x)*sqrt(a + b*x)*a**2*b**3*c*d*x + 504* 
sqrt(x)*sqrt(a + b*x)*a**2*b**3*d**2*x**2 - 800*sqrt(x)*sqrt(a + b*x)*a*b* 
*4*c**2*x - 1120*sqrt(x)*sqrt(a + b*x)*a*b**4*c*d*x**2 - 432*sqrt(x)*sqrt( 
a + b*x)*a*b**4*d**2*x**3 + 640*sqrt(x)*sqrt(a + b*x)*b**5*c**2*x**2 + 960 
*sqrt(x)*sqrt(a + b*x)*b**5*c*d*x**3 + 384*sqrt(x)*sqrt(a + b*x)*b**5*d**2 
*x**4 - 945*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**5*d* 
*2 + 2100*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**4*b*c* 
d - 1200*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**3*b**2* 
c**2)/(1920*b**6)