3.56 \(\int x^7 (a^2+2 a b x^3+b^2 x^6)^{5/2} \, dx\)

Optimal. Leaf size=255 \[ \frac{b^5 x^{23} \sqrt{a^2+2 a b x^3+b^2 x^6}}{23 \left (a+b x^3\right )}+\frac{a b^4 x^{20} \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{5 a^4 b x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{a^5 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )} \]

[Out]

(a^5*x^8*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(8*(a + b*x^3)) + (5*a^4*b*x^11*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(11
*(a + b*x^3)) + (5*a^3*b^2*x^14*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*(a + b*x^3)) + (10*a^2*b^3*x^17*Sqrt[a^2 +
2*a*b*x^3 + b^2*x^6])/(17*(a + b*x^3)) + (a*b^4*x^20*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*(a + b*x^3)) + (b^5*
x^23*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(23*(a + b*x^3))

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Rubi [A]  time = 0.0602653, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1355, 270} \[ \frac{b^5 x^{23} \sqrt{a^2+2 a b x^3+b^2 x^6}}{23 \left (a+b x^3\right )}+\frac{a b^4 x^{20} \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{5 a^4 b x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{a^5 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^7*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

(a^5*x^8*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(8*(a + b*x^3)) + (5*a^4*b*x^11*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(11
*(a + b*x^3)) + (5*a^3*b^2*x^14*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*(a + b*x^3)) + (10*a^2*b^3*x^17*Sqrt[a^2 +
2*a*b*x^3 + b^2*x^6])/(17*(a + b*x^3)) + (a*b^4*x^20*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*(a + b*x^3)) + (b^5*
x^23*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(23*(a + b*x^3))

Rule 1355

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x^7 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int x^7 \left (a b+b^2 x^3\right )^5 \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \left (a^5 b^5 x^7+5 a^4 b^6 x^{10}+10 a^3 b^7 x^{13}+10 a^2 b^8 x^{16}+5 a b^9 x^{19}+b^{10} x^{22}\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{a^5 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{5 a^4 b x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac{a b^4 x^{20} \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{b^5 x^{23} \sqrt{a^2+2 a b x^3+b^2 x^6}}{23 \left (a+b x^3\right )}\\ \end{align*}

Mathematica [A]  time = 0.022395, size = 83, normalized size = 0.33 \[ \frac{x^8 \sqrt{\left (a+b x^3\right )^2} \left (141680 a^2 b^3 x^9+172040 a^3 b^2 x^6+109480 a^4 b x^3+30107 a^5+60214 a b^4 x^{12}+10472 b^5 x^{15}\right )}{240856 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

(x^8*Sqrt[(a + b*x^3)^2]*(30107*a^5 + 109480*a^4*b*x^3 + 172040*a^3*b^2*x^6 + 141680*a^2*b^3*x^9 + 60214*a*b^4
*x^12 + 10472*b^5*x^15))/(240856*(a + b*x^3))

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Maple [A]  time = 0.007, size = 80, normalized size = 0.3 \begin{align*}{\frac{{x}^{8} \left ( 10472\,{b}^{5}{x}^{15}+60214\,a{b}^{4}{x}^{12}+141680\,{a}^{2}{b}^{3}{x}^{9}+172040\,{a}^{3}{b}^{2}{x}^{6}+109480\,{a}^{4}b{x}^{3}+30107\,{a}^{5} \right ) }{240856\, \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)

[Out]

1/240856*x^8*(10472*b^5*x^15+60214*a*b^4*x^12+141680*a^2*b^3*x^9+172040*a^3*b^2*x^6+109480*a^4*b*x^3+30107*a^5
)*((b*x^3+a)^2)^(5/2)/(b*x^3+a)^5

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Maxima [A]  time = 1.02469, size = 77, normalized size = 0.3 \begin{align*} \frac{1}{23} \, b^{5} x^{23} + \frac{1}{4} \, a b^{4} x^{20} + \frac{10}{17} \, a^{2} b^{3} x^{17} + \frac{5}{7} \, a^{3} b^{2} x^{14} + \frac{5}{11} \, a^{4} b x^{11} + \frac{1}{8} \, a^{5} x^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="maxima")

[Out]

1/23*b^5*x^23 + 1/4*a*b^4*x^20 + 10/17*a^2*b^3*x^17 + 5/7*a^3*b^2*x^14 + 5/11*a^4*b*x^11 + 1/8*a^5*x^8

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Fricas [A]  time = 1.66509, size = 140, normalized size = 0.55 \begin{align*} \frac{1}{23} \, b^{5} x^{23} + \frac{1}{4} \, a b^{4} x^{20} + \frac{10}{17} \, a^{2} b^{3} x^{17} + \frac{5}{7} \, a^{3} b^{2} x^{14} + \frac{5}{11} \, a^{4} b x^{11} + \frac{1}{8} \, a^{5} x^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/23*b^5*x^23 + 1/4*a*b^4*x^20 + 10/17*a^2*b^3*x^17 + 5/7*a^3*b^2*x^14 + 5/11*a^4*b*x^11 + 1/8*a^5*x^8

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{7} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

Integral(x**7*((a + b*x**3)**2)**(5/2), x)

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Giac [A]  time = 1.09516, size = 142, normalized size = 0.56 \begin{align*} \frac{1}{23} \, b^{5} x^{23} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{1}{4} \, a b^{4} x^{20} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{10}{17} \, a^{2} b^{3} x^{17} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{7} \, a^{3} b^{2} x^{14} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{11} \, a^{4} b x^{11} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{1}{8} \, a^{5} x^{8} \mathrm{sgn}\left (b x^{3} + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="giac")

[Out]

1/23*b^5*x^23*sgn(b*x^3 + a) + 1/4*a*b^4*x^20*sgn(b*x^3 + a) + 10/17*a^2*b^3*x^17*sgn(b*x^3 + a) + 5/7*a^3*b^2
*x^14*sgn(b*x^3 + a) + 5/11*a^4*b*x^11*sgn(b*x^3 + a) + 1/8*a^5*x^8*sgn(b*x^3 + a)