3.792 \(\int \frac{(a+c x^4)^{3/2}}{x^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0410732, 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 = {275, 277, 217, 206} \[ \frac{1}{2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )-\frac{c \sqrt{a+c x^4}}{2 x^2}-\frac{\left (a+c x^4\right )^{3/2}}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^(3/2)/x^7,x]

[Out]

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

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [C]  time = 0.0118768, size = 52, normalized size = 0.76 \[ -\frac{a \sqrt{a+c x^4} \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{c x^4}{a}\right )}{6 x^6 \sqrt{\frac{c x^4}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^(3/2)/x^7,x]

[Out]

-(a*Sqrt[a + c*x^4]*Hypergeometric2F1[-3/2, -3/2, -1/2, -((c*x^4)/a)])/(6*x^6*Sqrt[1 + (c*x^4)/a])

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Maple [A]  time = 0.016, size = 55, normalized size = 0.8 \begin{align*}{\frac{1}{2}{c}^{{\frac{3}{2}}}\ln \left ({x}^{2}\sqrt{c}+\sqrt{c{x}^{4}+a} \right ) }-{\frac{a}{6\,{x}^{6}}\sqrt{c{x}^{4}+a}}-{\frac{2\,c}{3\,{x}^{2}}\sqrt{c{x}^{4}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)^(3/2)/x^7,x)

[Out]

1/2*c^(3/2)*ln(x^2*c^(1/2)+(c*x^4+a)^(1/2))-1/6*a/x^6*(c*x^4+a)^(1/2)-2/3*c*(c*x^4+a)^(1/2)/x^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((c*x^4+a)^(3/2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.57401, size = 286, normalized size = 4.21 \begin{align*} \left [\frac{3 \, c^{\frac{3}{2}} x^{6} \log \left (-2 \, c x^{4} - 2 \, \sqrt{c x^{4} + a} \sqrt{c} x^{2} - a\right ) - 2 \,{\left (4 \, c x^{4} + a\right )} \sqrt{c x^{4} + a}}{12 \, x^{6}}, -\frac{3 \, \sqrt{-c} c x^{6} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + a}}\right ) +{\left (4 \, c x^{4} + a\right )} \sqrt{c x^{4} + a}}{6 \, x^{6}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x^7,x, algorithm="fricas")

[Out]

[1/12*(3*c^(3/2)*x^6*log(-2*c*x^4 - 2*sqrt(c*x^4 + a)*sqrt(c)*x^2 - a) - 2*(4*c*x^4 + a)*sqrt(c*x^4 + a))/x^6,
-1/6*(3*sqrt(-c)*c*x^6*arctan(sqrt(-c)*x^2/sqrt(c*x^4 + a)) + (4*c*x^4 + a)*sqrt(c*x^4 + a))/x^6]

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Sympy [A]  time = 3.13001, size = 80, normalized size = 1.18 \begin{align*} - \frac{a \sqrt{c} \sqrt{\frac{a}{c x^{4}} + 1}}{6 x^{4}} - \frac{2 c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{4}} + 1}}{3} - \frac{c^{\frac{3}{2}} \log{\left (\frac{a}{c x^{4}} \right )}}{4} + \frac{c^{\frac{3}{2}} \log{\left (\sqrt{\frac{a}{c x^{4}} + 1} + 1 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+a)**(3/2)/x**7,x)

[Out]

-a*sqrt(c)*sqrt(a/(c*x**4) + 1)/(6*x**4) - 2*c**(3/2)*sqrt(a/(c*x**4) + 1)/3 - c**(3/2)*log(a/(c*x**4))/4 + c*
*(3/2)*log(sqrt(a/(c*x**4) + 1) + 1)/2

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Giac [A]  time = 1.11822, size = 68, normalized size = 1. \begin{align*} -\frac{c^{2} \arctan \left (\frac{\sqrt{c + \frac{a}{x^{4}}}}{\sqrt{-c}}\right )}{2 \, \sqrt{-c}} - \frac{1}{6} \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{c + \frac{a}{x^{4}}} c \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x^7,x, algorithm="giac")

[Out]

-1/2*c^2*arctan(sqrt(c + a/x^4)/sqrt(-c))/sqrt(-c) - 1/6*(c + a/x^4)^(3/2) - 1/2*sqrt(c + a/x^4)*c