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

Optimal. Leaf size=152 \[ \sqrt {2} c^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{a x^5-b}}{\sqrt {a x^5-b}-\sqrt {c} x^2}\right )+\sqrt {2} c^{3/4} \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^5-b}}{\sqrt {2} \sqrt [4]{c}}+\frac {\sqrt [4]{c} x^2}{\sqrt {2}}}{x \sqrt [4]{a x^5-b}}\right )+\frac {4 \left (a x^5-b\right )^{3/4}}{3 x^3} \]

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Rubi [F]  time = 1.23, 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 (-b+a x^5\right )^{3/4} \left (4 b+a x^5\right )}{x^4 \left (-b+c x^4+a x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.73, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-b+a x^5\right )^{3/4} \left (4 b+a x^5\right )}{x^4 \left (-b+c x^4+a x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.18, size = 152, normalized size = 1.00 \begin {gather*} \frac {4 \left (-b+a x^5\right )^{3/4}}{3 x^3}+\sqrt {2} c^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^5}}{-\sqrt {c} x^2+\sqrt {-b+a x^5}}\right )+\sqrt {2} c^{3/4} \tanh ^{-1}\left (\frac {\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {-b+a x^5}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{-b+a x^5}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(-b + a*x^5)^(3/4))/(3*x^3) + Sqrt[2]*c^(3/4)*ArcTan[(Sqrt[2]*c^(1/4)*x*(-b + a*x^5)^(1/4))/(-(Sqrt[c]*x^2)
+ Sqrt[-b + a*x^5])] + Sqrt[2]*c^(3/4)*ArcTanh[((c^(1/4)*x^2)/Sqrt[2] + Sqrt[-b + a*x^5]/(Sqrt[2]*c^(1/4)))/(
x*(-b + a*x^5)^(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-b)^(3/4)*(a*x^5+4*b)/x^4/(a*x^5+c*x^4-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} + 4 \, b\right )} {\left (a x^{5} - b\right )}^{\frac {3}{4}}}{{\left (a x^{5} + c x^{4} - b\right )} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{5}-b \right )^{\frac {3}{4}} \left (a \,x^{5}+4 b \right )}{x^{4} \left (a \,x^{5}+c \,x^{4}-b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*x^5-b)^(3/4)*(a*x^5+4*b)/x^4/(a*x^5+c*x^4-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} + 4 \, b\right )} {\left (a x^{5} - b\right )}^{\frac {3}{4}}}{{\left (a x^{5} + c x^{4} - b\right )} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x^5 + 4*b)*(a*x^5 - b)^(3/4)/((a*x^5 + c*x^4 - 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 (a\,x^5+4\,b\right )}{x^4\,\left (a\,x^5+c\,x^4-b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(((a*x^5 - b)^(3/4)*(4*b + a*x^5))/(x^4*(a*x^5 - b + 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} - b\right )^{\frac {3}{4}} \left (a x^{5} + 4 b\right )}{x^{4} \left (a x^{5} - b + c x^{4}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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