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

Optimal. Leaf size=251 \[ \frac{b^5 x^{15} \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 \left (a+b x^3\right )}+\frac{5 a b^4 x^{12} \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{5 a^4 b x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{a^5 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]

[Out]

(5*a^4*b*x^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*(a + b*x^3)) + (5*a^3*b^2*x^6*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]
)/(3*(a + b*x^3)) + (10*a^2*b^3*x^9*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(9*(a + b*x^3)) + (5*a*b^4*x^12*Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])/(12*(a + b*x^3)) + (b^5*x^15*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(15*(a + b*x^3)) + (a^5*
Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*Log[x])/(a + b*x^3)

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Rubi [A]  time = 0.0686626, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1355, 266, 43} \[ \frac{b^5 x^{15} \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 \left (a+b x^3\right )}+\frac{5 a b^4 x^{12} \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{5 a^4 b x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{a^5 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*a^4*b*x^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*(a + b*x^3)) + (5*a^3*b^2*x^6*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]
)/(3*(a + b*x^3)) + (10*a^2*b^3*x^9*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(9*(a + b*x^3)) + (5*a*b^4*x^12*Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])/(12*(a + b*x^3)) + (b^5*x^15*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(15*(a + b*x^3)) + (a^5*
Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*Log[x])/(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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \frac{\left (a b+b^2 x^3\right )^5}{x} \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^5}{x} \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \operatorname{Subst}\left (\int \left (5 a^4 b^6+\frac{a^5 b^5}{x}+10 a^3 b^7 x+10 a^2 b^8 x^2+5 a b^9 x^3+b^{10} x^4\right ) \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{5 a^4 b x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{5 a b^4 x^{12} \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac{b^5 x^{15} \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 \left (a+b x^3\right )}+\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6} \log (x)}{a+b x^3}\\ \end{align*}

Mathematica [A]  time = 0.0260967, size = 82, normalized size = 0.33 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (b x^3 \left (200 a^2 b^2 x^6+300 a^3 b x^3+300 a^4+75 a b^3 x^9+12 b^4 x^{12}\right )+180 a^5 \log (x)\right )}{180 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a + b*x^3)^2]*(b*x^3*(300*a^4 + 300*a^3*b*x^3 + 200*a^2*b^2*x^6 + 75*a*b^3*x^9 + 12*b^4*x^12) + 180*a^5
*Log[x]))/(180*(a + b*x^3))

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Maple [A]  time = 0.009, size = 79, normalized size = 0.3 \begin{align*}{\frac{12\,{b}^{5}{x}^{15}+75\,a{b}^{4}{x}^{12}+200\,{a}^{2}{b}^{3}{x}^{9}+300\,{a}^{3}{b}^{2}{x}^{6}+300\,{a}^{4}b{x}^{3}+180\,{a}^{5}\ln \left ( x \right ) }{180\, \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((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x,x)

[Out]

1/180*((b*x^3+a)^2)^(5/2)*(12*b^5*x^15+75*a*b^4*x^12+200*a^2*b^3*x^9+300*a^3*b^2*x^6+300*a^4*b*x^3+180*a^5*ln(
x))/(b*x^3+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((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.63602, size = 134, normalized size = 0.53 \begin{align*} \frac{1}{15} \, b^{5} x^{15} + \frac{5}{12} \, a b^{4} x^{12} + \frac{10}{9} \, a^{2} b^{3} x^{9} + \frac{5}{3} \, a^{3} b^{2} x^{6} + \frac{5}{3} \, a^{4} b x^{3} + a^{5} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*b^5*x^15 + 5/12*a*b^4*x^12 + 10/9*a^2*b^3*x^9 + 5/3*a^3*b^2*x^6 + 5/3*a^4*b*x^3 + a^5*log(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.09832, size = 140, normalized size = 0.56 \begin{align*} \frac{1}{15} \, b^{5} x^{15} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{12} \, a b^{4} x^{12} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{10}{9} \, a^{2} b^{3} x^{9} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{3} \, a^{3} b^{2} x^{6} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{3} \, a^{4} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + a^{5} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (b x^{3} + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*b^5*x^15*sgn(b*x^3 + a) + 5/12*a*b^4*x^12*sgn(b*x^3 + a) + 10/9*a^2*b^3*x^9*sgn(b*x^3 + a) + 5/3*a^3*b^2*
x^6*sgn(b*x^3 + a) + 5/3*a^4*b*x^3*sgn(b*x^3 + a) + a^5*log(abs(x))*sgn(b*x^3 + a)