3.151 $$\int x^2 (a^2+2 a b x+b^2 x^2)^{3/2} \, dx$$

Optimal. Leaf size=96 $\frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b^3}-\frac{2 a \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^3}+\frac{a^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b^3}$

[Out]

(a^2*(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(4*b^3) - (2*a*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(5*b^3) + ((a
+ b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(6*b^3)

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Rubi [A]  time = 0.0310885, antiderivative size = 107, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 1, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.042, Rules used = {645} $\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^3}-\frac{2 a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4}{5 b^3}+\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3}{4 b^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 645

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[ExpandLinearProduct[(b/2 + c*x)^(2*p), (d + e*x)^m, b
/2, c, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*
e, 0] && IGtQ[m, 0] && EqQ[m - 2*p + 1, 0]

Rubi steps

\begin{align*} \int x^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{a^2 \left (a b+b^2 x\right )^3}{b^2}-\frac{2 a \left (a b+b^2 x\right )^4}{b^3}+\frac{\left (a b+b^2 x\right )^5}{b^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{a^2 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 b^3}-\frac{2 a (a+b x)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{5 b^3}+\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0137966, size = 55, normalized size = 0.57 $\frac{x^3 \sqrt{(a+b x)^2} \left (45 a^2 b x+20 a^3+36 a b^2 x^2+10 b^3 x^3\right )}{60 (a+b x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x^3*Sqrt[(a + b*x)^2]*(20*a^3 + 45*a^2*b*x + 36*a*b^2*x^2 + 10*b^3*x^3))/(60*(a + b*x))

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Maple [A]  time = 0.184, size = 52, normalized size = 0.5 \begin{align*}{\frac{{x}^{3} \left ( 10\,{b}^{3}{x}^{3}+36\,a{b}^{2}{x}^{2}+45\,{a}^{2}bx+20\,{a}^{3} \right ) }{60\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*x^3*(10*b^3*x^3+36*a*b^2*x^2+45*a^2*b*x+20*a^3)*((b*x+a)^2)^(3/2)/(b*x+a)^3

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.965, size = 80, normalized size = 0.83 \begin{align*} \frac{1}{6} \, b^{3} x^{6} + \frac{3}{5} \, a b^{2} x^{5} + \frac{3}{4} \, a^{2} b x^{4} + \frac{1}{3} \, a^{3} x^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^3*x^6 + 3/5*a*b^2*x^5 + 3/4*a^2*b*x^4 + 1/3*a^3*x^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*((a + b*x)**2)**(3/2), x)

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Giac [A]  time = 1.30003, size = 99, normalized size = 1.03 \begin{align*} \frac{1}{6} \, b^{3} x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{5} \, a b^{2} x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{4} \, a^{2} b x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, a^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{a^{6} \mathrm{sgn}\left (b x + a\right )}{60 \, b^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^3*x^6*sgn(b*x + a) + 3/5*a*b^2*x^5*sgn(b*x + a) + 3/4*a^2*b*x^4*sgn(b*x + a) + 1/3*a^3*x^3*sgn(b*x + a)
+ 1/60*a^6*sgn(b*x + a)/b^3