3.67 \(\int \frac{(a^2+2 a b x^3+b^2 x^6)^{5/2}}{x^4} \, dx\)

Optimal. Leaf size=252 \[ \frac{b^5 x^{12} \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac{5 a b^4 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac{5 a^4 b \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]

[Out]

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

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Rubi [A]  time = 0.0735974, antiderivative size = 252, 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} \[ \frac{b^5 x^{12} \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac{5 a b^4 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac{5 a^4 b \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*x^3*(a + b*x^3)) + (10*a^3*b^2*x^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/
(3*(a + b*x^3)) + (5*a^2*b^3*x^6*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*(a + b*x^3)) + (5*a*b^4*x^9*Sqrt[a^2 + 2*
a*b*x^3 + b^2*x^6])/(9*(a + b*x^3)) + (b^5*x^12*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(12*(a + b*x^3)) + (5*a^4*b*S
qrt[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^4} \, 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^4} \, 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^2} \, 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 (10 a^3 b^7+\frac{a^5 b^5}{x^2}+\frac{5 a^4 b^6}{x}+10 a^2 b^8 x+5 a b^9 x^2+b^{10} x^3\right ) \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )}\\ &=-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{5 a b^4 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{b^5 x^{12} \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6} \log (x)}{a+b x^3}\\ \end{align*}

Mathematica [A]  time = 0.0261356, size = 85, normalized size = 0.34 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (60 a^2 b^3 x^9+120 a^3 b^2 x^6+180 a^4 b x^3 \log (x)-12 a^5+20 a b^4 x^{12}+3 b^5 x^{15}\right )}{36 x^3 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a + b*x^3)^2]*(-12*a^5 + 120*a^3*b^2*x^6 + 60*a^2*b^3*x^9 + 20*a*b^4*x^12 + 3*b^5*x^15 + 180*a^4*b*x^3*
Log[x]))/(36*x^3*(a + b*x^3))

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Maple [A]  time = 0.011, size = 82, normalized size = 0.3 \begin{align*}{\frac{3\,{b}^{5}{x}^{15}+20\,a{b}^{4}{x}^{12}+60\,{a}^{2}{b}^{3}{x}^{9}+120\,{a}^{3}{b}^{2}{x}^{6}+180\,{a}^{4}b\ln \left ( x \right ){x}^{3}-12\,{a}^{5}}{36\, \left ( b{x}^{3}+a \right ) ^{5}{x}^{3}} \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^4,x)

[Out]

1/36*((b*x^3+a)^2)^(5/2)*(3*b^5*x^15+20*a*b^4*x^12+60*a^2*b^3*x^9+120*a^3*b^2*x^6+180*a^4*b*ln(x)*x^3-12*a^5)/
(b*x^3+a)^5/x^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((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.81542, size = 143, normalized size = 0.57 \begin{align*} \frac{3 \, b^{5} x^{15} + 20 \, a b^{4} x^{12} + 60 \, a^{2} b^{3} x^{9} + 120 \, a^{3} b^{2} x^{6} + 180 \, a^{4} b x^{3} \log \left (x\right ) - 12 \, a^{5}}{36 \, x^{3}} \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^4,x, algorithm="fricas")

[Out]

1/36*(3*b^5*x^15 + 20*a*b^4*x^12 + 60*a^2*b^3*x^9 + 120*a^3*b^2*x^6 + 180*a^4*b*x^3*log(x) - 12*a^5)/x^3

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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^{4}}\, 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**4,x)

[Out]

Integral(((a + b*x**3)**2)**(5/2)/x**4, x)

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Giac [A]  time = 1.1099, size = 167, normalized size = 0.66 \begin{align*} \frac{1}{12} \, b^{5} x^{12} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{9} \, a b^{4} x^{9} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{3} \, a^{2} b^{3} x^{6} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{10}{3} \, a^{3} b^{2} x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + 5 \, a^{4} b \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (b x^{3} + a\right ) - \frac{5 \, a^{4} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + a^{5} \mathrm{sgn}\left (b x^{3} + a\right )}{3 \, x^{3}} \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^4,x, algorithm="giac")

[Out]

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