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

Optimal. Leaf size=68 \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{b \sqrt{a+b x^3}}{4 x^3}-\frac{\left (a+b x^3\right )^{3/2}}{6 x^6} \]

[Out]

-(b*Sqrt[a + b*x^3])/(4*x^3) - (a + b*x^3)^(3/2)/(6*x^6) - (b^2*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(4*Sqrt[a])

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Rubi [A]  time = 0.039009, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 47, 63, 208} \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{b \sqrt{a+b x^3}}{4 x^3}-\frac{\left (a+b x^3\right )^{3/2}}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^(3/2)/x^7,x]

[Out]

-(b*Sqrt[a + b*x^3])/(4*x^3) - (a + b*x^3)^(3/2)/(6*x^6) - (b^2*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(4*Sqrt[a])

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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^{3/2}}{x^7} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^3} \, dx,x,x^3\right )\\ &=-\frac{\left (a+b x^3\right )^{3/2}}{6 x^6}+\frac{1}{4} b \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^2} \, dx,x,x^3\right )\\ &=-\frac{b \sqrt{a+b x^3}}{4 x^3}-\frac{\left (a+b x^3\right )^{3/2}}{6 x^6}+\frac{1}{8} b^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )\\ &=-\frac{b \sqrt{a+b x^3}}{4 x^3}-\frac{\left (a+b x^3\right )^{3/2}}{6 x^6}+\frac{1}{4} b \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )\\ &=-\frac{b \sqrt{a+b x^3}}{4 x^3}-\frac{\left (a+b x^3\right )^{3/2}}{6 x^6}-\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 \sqrt{a}}\\ \end{align*}

Mathematica [A]  time = 0.0372222, size = 76, normalized size = 1.12 \[ -\frac{2 a^2+3 b^2 x^6 \sqrt{\frac{b x^3}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )+7 a b x^3+5 b^2 x^6}{12 x^6 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^(3/2)/x^7,x]

[Out]

-(2*a^2 + 7*a*b*x^3 + 5*b^2*x^6 + 3*b^2*x^6*Sqrt[1 + (b*x^3)/a]*ArcTanh[Sqrt[1 + (b*x^3)/a]])/(12*x^6*Sqrt[a +
b*x^3])

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Maple [A]  time = 0.019, size = 54, normalized size = 0.8 \begin{align*} -{\frac{a}{6\,{x}^{6}}\sqrt{b{x}^{3}+a}}-{\frac{5\,b}{12\,{x}^{3}}\sqrt{b{x}^{3}+a}}-{\frac{{b}^{2}}{4}{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*a*(b*x^3+a)^(1/2)/x^6-5/12*b*(b*x^3+a)^(1/2)/x^3-1/4*b^2*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.54111, size = 320, normalized size = 4.71 \begin{align*} \left [\frac{3 \, \sqrt{a} b^{2} x^{6} \log \left (\frac{b x^{3} - 2 \, \sqrt{b x^{3} + a} \sqrt{a} + 2 \, a}{x^{3}}\right ) - 2 \,{\left (5 \, a b x^{3} + 2 \, a^{2}\right )} \sqrt{b x^{3} + a}}{24 \, a x^{6}}, \frac{3 \, \sqrt{-a} b^{2} x^{6} \arctan \left (\frac{\sqrt{b x^{3} + a} \sqrt{-a}}{a}\right ) -{\left (5 \, a b x^{3} + 2 \, a^{2}\right )} \sqrt{b x^{3} + a}}{12 \, a x^{6}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/24*(3*sqrt(a)*b^2*x^6*log((b*x^3 - 2*sqrt(b*x^3 + a)*sqrt(a) + 2*a)/x^3) - 2*(5*a*b*x^3 + 2*a^2)*sqrt(b*x^3
+ a))/(a*x^6), 1/12*(3*sqrt(-a)*b^2*x^6*arctan(sqrt(b*x^3 + a)*sqrt(-a)/a) - (5*a*b*x^3 + 2*a^2)*sqrt(b*x^3 +
a))/(a*x^6)]

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Sympy [A]  time = 3.66908, size = 78, normalized size = 1.15 \begin{align*} - \frac{a \sqrt{b} \sqrt{\frac{a}{b x^{3}} + 1}}{6 x^{\frac{9}{2}}} - \frac{5 b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}}{12 x^{\frac{3}{2}}} - \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{4 \sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*sqrt(b)*sqrt(a/(b*x**3) + 1)/(6*x**(9/2)) - 5*b**(3/2)*sqrt(a/(b*x**3) + 1)/(12*x**(3/2)) - b**2*asinh(sqrt
(a)/(sqrt(b)*x**(3/2)))/(4*sqrt(a))

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Giac [A]  time = 1.09608, size = 82, normalized size = 1.21 \begin{align*} \frac{1}{12} \, b^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x^{3} + a} a}{b^{2} x^{6}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b^2*(3*arctan(sqrt(b*x^3 + a)/sqrt(-a))/sqrt(-a) - (5*(b*x^3 + a)^(3/2) - 3*sqrt(b*x^3 + a)*a)/(b^2*x^6))