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

Optimal. Leaf size=142 \[ 2 \sqrt {2} \sqrt [4]{c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{a x^5+b}}{\sqrt {a x^5+b}-\sqrt {c} x^2}\right )-2 \sqrt {2} \sqrt [4]{c} \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^5+b}}{\sqrt {2} \sqrt [4]{c}}+\frac {\sqrt [4]{c} x^2}{\sqrt {2}}}{x \sqrt [4]{a x^5+b}}\right )+\frac {4 \sqrt [4]{a x^5+b}}{x} \]

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Rubi [F]  time = 3.46, 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-c x^4+a x^5\right )}{x^2 \left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(4*b*(1 + (a*x^5)/b)^(3/4)*Hypergeometric2F1[-1/5, 3/4, 4/5, -((a*x^5)/b)])/(x*(b + a*x^5)^(3/4)) - (2*c^3*x*(
1 + (a*x^5)/b)^(3/4)*Hypergeometric2F1[1/5, 3/4, 6/5, -((a*x^5)/b)])/(a^2*(b + a*x^5)^(3/4)) + (c^2*x^2*(1 + (
a*x^5)/b)^(3/4)*Hypergeometric2F1[2/5, 3/4, 7/5, -((a*x^5)/b)])/(a*(b + a*x^5)^(3/4)) - (2*c*x^3*(1 + (a*x^5)/
b)^(3/4)*Hypergeometric2F1[3/5, 3/4, 8/5, -((a*x^5)/b)])/(3*(b + a*x^5)^(3/4)) + (a*x^4*(1 + (a*x^5)/b)^(3/4)*
Hypergeometric2F1[3/4, 4/5, 9/5, -((a*x^5)/b)])/(4*(b + a*x^5)^(3/4)) + (2*b*c^3*Defer[Int][1/((b + a*x^5)^(3/
4)*(b + c*x^4 + a*x^5)), x])/a^2 - (2*b*c^2*Defer[Int][x/((b + a*x^5)^(3/4)*(b + c*x^4 + a*x^5)), x])/a + 10*b
*c*Defer[Int][x^2/((b + a*x^5)^(3/4)*(b + c*x^4 + a*x^5)), x] + (2*c^4*Defer[Int][x^4/((b + a*x^5)^(3/4)*(b +
c*x^4 + a*x^5)), x])/a^2

Rubi steps

\begin {align*} \int \frac {\left (-4 b+a x^5\right ) \left (b-c x^4+a x^5\right )}{x^2 \left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )} \, dx &=\int \left (-\frac {2 c^3}{a^2 \left (b+a x^5\right )^{3/4}}-\frac {4 b}{x^2 \left (b+a x^5\right )^{3/4}}+\frac {2 c^2 x}{a \left (b+a x^5\right )^{3/4}}-\frac {2 c x^2}{\left (b+a x^5\right )^{3/4}}+\frac {a x^3}{\left (b+a x^5\right )^{3/4}}+\frac {2 \left (b c^3-a b c^2 x+5 a^2 b c x^2+c^4 x^4\right )}{a^2 \left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )}\right ) \, dx\\ &=\frac {2 \int \frac {b c^3-a b c^2 x+5 a^2 b c x^2+c^4 x^4}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )} \, dx}{a^2}+a \int \frac {x^3}{\left (b+a x^5\right )^{3/4}} \, dx-(4 b) \int \frac {1}{x^2 \left (b+a x^5\right )^{3/4}} \, dx-(2 c) \int \frac {x^2}{\left (b+a x^5\right )^{3/4}} \, dx+\frac {\left (2 c^2\right ) \int \frac {x}{\left (b+a x^5\right )^{3/4}} \, dx}{a}-\frac {\left (2 c^3\right ) \int \frac {1}{\left (b+a x^5\right )^{3/4}} \, dx}{a^2}\\ &=\frac {2 \int \left (\frac {b c^3}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )}-\frac {a b c^2 x}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )}+\frac {5 a^2 b c x^2}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )}+\frac {c^4 x^4}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )}\right ) \, dx}{a^2}+\frac {\left (a \left (1+\frac {a x^5}{b}\right )^{3/4}\right ) \int \frac {x^3}{\left (1+\frac {a x^5}{b}\right )^{3/4}} \, dx}{\left (b+a x^5\right )^{3/4}}-\frac {\left (4 b \left (1+\frac {a x^5}{b}\right )^{3/4}\right ) \int \frac {1}{x^2 \left (1+\frac {a x^5}{b}\right )^{3/4}} \, dx}{\left (b+a x^5\right )^{3/4}}-\frac {\left (2 c \left (1+\frac {a x^5}{b}\right )^{3/4}\right ) \int \frac {x^2}{\left (1+\frac {a x^5}{b}\right )^{3/4}} \, dx}{\left (b+a x^5\right )^{3/4}}+\frac {\left (2 c^2 \left (1+\frac {a x^5}{b}\right )^{3/4}\right ) \int \frac {x}{\left (1+\frac {a x^5}{b}\right )^{3/4}} \, dx}{a \left (b+a x^5\right )^{3/4}}-\frac {\left (2 c^3 \left (1+\frac {a x^5}{b}\right )^{3/4}\right ) \int \frac {1}{\left (1+\frac {a x^5}{b}\right )^{3/4}} \, dx}{a^2 \left (b+a x^5\right )^{3/4}}\\ &=\frac {4 b \left (1+\frac {a x^5}{b}\right )^{3/4} \, _2F_1\left (-\frac {1}{5},\frac {3}{4};\frac {4}{5};-\frac {a x^5}{b}\right )}{x \left (b+a x^5\right )^{3/4}}-\frac {2 c^3 x \left (1+\frac {a x^5}{b}\right )^{3/4} \, _2F_1\left (\frac {1}{5},\frac {3}{4};\frac {6}{5};-\frac {a x^5}{b}\right )}{a^2 \left (b+a x^5\right )^{3/4}}+\frac {c^2 x^2 \left (1+\frac {a x^5}{b}\right )^{3/4} \, _2F_1\left (\frac {2}{5},\frac {3}{4};\frac {7}{5};-\frac {a x^5}{b}\right )}{a \left (b+a x^5\right )^{3/4}}-\frac {2 c x^3 \left (1+\frac {a x^5}{b}\right )^{3/4} \, _2F_1\left (\frac {3}{5},\frac {3}{4};\frac {8}{5};-\frac {a x^5}{b}\right )}{3 \left (b+a x^5\right )^{3/4}}+\frac {a x^4 \left (1+\frac {a x^5}{b}\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {4}{5};\frac {9}{5};-\frac {a x^5}{b}\right )}{4 \left (b+a x^5\right )^{3/4}}+(10 b c) \int \frac {x^2}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )} \, dx-\frac {\left (2 b c^2\right ) \int \frac {x}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )} \, dx}{a}+\frac {\left (2 b c^3\right ) \int \frac {1}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )} \, dx}{a^2}+\frac {\left (2 c^4\right ) \int \frac {x^4}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )} \, dx}{a^2}\\ \end {align*}

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Mathematica [F]  time = 0.86, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-4 b+a x^5\right ) \left (b-c x^4+a x^5\right )}{x^2 \left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 13.14, size = 142, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{b+a x^5}}{x}+2 \sqrt {2} \sqrt [4]{c} \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 )-2 \sqrt {2} \sqrt [4]{c} \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[((-4*b + a*x^5)*(b - c*x^4 + a*x^5))/(x^2*(b + a*x^5)^(3/4)*(b + c*x^4 + a*x^5)),x]

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*x^5-4*b)*(a*x^5-c*x^4+b)/x^2/(a*x^5+b)^(3/4)/(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} - c x^{4} + b\right )} {\left (a x^{5} - 4 \, b\right )}}{{\left (a x^{5} + c x^{4} + b\right )} {\left (a x^{5} + b\right )}^{\frac {3}{4}} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

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

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