3.318 \(\int \frac{1}{x^4 \sqrt{a x^3+b x^4}} \, dx\)

Optimal. Leaf size=136 \[ -\frac{256 b^4 \sqrt{a x^3+b x^4}}{315 a^5 x^2}+\frac{128 b^3 \sqrt{a x^3+b x^4}}{315 a^4 x^3}-\frac{32 b^2 \sqrt{a x^3+b x^4}}{105 a^3 x^4}+\frac{16 b \sqrt{a x^3+b x^4}}{63 a^2 x^5}-\frac{2 \sqrt{a x^3+b x^4}}{9 a x^6} \]

[Out]

(-2*Sqrt[a*x^3 + b*x^4])/(9*a*x^6) + (16*b*Sqrt[a*x^3 + b*x^4])/(63*a^2*x^5) - (32*b^2*Sqrt[a*x^3 + b*x^4])/(1
05*a^3*x^4) + (128*b^3*Sqrt[a*x^3 + b*x^4])/(315*a^4*x^3) - (256*b^4*Sqrt[a*x^3 + b*x^4])/(315*a^5*x^2)

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Rubi [A]  time = 0.174421, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2016, 2000} \[ -\frac{256 b^4 \sqrt{a x^3+b x^4}}{315 a^5 x^2}+\frac{128 b^3 \sqrt{a x^3+b x^4}}{315 a^4 x^3}-\frac{32 b^2 \sqrt{a x^3+b x^4}}{105 a^3 x^4}+\frac{16 b \sqrt{a x^3+b x^4}}{63 a^2 x^5}-\frac{2 \sqrt{a x^3+b x^4}}{9 a x^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[a*x^3 + b*x^4]),x]

[Out]

(-2*Sqrt[a*x^3 + b*x^4])/(9*a*x^6) + (16*b*Sqrt[a*x^3 + b*x^4])/(63*a^2*x^5) - (32*b^2*Sqrt[a*x^3 + b*x^4])/(1
05*a^3*x^4) + (128*b^3*Sqrt[a*x^3 + b*x^4])/(315*a^4*x^3) - (256*b^4*Sqrt[a*x^3 + b*x^4])/(315*a^5*x^2)

Rule 2016

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

Rule 2000

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(b*(n - j)*(p + 1)*x
^(n - 1)), x] /; FreeQ[{a, b, j, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && EqQ[j*p - n + j + 1, 0]

Rubi steps

\begin{align*} \int \frac{1}{x^4 \sqrt{a x^3+b x^4}} \, dx &=-\frac{2 \sqrt{a x^3+b x^4}}{9 a x^6}-\frac{(8 b) \int \frac{1}{x^3 \sqrt{a x^3+b x^4}} \, dx}{9 a}\\ &=-\frac{2 \sqrt{a x^3+b x^4}}{9 a x^6}+\frac{16 b \sqrt{a x^3+b x^4}}{63 a^2 x^5}+\frac{\left (16 b^2\right ) \int \frac{1}{x^2 \sqrt{a x^3+b x^4}} \, dx}{21 a^2}\\ &=-\frac{2 \sqrt{a x^3+b x^4}}{9 a x^6}+\frac{16 b \sqrt{a x^3+b x^4}}{63 a^2 x^5}-\frac{32 b^2 \sqrt{a x^3+b x^4}}{105 a^3 x^4}-\frac{\left (64 b^3\right ) \int \frac{1}{x \sqrt{a x^3+b x^4}} \, dx}{105 a^3}\\ &=-\frac{2 \sqrt{a x^3+b x^4}}{9 a x^6}+\frac{16 b \sqrt{a x^3+b x^4}}{63 a^2 x^5}-\frac{32 b^2 \sqrt{a x^3+b x^4}}{105 a^3 x^4}+\frac{128 b^3 \sqrt{a x^3+b x^4}}{315 a^4 x^3}+\frac{\left (128 b^4\right ) \int \frac{1}{\sqrt{a x^3+b x^4}} \, dx}{315 a^4}\\ &=-\frac{2 \sqrt{a x^3+b x^4}}{9 a x^6}+\frac{16 b \sqrt{a x^3+b x^4}}{63 a^2 x^5}-\frac{32 b^2 \sqrt{a x^3+b x^4}}{105 a^3 x^4}+\frac{128 b^3 \sqrt{a x^3+b x^4}}{315 a^4 x^3}-\frac{256 b^4 \sqrt{a x^3+b x^4}}{315 a^5 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0189052, size = 64, normalized size = 0.47 \[ -\frac{2 \sqrt{x^3 (a+b x)} \left (48 a^2 b^2 x^2-40 a^3 b x+35 a^4-64 a b^3 x^3+128 b^4 x^4\right )}{315 a^5 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[a*x^3 + b*x^4]),x]

[Out]

(-2*Sqrt[x^3*(a + b*x)]*(35*a^4 - 40*a^3*b*x + 48*a^2*b^2*x^2 - 64*a*b^3*x^3 + 128*b^4*x^4))/(315*a^5*x^6)

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Maple [A]  time = 0.003, size = 68, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( 128\,{b}^{4}{x}^{4}-64\,a{b}^{3}{x}^{3}+48\,{b}^{2}{x}^{2}{a}^{2}-40\,x{a}^{3}b+35\,{a}^{4} \right ) }{315\,{x}^{3}{a}^{5}}{\frac{1}{\sqrt{b{x}^{4}+a{x}^{3}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x^4+a*x^3)^(1/2),x)

[Out]

-2/315*(b*x+a)*(128*b^4*x^4-64*a*b^3*x^3+48*a^2*b^2*x^2-40*a^3*b*x+35*a^4)/x^3/a^5/(b*x^4+a*x^3)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{4} + a x^{3}} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^4+a*x^3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x^4 + a*x^3)*x^4), x)

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Fricas [A]  time = 0.821276, size = 143, normalized size = 1.05 \begin{align*} -\frac{2 \,{\left (128 \, b^{4} x^{4} - 64 \, a b^{3} x^{3} + 48 \, a^{2} b^{2} x^{2} - 40 \, a^{3} b x + 35 \, a^{4}\right )} \sqrt{b x^{4} + a x^{3}}}{315 \, a^{5} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^4+a*x^3)^(1/2),x, algorithm="fricas")

[Out]

-2/315*(128*b^4*x^4 - 64*a*b^3*x^3 + 48*a^2*b^2*x^2 - 40*a^3*b*x + 35*a^4)*sqrt(b*x^4 + a*x^3)/(a^5*x^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{4} \sqrt{x^{3} \left (a + b x\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x**4+a*x**3)**(1/2),x)

[Out]

Integral(1/(x**4*sqrt(x**3*(a + b*x))), x)

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Giac [A]  time = 1.27836, size = 96, normalized size = 0.71 \begin{align*} -\frac{2 \,{\left (35 \,{\left (b + \frac{a}{x}\right )}^{\frac{9}{2}} - 180 \,{\left (b + \frac{a}{x}\right )}^{\frac{7}{2}} b + 378 \,{\left (b + \frac{a}{x}\right )}^{\frac{5}{2}} b^{2} - 420 \,{\left (b + \frac{a}{x}\right )}^{\frac{3}{2}} b^{3} + 315 \, \sqrt{b + \frac{a}{x}} b^{4}\right )}}{315 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^4+a*x^3)^(1/2),x, algorithm="giac")

[Out]

-2/315*(35*(b + a/x)^(9/2) - 180*(b + a/x)^(7/2)*b + 378*(b + a/x)^(5/2)*b^2 - 420*(b + a/x)^(3/2)*b^3 + 315*s
qrt(b + a/x)*b^4)/a^5