3.22.23 \(\int \frac {(-4 b+a x^5) (b+a x^5)^{3/4}}{x^4 (2 b+c x^4+2 a x^5)} \, dx\)

Optimal. Leaf size=154 \[ \frac {c^{3/4} \tan ^{-1}\left (\frac {2^{3/4} \sqrt [4]{c} x \sqrt [4]{a x^5+b}}{\sqrt {2} \sqrt {a x^5+b}-\sqrt {c} x^2}\right )}{2 \sqrt [4]{2}}+\frac {c^{3/4} \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^5+b}}{\sqrt [4]{2} \sqrt [4]{c}}+\frac {\sqrt [4]{c} x^2}{2^{3/4}}}{x \sqrt [4]{a x^5+b}}\right )}{2 \sqrt [4]{2}}+\frac {2 \left (a x^5+b\right )^{3/4}}{3 x^3} \]

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Rubi [F]  time = 1.19, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-4*b + a*x^5)*(b + a*x^5)^(3/4))/(x^4*(2*b + c*x^4 + 2*a*x^5)),x]

[Out]

(2*(b + a*x^5)^(3/4)*Hypergeometric2F1[-3/4, -3/5, 2/5, -((a*x^5)/b)])/(3*x^3*(1 + (a*x^5)/b)^(3/4)) + 2*c*Def
er[Int][(b + a*x^5)^(3/4)/(2*b + c*x^4 + 2*a*x^5), x] + 5*a*Defer[Int][(x*(b + a*x^5)^(3/4))/(2*b + c*x^4 + 2*
a*x^5), x]

Rubi steps

\begin {align*} \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx &=\int \left (-\frac {2 \left (b+a x^5\right )^{3/4}}{x^4}+\frac {(2 c+5 a x) \left (b+a x^5\right )^{3/4}}{2 b+c x^4+2 a x^5}\right ) \, dx\\ &=-\left (2 \int \frac {\left (b+a x^5\right )^{3/4}}{x^4} \, dx\right )+\int \frac {(2 c+5 a x) \left (b+a x^5\right )^{3/4}}{2 b+c x^4+2 a x^5} \, dx\\ &=-\frac {\left (2 \left (b+a x^5\right )^{3/4}\right ) \int \frac {\left (1+\frac {a x^5}{b}\right )^{3/4}}{x^4} \, dx}{\left (1+\frac {a x^5}{b}\right )^{3/4}}+\int \left (\frac {2 c \left (b+a x^5\right )^{3/4}}{2 b+c x^4+2 a x^5}+\frac {5 a x \left (b+a x^5\right )^{3/4}}{2 b+c x^4+2 a x^5}\right ) \, dx\\ &=\frac {2 \left (b+a x^5\right )^{3/4} \, _2F_1\left (-\frac {3}{4},-\frac {3}{5};\frac {2}{5};-\frac {a x^5}{b}\right )}{3 x^3 \left (1+\frac {a x^5}{b}\right )^{3/4}}+(5 a) \int \frac {x \left (b+a x^5\right )^{3/4}}{2 b+c x^4+2 a x^5} \, dx+(2 c) \int \frac {\left (b+a x^5\right )^{3/4}}{2 b+c x^4+2 a x^5} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.57, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-4*b + a*x^5)*(b + a*x^5)^(3/4))/(x^4*(2*b + c*x^4 + 2*a*x^5)),x]

[Out]

Integrate[((-4*b + a*x^5)*(b + a*x^5)^(3/4))/(x^4*(2*b + c*x^4 + 2*a*x^5)), x]

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IntegrateAlgebraic [A]  time = 1.38, size = 154, normalized size = 1.00 \begin {gather*} \frac {2 \left (b+a x^5\right )^{3/4}}{3 x^3}-\frac {c^{3/4} \tan ^{-1}\left (\frac {-\frac {\sqrt [4]{c} x^2}{2^{3/4}}+\frac {\sqrt {b+a x^5}}{\sqrt [4]{2} \sqrt [4]{c}}}{x \sqrt [4]{b+a x^5}}\right )}{2 \sqrt [4]{2}}+\frac {c^{3/4} \tanh ^{-1}\left (\frac {\frac {\sqrt [4]{c} x^2}{2^{3/4}}+\frac {\sqrt {b+a x^5}}{\sqrt [4]{2} \sqrt [4]{c}}}{x \sqrt [4]{b+a x^5}}\right )}{2 \sqrt [4]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-4*b + a*x^5)*(b + a*x^5)^(3/4))/(x^4*(2*b + c*x^4 + 2*a*x^5)),x]

[Out]

(2*(b + a*x^5)^(3/4))/(3*x^3) - (c^(3/4)*ArcTan[(-((c^(1/4)*x^2)/2^(3/4)) + Sqrt[b + a*x^5]/(2^(1/4)*c^(1/4)))
/(x*(b + a*x^5)^(1/4))])/(2*2^(1/4)) + (c^(3/4)*ArcTanh[((c^(1/4)*x^2)/2^(3/4) + Sqrt[b + a*x^5]/(2^(1/4)*c^(1
/4)))/(x*(b + a*x^5)^(1/4))])/(2*2^(1/4))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^5-4*b)*(a*x^5+b)^(3/4)/x^4/(2*a*x^5+c*x^4+2*b),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{5} + b\right )}^{\frac {3}{4}} {\left (a x^{5} - 4 \, b\right )}}{{\left (2 \, a x^{5} + c x^{4} + 2 \, b\right )} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^5-4*b)*(a*x^5+b)^(3/4)/x^4/(2*a*x^5+c*x^4+2*b),x, algorithm="giac")

[Out]

integrate((a*x^5 + b)^(3/4)*(a*x^5 - 4*b)/((2*a*x^5 + c*x^4 + 2*b)*x^4), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{5}-4 b \right ) \left (a \,x^{5}+b \right )^{\frac {3}{4}}}{x^{4} \left (2 a \,x^{5}+c \,x^{4}+2 b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^5-4*b)*(a*x^5+b)^(3/4)/x^4/(2*a*x^5+c*x^4+2*b),x)

[Out]

int((a*x^5-4*b)*(a*x^5+b)^(3/4)/x^4/(2*a*x^5+c*x^4+2*b),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{5} + b\right )}^{\frac {3}{4}} {\left (a x^{5} - 4 \, b\right )}}{{\left (2 \, a x^{5} + c x^{4} + 2 \, b\right )} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^5-4*b)*(a*x^5+b)^(3/4)/x^4/(2*a*x^5+c*x^4+2*b),x, algorithm="maxima")

[Out]

integrate((a*x^5 + b)^(3/4)*(a*x^5 - 4*b)/((2*a*x^5 + c*x^4 + 2*b)*x^4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {{\left (a\,x^5+b\right )}^{3/4}\,\left (4\,b-a\,x^5\right )}{x^4\,\left (2\,a\,x^5+c\,x^4+2\,b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((b + a*x^5)^(3/4)*(4*b - a*x^5))/(x^4*(2*b + 2*a*x^5 + c*x^4)),x)

[Out]

int(-((b + a*x^5)^(3/4)*(4*b - a*x^5))/(x^4*(2*b + 2*a*x^5 + c*x^4)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x^{5} - 4 b\right ) \left (a x^{5} + b\right )^{\frac {3}{4}}}{x^{4} \left (2 a x^{5} + 2 b + c x^{4}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**5-4*b)*(a*x**5+b)**(3/4)/x**4/(2*a*x**5+c*x**4+2*b),x)

[Out]

Integral((a*x**5 - 4*b)*(a*x**5 + b)**(3/4)/(x**4*(2*a*x**5 + 2*b + c*x**4)), x)

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