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

Optimal. Leaf size=162 \[ -\frac {a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^3 \left (a+b x^3\right )}+\frac {3 a b^2 \log (x) \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {b^3 x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )} \]

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Rubi [A]  time = 0.05, antiderivative size = 162, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )}-\frac {a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^3 \left (a+b x^3\right )}+\frac {b^3 x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac {3 a b^2 \log (x) \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

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 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]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 61, normalized size = 0.38 \begin {gather*} -\frac {\sqrt {\left (a+b x^3\right )^2} \left (a^3+6 a^2 b x^3-18 a b^2 x^6 \log (x)-2 b^3 x^9\right )}{6 x^6 \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(Sqrt[(a + b*x^3)^2]*(a^3 + 6*a^2*b*x^3 - 2*b^3*x^9 - 18*a*b^2*x^6*Log[x]))/(x^6*(a + b*x^3))

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IntegrateAlgebraic [B]  time = 1.68, size = 1164, normalized size = 7.19 \begin {gather*} -\frac {b^2 \log \left (-\sqrt {b^2} x^3+a+\sqrt {b^2 x^6+2 a b x^3+a^2}\right ) a^5}{2 \left (-\sqrt {b^2} x^3-a+\sqrt {b^2 x^6+2 a b x^3+a^2}\right )^2 \left (-\sqrt {b^2} x^3+a+\sqrt {b^2 x^6+2 a b x^3+a^2}\right )^2}-\frac {b \sqrt {b^2} \log \left (-\sqrt {b^2} x^3+a+\sqrt {b^2 x^6+2 a b x^3+a^2}\right ) a^5}{2 \left (-\sqrt {b^2} x^3-a+\sqrt {b^2 x^6+2 a b x^3+a^2}\right )^2 \left (-\sqrt {b^2} x^3+a+\sqrt {b^2 x^6+2 a b x^3+a^2}\right )^2}+\frac {b^2 \left (\sqrt {b^2 x^6+2 a b x^3+a^2}-\sqrt {b^2} x^3\right )^2 \log \left (-\sqrt {b^2} x^3+a+\sqrt {b^2 x^6+2 a b x^3+a^2}\right ) a^3}{\left (-\sqrt {b^2} x^3-a+\sqrt {b^2 x^6+2 a b x^3+a^2}\right )^2 \left (-\sqrt {b^2} x^3+a+\sqrt {b^2 x^6+2 a b x^3+a^2}\right )^2}+\frac {b \sqrt {b^2} \left (\sqrt {b^2 x^6+2 a b x^3+a^2}-\sqrt {b^2} x^3\right )^2 \log \left (-\sqrt {b^2} x^3+a+\sqrt {b^2 x^6+2 a b x^3+a^2}\right ) a^3}{\left (-\sqrt {b^2} x^3-a+\sqrt {b^2 x^6+2 a b x^3+a^2}\right )^2 \left (-\sqrt {b^2} x^3+a+\sqrt {b^2 x^6+2 a b x^3+a^2}\right )^2}+\frac {1}{2} b^2 \log \left (-\sqrt {b^2} x^3-a+\sqrt {b^2 x^6+2 a b x^3+a^2}\right ) a-\frac {1}{2} b \sqrt {b^2} \log \left (-\sqrt {b^2} x^3-a+\sqrt {b^2 x^6+2 a b x^3+a^2}\right ) a-\frac {b^2 \left (\sqrt {b^2 x^6+2 a b x^3+a^2}-\sqrt {b^2} x^3\right )^4 \log \left (-\sqrt {b^2} x^3+a+\sqrt {b^2 x^6+2 a b x^3+a^2}\right ) a}{2 \left (-\sqrt {b^2} x^3-a+\sqrt {b^2 x^6+2 a b x^3+a^2}\right )^2 \left (-\sqrt {b^2} x^3+a+\sqrt {b^2 x^6+2 a b x^3+a^2}\right )^2}-\frac {b \sqrt {b^2} \left (\sqrt {b^2 x^6+2 a b x^3+a^2}-\sqrt {b^2} x^3\right )^4 \log \left (-\sqrt {b^2} x^3+a+\sqrt {b^2 x^6+2 a b x^3+a^2}\right ) a}{2 \left (-\sqrt {b^2} x^3-a+\sqrt {b^2 x^6+2 a b x^3+a^2}\right )^2 \left (-\sqrt {b^2} x^3+a+\sqrt {b^2 x^6+2 a b x^3+a^2}\right )^2}+\frac {\sqrt {b^2 x^6+2 a b x^3+a^2} \left (2 b^4 x^9+a b^3 x^6-6 a^2 b^2 x^3-a^3 b\right )+\sqrt {b^2} \left (-2 b^4 x^{12}-3 a b^3 x^9+5 a^2 b^2 x^6+7 a^3 b x^3+a^4\right )}{6 x^6 \left (b^2 x^3+a b\right )-6 \sqrt {b^2} x^6 \sqrt {b^2 x^6+2 a b x^3+a^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*(-(a^3*b) - 6*a^2*b^2*x^3 + a*b^3*x^6 + 2*b^4*x^9) + Sqrt[b^2]*(a^4 + 7*a^3*b
*x^3 + 5*a^2*b^2*x^6 - 3*a*b^3*x^9 - 2*b^4*x^12))/(6*x^6*(a*b + b^2*x^3) - 6*Sqrt[b^2]*x^6*Sqrt[a^2 + 2*a*b*x^
3 + b^2*x^6]) + (a*b^2*Log[-a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]])/2 - (a*b*Sqrt[b^2]*Log[-a -
Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]])/2 - (a^5*b^2*Log[a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 +
b^2*x^6]])/(2*(-a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^2*(a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x
^3 + b^2*x^6])^2) - (a^5*b*Sqrt[b^2]*Log[a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]])/(2*(-a - Sqrt[b
^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^2*(a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^2) + (a^3*b
^2*(-(Sqrt[b^2]*x^3) + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^2*Log[a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x
^6]])/((-a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^2*(a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^
2*x^6])^2) + (a^3*b*Sqrt[b^2]*(-(Sqrt[b^2]*x^3) + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^2*Log[a - Sqrt[b^2]*x^3 + S
qrt[a^2 + 2*a*b*x^3 + b^2*x^6]])/((-a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^2*(a - Sqrt[b^2]*x^3
+ Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^2) - (a*b^2*(-(Sqrt[b^2]*x^3) + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^4*Log[a -
Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]])/(2*(-a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^2*
(a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^2) - (a*b*Sqrt[b^2]*(-(Sqrt[b^2]*x^3) + Sqrt[a^2 + 2*a*b
*x^3 + b^2*x^6])^4*Log[a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]])/(2*(-a - Sqrt[b^2]*x^3 + Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])^2*(a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^2)

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fricas [A]  time = 1.30, size = 39, normalized size = 0.24 \begin {gather*} \frac {2 \, b^{3} x^{9} + 18 \, a b^{2} x^{6} \log \relax (x) - 6 \, a^{2} b x^{3} - a^{3}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*b^3*x^9 + 18*a*b^2*x^6*log(x) - 6*a^2*b*x^3 - a^3)/x^6

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giac [A]  time = 0.34, size = 86, normalized size = 0.53 \begin {gather*} \frac {1}{3} \, b^{3} x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 3 \, a b^{2} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {9 \, a b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 6 \, a^{2} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + a^{3} \mathrm {sgn}\left (b x^{3} + a\right )}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 60, normalized size = 0.37 \begin {gather*} \frac {\left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {3}{2}} \left (2 b^{3} x^{9}+18 a \,b^{2} x^{6} \ln \relax (x )-6 a^{2} b \,x^{3}-a^{3}\right )}{6 \left (b \,x^{3}+a \right )^{3} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*((b*x^3+a)^2)^(3/2)*(2*b^3*x^9+18*a*b^2*ln(x)*x^6-6*a^2*b*x^3-a^3)/(b*x^3+a)^3/x^6

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maxima [A]  time = 0.70, size = 220, normalized size = 1.36 \begin {gather*} \frac {\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} b^{3} x^{3}}{2 \, a} + \left (-1\right )^{2 \, b^{2} x^{3} + 2 \, a b} a b^{2} \log \left (2 \, b^{2} x^{3} + 2 \, a b\right ) - \left (-1\right )^{2 \, a b x^{3} + 2 \, a^{2}} a b^{2} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{x^{2} {\left | x \right |}}\right ) + \frac {3}{2} \, \sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} b^{2} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} b^{2}}{6 \, a^{2}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} b}{6 \, a x^{3}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}}}{6 \, a^{2} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(b^2*x^6 + 2*a*b*x^3 + a^2)*b^3*x^3/a + (-1)^(2*b^2*x^3 + 2*a*b)*a*b^2*log(2*b^2*x^3 + 2*a*b) - (-1)^(
2*a*b*x^3 + 2*a^2)*a*b^2*log(2*a*b*x/abs(x) + 2*a^2/(x^2*abs(x))) + 3/2*sqrt(b^2*x^6 + 2*a*b*x^3 + a^2)*b^2 +
1/6*(b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*b^2/a^2 - 1/6*(b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*b/(a*x^3) - 1/6*(b^2*x^6
+ 2*a*b*x^3 + a^2)^(5/2)/(a^2*x^6)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2}}{x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}}{x^{7}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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