3.4.18 \(\int \frac {1}{x^4 \sqrt {a x^3+b x^4}} \, dx\) [318]

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

[Out]

-2/9*(b*x^4+a*x^3)^(1/2)/a/x^6+16/63*b*(b*x^4+a*x^3)^(1/2)/a^2/x^5-32/105*b^2*(b*x^4+a*x^3)^(1/2)/a^3/x^4+128/
315*b^3*(b*x^4+a*x^3)^(1/2)/a^4/x^3-256/315*b^4*(b*x^4+a*x^3)^(1/2)/a^5/x^2

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Rubi [A]
time = 0.12, antiderivative size = 136, normalized size of antiderivative = 1.00, 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 = {2041, 2025} \begin {gather*} -\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} \end {gather*}

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 2025

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]

Rule 2041

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])

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*}

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Mathematica [A]
time = 0.09, size = 64, normalized size = 0.47 \begin {gather*} -\frac {2 \sqrt {x^3 (a+b x)} \left (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\right )}{315 a^5 x^6} \end {gather*}

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 = 0.49, size = 83, normalized size = 0.61

method	result	size
trager	\(-\frac {2 \left (128 b^{4} x^{4}-64 a \,b^{3} x^{3}+48 a^{2} b^{2} x^{2}-40 a^{3} b x +35 a^{4}\right ) \sqrt {b \,x^{4}+a \,x^{3}}}{315 a^{5} x^{6}}\)	\(63\)
risch	\(-\frac {2 \left (b x +a \right ) \left (128 b^{4} x^{4}-64 a \,b^{3} x^{3}+48 a^{2} b^{2} x^{2}-40 a^{3} b x +35 a^{4}\right )}{315 x^{3} \sqrt {x^{3} \left (b x +a \right )}\, a^{5}}\)	\(66\)
gosper	\(-\frac {2 \left (b x +a \right ) \left (128 b^{4} x^{4}-64 a \,b^{3} x^{3}+48 a^{2} b^{2} x^{2}-40 a^{3} b x +35 a^{4}\right )}{315 x^{3} a^{5} \sqrt {b \,x^{4}+a \,x^{3}}}\)	\(68\)
default	\(-\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b \,x^{2}+a x}\, \left (128 b^{4} x^{4}-64 a \,b^{3} x^{3}+48 a^{2} b^{2} x^{2}-40 a^{3} b x +35 a^{4}\right )}{315 x^{4} \sqrt {b \,x^{4}+a \,x^{3}}\, a^{5}}\)	\(83\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315/x^4*(x*(b*x+a))^(1/2)*(b*x^2+a*x)^(1/2)*(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)/(b*
x^4+a*x^3)^(1/2)/a^5

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 3.32, size = 62, normalized size = 0.46 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \sqrt {x^{3} \left (a + b x\right )}}\, dx \end {gather*}

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 = 0.61, size = 140, normalized size = 1.03 \begin {gather*} \frac {2 \, {\left (1008 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} b^{2} + 1680 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a b^{\frac {3}{2}} + 1080 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{2} b + 315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{3} \sqrt {b} + 35 \, a^{4}\right )}}{315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{9} \mathrm {sgn}\left (x\right )} \end {gather*}

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*(1008*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*b^2 + 1680*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a*b^(3/2) + 1080*(s
qrt(b)*x - sqrt(b*x^2 + a*x))^2*a^2*b + 315*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^3*sqrt(b) + 35*a^4)/((sqrt(b)*x
- sqrt(b*x^2 + a*x))^9*sgn(x))

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Mupad [B]
time = 5.14, size = 116, normalized size = 0.85 \begin {gather*} \frac {16\,b\,\sqrt {b\,x^4+a\,x^3}}{63\,a^2\,x^5}-\frac {2\,\sqrt {b\,x^4+a\,x^3}}{9\,a\,x^6}-\frac {32\,b^2\,\sqrt {b\,x^4+a\,x^3}}{105\,a^3\,x^4}+\frac {128\,b^3\,\sqrt {b\,x^4+a\,x^3}}{315\,a^4\,x^3}-\frac {256\,b^4\,\sqrt {b\,x^4+a\,x^3}}{315\,a^5\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a*x^3 + b*x^4)^(1/2)),x)

[Out]

(16*b*(a*x^3 + b*x^4)^(1/2))/(63*a^2*x^5) - (2*(a*x^3 + b*x^4)^(1/2))/(9*a*x^6) - (32*b^2*(a*x^3 + b*x^4)^(1/2
))/(105*a^3*x^4) + (128*b^3*(a*x^3 + b*x^4)^(1/2))/(315*a^4*x^3) - (256*b^4*(a*x^3 + b*x^4)^(1/2))/(315*a^5*x^
2)

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