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

Optimal. Leaf size=75 \[ c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )-\frac{c \sqrt{b x^2+c x^4}}{x^2}-\frac{\left (b x^2+c x^4\right )^{3/2}}{3 x^6} \]

[Out]

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

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Rubi [A]  time = 0.103917, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2018, 662, 620, 206} \[ c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )-\frac{c \sqrt{b x^2+c x^4}}{x^2}-\frac{\left (b x^2+c x^4\right )^{3/2}}{3 x^6} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^(3/2)/x^7,x]

[Out]

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

Rule 2018

Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)
/n] - 1)*(a*x^Simplify[j/n] + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]

Rule 662

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + p + 1)), x] - Dist[(c*p)/(e^2*(m + p + 1)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2
)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[
p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] && IntegerQ[2*p]

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

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

Mathematica [C]  time = 0.0176505, size = 56, normalized size = 0.75 \[ -\frac{b \sqrt{x^2 \left (b+c x^2\right )} \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{c x^2}{b}\right )}{3 x^4 \sqrt{\frac{c x^2}{b}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^(3/2)/x^7,x]

[Out]

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

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Maple [B]  time = 0.054, size = 129, normalized size = 1.7 \begin{align*}{\frac{1}{3\,{b}^{2}{x}^{6}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 2\,{c}^{5/2} \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{4}+3\,{c}^{5/2}\sqrt{c{x}^{2}+b}{x}^{4}b-2\,{c}^{3/2} \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{2}+3\,\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){x}^{3}{b}^{2}{c}^{2}- \left ( c{x}^{2}+b \right ) ^{{\frac{5}{2}}}b\sqrt{c} \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(c*x^4+b*x^2)^(3/2)*(2*c^(5/2)*(c*x^2+b)^(3/2)*x^4+3*c^(5/2)*(c*x^2+b)^(1/2)*x^4*b-2*c^(3/2)*(c*x^2+b)^(5/
2)*x^2+3*ln(x*c^(1/2)+(c*x^2+b)^(1/2))*x^3*b^2*c^2-(c*x^2+b)^(5/2)*b*c^(1/2))/x^6/(c*x^2+b)^(3/2)/b^2/c^(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((c*x^4+b*x^2)^(3/2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.29729, size = 312, normalized size = 4.16 \begin{align*} \left [\frac{3 \, c^{\frac{3}{2}} x^{4} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right ) - 2 \, \sqrt{c x^{4} + b x^{2}}{\left (4 \, c x^{2} + b\right )}}{6 \, x^{4}}, -\frac{3 \, \sqrt{-c} c x^{4} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}{c x^{2} + b}\right ) + \sqrt{c x^{4} + b x^{2}}{\left (4 \, c x^{2} + b\right )}}{3 \, x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}{x^{7}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.79774, size = 165, normalized size = 2.2 \begin{align*} -\frac{1}{2} \, c^{\frac{3}{2}} \log \left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2}\right ) \mathrm{sgn}\left (x\right ) + \frac{4 \,{\left (3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} b c^{\frac{3}{2}} \mathrm{sgn}\left (x\right ) - 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} b^{2} c^{\frac{3}{2}} \mathrm{sgn}\left (x\right ) + 2 \, b^{3} c^{\frac{3}{2}} \mathrm{sgn}\left (x\right )\right )}}{3 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*c^(3/2)*log((sqrt(c)*x - sqrt(c*x^2 + b))^2)*sgn(x) + 4/3*(3*(sqrt(c)*x - sqrt(c*x^2 + b))^4*b*c^(3/2)*sg
n(x) - 3*(sqrt(c)*x - sqrt(c*x^2 + b))^2*b^2*c^(3/2)*sgn(x) + 2*b^3*c^(3/2)*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 +
b))^2 - b)^3