### 3.164 $$\int x^4 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx$$

Optimal. Leaf size=181 $\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5}-\frac{4 a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^5}+\frac{3 a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{4 b^5}-\frac{4 a^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^5}+\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^5}$

[Out]

(a^4*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*b^5) - (4*a^3*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7
*b^5) + (3*a^2*(a + b*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*b^5) - (4*a*(a + b*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*
x^2])/(9*b^5) + ((a + b*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*b^5)

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Rubi [A]  time = 0.051446, antiderivative size = 181, normalized size of antiderivative = 1., 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} $\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5}-\frac{4 a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^5}+\frac{3 a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{4 b^5}-\frac{4 a^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^5}+\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^4*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(a^4*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*b^5) - (4*a^3*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7
*b^5) + (3*a^2*(a + b*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*b^5) - (4*a*(a + b*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*
x^2])/(9*b^5) + ((a + b*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*b^5)

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^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{a^4 \left (a b+b^2 x\right )^5}{b^4}-\frac{4 a^3 \left (a b+b^2 x\right )^6}{b^5}+\frac{6 a^2 \left (a b+b^2 x\right )^7}{b^6}-\frac{4 a \left (a b+b^2 x\right )^8}{b^7}+\frac{\left (a b+b^2 x\right )^9}{b^8}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{a^4 (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b^5}-\frac{4 a^3 (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 b^5}+\frac{3 a^2 (a+b x)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{4 b^5}-\frac{4 a (a+b x)^8 \sqrt{a^2+2 a b x+b^2 x^2}}{9 b^5}+\frac{(a+b x)^9 \sqrt{a^2+2 a b x+b^2 x^2}}{10 b^5}\\ \end{align*}

Mathematica [A]  time = 0.0186447, size = 77, normalized size = 0.43 $\frac{x^5 \sqrt{(a+b x)^2} \left (1800 a^3 b^2 x^2+1575 a^2 b^3 x^3+1050 a^4 b x+252 a^5+700 a b^4 x^4+126 b^5 x^5\right )}{1260 (a+b x)}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^4*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x^5*Sqrt[(a + b*x)^2]*(252*a^5 + 1050*a^4*b*x + 1800*a^3*b^2*x^2 + 1575*a^2*b^3*x^3 + 700*a*b^4*x^4 + 126*b^5
*x^5))/(1260*(a + b*x))

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Maple [A]  time = 0.174, size = 74, normalized size = 0.4 \begin{align*}{\frac{{x}^{5} \left ( 126\,{b}^{5}{x}^{5}+700\,a{b}^{4}{x}^{4}+1575\,{a}^{2}{b}^{3}{x}^{3}+1800\,{a}^{3}{b}^{2}{x}^{2}+1050\,{a}^{4}bx+252\,{a}^{5} \right ) }{1260\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/1260*x^5*(126*b^5*x^5+700*a*b^4*x^4+1575*a^2*b^3*x^3+1800*a^3*b^2*x^2+1050*a^4*b*x+252*a^5)*((b*x+a)^2)^(5/2
)/(b*x+a)^5

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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^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.65244, size = 132, normalized size = 0.73 \begin{align*} \frac{1}{10} \, b^{5} x^{10} + \frac{5}{9} \, a b^{4} x^{9} + \frac{5}{4} \, a^{2} b^{3} x^{8} + \frac{10}{7} \, a^{3} b^{2} x^{7} + \frac{5}{6} \, a^{4} b x^{6} + \frac{1}{5} \, a^{5} x^{5} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/10*b^5*x^10 + 5/9*a*b^4*x^9 + 5/4*a^2*b^3*x^8 + 10/7*a^3*b^2*x^7 + 5/6*a^4*b*x^6 + 1/5*a^5*x^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral(x**4*((a + b*x)**2)**(5/2), x)

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Giac [A]  time = 1.3708, size = 144, normalized size = 0.8 \begin{align*} \frac{1}{10} \, b^{5} x^{10} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{9} \, a b^{4} x^{9} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{4} \, a^{2} b^{3} x^{8} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{7} \, a^{3} b^{2} x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{6} \, a^{4} b x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{5} \, a^{5} x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{a^{10} \mathrm{sgn}\left (b x + a\right )}{1260 \, b^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

1/10*b^5*x^10*sgn(b*x + a) + 5/9*a*b^4*x^9*sgn(b*x + a) + 5/4*a^2*b^3*x^8*sgn(b*x + a) + 10/7*a^3*b^2*x^7*sgn(
b*x + a) + 5/6*a^4*b*x^6*sgn(b*x + a) + 1/5*a^5*x^5*sgn(b*x + a) + 1/1260*a^10*sgn(b*x + a)/b^5