\(\int \frac {x^2 (c+d x)^2}{(a x+b x^2)^{5/2}} \, dx\) [179]

Optimal result

Integrand size = 24, antiderivative size = 108 \[ \int \frac {x^2 (c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=\frac {2 (b c-a d)^2 x^2}{3 a b^2 \left (a x+b x^2\right )^{3/2}}+\frac {4 (b c-a d) (b c+2 a d) x}{3 a^2 b^2 \sqrt {a x+b x^2}}+\frac {2 d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{b^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.03 \[ \int \frac {x^2 (c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=\frac {x \left (2 \sqrt {b} (b c-a d) x \left (3 a^2 d+2 b^2 c x+a b (3 c+4 d x)\right )-6 a^2 d^2 \sqrt {x} (a+b x)^{3/2} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )\right )}{3 a^2 b^{5/2} (x (a+b x))^{3/2}} \] Input:

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

Output:

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

Rubi [F]

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.

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

\(\Big \downarrow \) 1242

\(\displaystyle \frac {2 x^2 \text {PolynomialRemainder}\left [(c+d x)^2,0,x\right ]}{3 a \left (a x+b x^2\right )^{3/2}}-\frac {2 \int -\frac {a x \text {PolynomialRemainder}\left [(c+d x)^2,0,x\right ]}{\left (b x^2+a x\right )^{3/2}}dx}{3 a^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \int \frac {a x \text {PolynomialRemainder}\left [(c+d x)^2,0,x\right ]}{\left (b x^2+a x\right )^{3/2}}dx}{3 a^2}+\frac {2 x^2 \text {PolynomialRemainder}\left [(c+d x)^2,0,x\right ]}{3 a \left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \int \frac {x \text {PolynomialRemainder}\left [(c+d x)^2,0,x\right ]}{\left (b x^2+a x\right )^{3/2}}dx}{3 a}+\frac {2 x^2 \text {PolynomialRemainder}\left [(c+d x)^2,0,x\right ]}{3 a \left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {2 \sqrt {x} \sqrt {a+b x} \int \frac {\text {PolynomialRemainder}\left [(c+d x)^2,0,x\right ]}{\sqrt {x} (a+b x)^{3/2}}dx}{3 a \sqrt {a x+b x^2}}+\frac {2 x^2 \text {PolynomialRemainder}\left [(c+d x)^2,0,x\right ]}{3 a \left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 7284

\(\displaystyle \frac {4 \sqrt {x} \sqrt {a+b x} \int \frac {\text {PolynomialRemainder}\left [(c+d x)^2,0,x\right ]}{(a+b x)^{3/2}}d\sqrt {x}}{3 a \sqrt {a x+b x^2}}+\frac {2 x^2 \text {PolynomialRemainder}\left [(c+d x)^2,0,x\right ]}{3 a \left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 7299

\(\displaystyle \frac {4 \sqrt {x} \sqrt {a+b x} \int \frac {\text {PolynomialRemainder}\left [(c+d x)^2,0,x\right ]}{(a+b x)^{3/2}}d\sqrt {x}}{3 a \sqrt {a x+b x^2}}+\frac {2 x^2 \text {PolynomialRemainder}\left [(c+d x)^2,0,x\right ]}{3 a \left (a x+b x^2\right )^{3/2}}\)

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 1242

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(f + g*x) 
^n, a*e + c*d*x, x], R = PolynomialRemainder[(f + g*x)^n, a*e + c*d*x, x]}, 
 Simp[R*(2*c*d - b*e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^ 
2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 1)*( 
a + b*x + c*x^2)^(p + 1)*ExpandToSum[d*e*(p + 1)*(b^2 - 4*a*c)*Q - R*(2*c*d 
 - b*e)*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IG 
tQ[n, 1] && IGtQ[m, 0] && LtQ[p, -1] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

rule 7284

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.04

method	result	size
pseudoelliptic	\(\frac {2 a^{2} \sqrt {x \left (b x +a \right )}\, d^{2} \left (b x +a \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )-2 x \left (-\left (\frac {2 d x}{3}+c \right ) a c \,b^{\frac {5}{2}}+\sqrt {b}\, a^{3} d^{2}+\frac {4 a^{2} d^{2} x \,b^{\frac {3}{2}}}{3}-\frac {2 b^{\frac {7}{2}} c^{2} x}{3}\right )}{b^{\frac {5}{2}} \left (b x +a \right ) \sqrt {x \left (b x +a \right )}\, a^{2}}\)	\(112\)
default	\(c^{2} \left (-\frac {x}{2 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {1}{3 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {2 \left (2 b x +a \right )}{3 a^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {16 b \left (2 b x +a \right )}{3 a^{4} \sqrt {b \,x^{2}+a x}}\right )}{2 b}\right )}{4 b}\right )+d^{2} \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {x^{2}}{b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {a \left (-\frac {x}{2 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {1}{3 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {2 \left (2 b x +a \right )}{3 a^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {16 b \left (2 b x +a \right )}{3 a^{4} \sqrt {b \,x^{2}+a x}}\right )}{2 b}\right )}{4 b}\right )}{2 b}\right )}{2 b}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a x}}-\frac {a \left (-\frac {1}{b \sqrt {b \,x^{2}+a x}}+\frac {2 b x +a}{a b \sqrt {b \,x^{2}+a x}}\right )}{2 b}+\frac {\ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{b^{\frac {3}{2}}}}{b}\right )+2 c d \left (-\frac {x^{2}}{b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {a \left (-\frac {x}{2 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {1}{3 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {2 \left (2 b x +a \right )}{3 a^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {16 b \left (2 b x +a \right )}{3 a^{4} \sqrt {b \,x^{2}+a x}}\right )}{2 b}\right )}{4 b}\right )}{2 b}\right )\)	\(468\)

Input:

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

Output:

2/(x*(b*x+a))^(1/2)/b^(5/2)*(a^2*(x*(b*x+a))^(1/2)*d^2*(b*x+a)*arctanh((x* 
(b*x+a))^(1/2)/x/b^(1/2))-x*(-(2/3*d*x+c)*a*c*b^(5/2)+b^(1/2)*a^3*d^2+4/3* 
a^2*d^2*x*b^(3/2)-2/3*b^(7/2)*c^2*x))/(b*x+a)/a^2
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.86 \[ \int \frac {x^2 (c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (a^{2} b^{2} d^{2} x^{2} + 2 \, a^{3} b d^{2} x + a^{4} d^{2}\right )} \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + 2 \, {\left (3 \, a b^{3} c^{2} - 3 \, a^{3} b d^{2} + 2 \, {\left (b^{4} c^{2} + a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}}{3 \, {\left (a^{2} b^{5} x^{2} + 2 \, a^{3} b^{4} x + a^{4} b^{3}\right )}}, -\frac {2 \, {\left (3 \, {\left (a^{2} b^{2} d^{2} x^{2} + 2 \, a^{3} b d^{2} x + a^{4} d^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) - {\left (3 \, a b^{3} c^{2} - 3 \, a^{3} b d^{2} + 2 \, {\left (b^{4} c^{2} + a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}\right )}}{3 \, {\left (a^{2} b^{5} x^{2} + 2 \, a^{3} b^{4} x + a^{4} b^{3}\right )}}\right ] \] Input:

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

Output:

[1/3*(3*(a^2*b^2*d^2*x^2 + 2*a^3*b*d^2*x + a^4*d^2)*sqrt(b)*log(2*b*x + a 
+ 2*sqrt(b*x^2 + a*x)*sqrt(b)) + 2*(3*a*b^3*c^2 - 3*a^3*b*d^2 + 2*(b^4*c^2 
 + a*b^3*c*d - 2*a^2*b^2*d^2)*x)*sqrt(b*x^2 + a*x))/(a^2*b^5*x^2 + 2*a^3*b 
^4*x + a^4*b^3), -2/3*(3*(a^2*b^2*d^2*x^2 + 2*a^3*b*d^2*x + a^4*d^2)*sqrt( 
-b)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a)) - (3*a*b^3*c^2 - 3*a^3*b* 
d^2 + 2*(b^4*c^2 + a*b^3*c*d - 2*a^2*b^2*d^2)*x)*sqrt(b*x^2 + a*x))/(a^2*b 
^5*x^2 + 2*a^3*b^4*x + a^4*b^3)]
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (94) = 188\).

Time = 0.04 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.73 \[ \int \frac {x^2 (c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=-\frac {1}{3} \, d^{2} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} + \frac {a x}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} - \frac {2 \, x}{\sqrt {b x^{2} + a x} a b} - \frac {1}{\sqrt {b x^{2} + a x} b^{2}}\right )} - \frac {2 \, c d x^{2}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} + \frac {4 \, c^{2} x}{3 \, \sqrt {b x^{2} + a x} a^{2}} - \frac {2 \, c^{2} x}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} - \frac {2 \, a c d x}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} + \frac {4 \, c d x}{3 \, \sqrt {b x^{2} + a x} a b} - \frac {4 \, d^{2} x}{3 \, \sqrt {b x^{2} + a x} b^{2}} + \frac {d^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{b^{\frac {5}{2}}} + \frac {2 \, c^{2}}{3 \, \sqrt {b x^{2} + a x} a b} + \frac {2 \, c d}{3 \, \sqrt {b x^{2} + a x} b^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} d^{2}}{3 \, a b^{2}} \] Input:

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

Output:

-1/3*d^2*x*(3*x^2/((b*x^2 + a*x)^(3/2)*b) + a*x/((b*x^2 + a*x)^(3/2)*b^2) 
- 2*x/(sqrt(b*x^2 + a*x)*a*b) - 1/(sqrt(b*x^2 + a*x)*b^2)) - 2*c*d*x^2/((b 
*x^2 + a*x)^(3/2)*b) + 4/3*c^2*x/(sqrt(b*x^2 + a*x)*a^2) - 2/3*c^2*x/((b*x 
^2 + a*x)^(3/2)*b) - 2/3*a*c*d*x/((b*x^2 + a*x)^(3/2)*b^2) + 4/3*c*d*x/(sq 
rt(b*x^2 + a*x)*a*b) - 4/3*d^2*x/(sqrt(b*x^2 + a*x)*b^2) + d^2*log(2*b*x + 
 a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(5/2) + 2/3*c^2/(sqrt(b*x^2 + a*x)*a*b 
) + 2/3*c*d/(sqrt(b*x^2 + a*x)*b^2) - 2/3*sqrt(b*x^2 + a*x)*d^2/(a*b^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (94) = 188\).

Time = 0.26 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.18 \[ \int \frac {x^2 (c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=-\frac {d^{2} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{b^{\frac {5}{2}}} + \frac {2 \, {\left (6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} b^{2} c d - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a b d^{2} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} b^{\frac {5}{2}} c^{2} + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a b^{\frac {3}{2}} c d - 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{2} \sqrt {b} d^{2} + 2 \, a b^{2} c^{2} + 2 \, a^{2} b c d - 4 \, a^{3} d^{2}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a\right )}^{3} b^{\frac {5}{2}}} \] Input:

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

Output:

-d^2*log(abs(2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) + a))/b^(5/2) + 2/3 
*(6*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*b^2*c*d - 6*(sqrt(b)*x - sqrt(b*x^2 
+ a*x))^2*a*b*d^2 + 3*(sqrt(b)*x - sqrt(b*x^2 + a*x))*b^(5/2)*c^2 + 6*(sqr 
t(b)*x - sqrt(b*x^2 + a*x))*a*b^(3/2)*c*d - 9*(sqrt(b)*x - sqrt(b*x^2 + a* 
x))*a^2*sqrt(b)*d^2 + 2*a*b^2*c^2 + 2*a^2*b*c*d - 4*a^3*d^2)/(((sqrt(b)*x 
- sqrt(b*x^2 + a*x))*sqrt(b) + a)^3*b^(5/2))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.03 \[ \int \frac {x^2 (c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{3} d^{2}+2 \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{2} b \,d^{2} x +\frac {4 \sqrt {b}\, \sqrt {b x +a}\, a^{2} b c d}{3}-\frac {4 \sqrt {b}\, \sqrt {b x +a}\, a \,b^{2} c^{2}}{3}+\frac {4 \sqrt {b}\, \sqrt {b x +a}\, a \,b^{2} c d x}{3}-\frac {4 \sqrt {b}\, \sqrt {b x +a}\, b^{3} c^{2} x}{3}-2 \sqrt {x}\, a^{3} b \,d^{2}-\frac {8 \sqrt {x}\, a^{2} b^{2} d^{2} x}{3}+2 \sqrt {x}\, a \,b^{3} c^{2}+\frac {4 \sqrt {x}\, a \,b^{3} c d x}{3}+\frac {4 \sqrt {x}\, b^{4} c^{2} x}{3}}{\sqrt {b x +a}\, a^{2} b^{3} \left (b x +a \right )} \] Input:

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

Output:

(2*(3*sqrt(b)*sqrt(a + b*x)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a)) 
*a**3*d**2 + 3*sqrt(b)*sqrt(a + b*x)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b)) 
/sqrt(a))*a**2*b*d**2*x + 2*sqrt(b)*sqrt(a + b*x)*a**2*b*c*d - 2*sqrt(b)*s 
qrt(a + b*x)*a*b**2*c**2 + 2*sqrt(b)*sqrt(a + b*x)*a*b**2*c*d*x - 2*sqrt(b 
)*sqrt(a + b*x)*b**3*c**2*x - 3*sqrt(x)*a**3*b*d**2 - 4*sqrt(x)*a**2*b**2* 
d**2*x + 3*sqrt(x)*a*b**3*c**2 + 2*sqrt(x)*a*b**3*c*d*x + 2*sqrt(x)*b**4*c 
**2*x))/(3*sqrt(a + b*x)*a**2*b**3*(a + b*x))