3.24.12 \(\int \frac {c x^6 (-4 b+a x^5)}{(b+a x^5)^{3/4} (b^2+2 a b x^5-c^2 x^8+a^2 x^{10})} \, dx\)

Optimal. Leaf size=177 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a x^5+b}}\right )}{c^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a x^5+b}}\right )}{c^{3/4}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt {a x^5+b}}{\sqrt {2} \sqrt [4]{c}}-\frac {\sqrt [4]{c} x^2}{\sqrt {2}}}{x \sqrt [4]{a x^5+b}}\right )}{\sqrt {2} c^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{a x^5+b}}{\sqrt {a x^5+b}+\sqrt {c} x^2}\right )}{\sqrt {2} c^{3/4}} \]

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Rubi [F]  time = 4.26, 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 {c x^6 \left (-4 b+a x^5\right )}{\left (b+a x^5\right )^{3/4} \left (b^2+2 a b x^5-c^2 x^8+a^2 x^{10}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(c*x^2*(1 + (a*x^5)/b)^(3/4)*Hypergeometric2F1[2/5, 3/4, 7/5, -((a*x^5)/b)])/(2*a*(b + a*x^5)^(3/4)) - (c^3*De
fer[Int][x^4/((-b + c*x^4 - a*x^5)*(b + a*x^5)^(3/4)), x])/(2*a^2) - (b*c^2*Defer[Int][1/((b + a*x^5)^(3/4)*(b
- c*x^4 + a*x^5)), x])/(2*a^2) - (b*c*Defer[Int][x/((b + a*x^5)^(3/4)*(b - c*x^4 + a*x^5)), x])/(2*a) - (5*b*
Defer[Int][x^2/((b + a*x^5)^(3/4)*(b - c*x^4 + a*x^5)), x])/2 + (b*c^2*Defer[Int][1/((b + a*x^5)^(3/4)*(b + c*
x^4 + a*x^5)), x])/(2*a^2) - (b*c*Defer[Int][x/((b + a*x^5)^(3/4)*(b + c*x^4 + a*x^5)), x])/(2*a) + (5*b*Defer
[Int][x^2/((b + a*x^5)^(3/4)*(b + c*x^4 + a*x^5)), x])/2 + (c^3*Defer[Int][x^4/((b + a*x^5)^(3/4)*(b + c*x^4 +
a*x^5)), x])/(2*a^2)

Rubi steps

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

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Mathematica [F]  time = 1.21, size = 56, normalized size = 0.32 \begin {gather*} c \int \frac {x^6 \left (a x^5-4 b\right )}{\left (a x^5+b\right )^{3/4} \left (a^2 x^{10}+2 a b x^5+b^2-c^2 x^8\right )} \, dx \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*x^6*(-4*b + a*x^5))/((b + a*x^5)^(3/4)*(b^2 + 2*a*b*x^5 - c^2*x^8 + a^2*x^10)),x]

[Out]

c*Integrate[(x^6*(-4*b + a*x^5))/((b + a*x^5)^(3/4)*(b^2 + 2*a*b*x^5 - c^2*x^8 + a^2*x^10)), x]

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IntegrateAlgebraic [A]  time = 14.27, size = 176, normalized size = 0.99 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b+a x^5}}\right )}{c^{3/4}}+\frac {\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}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b+a x^5}}\right )}{c^{3/4}}-\frac {\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 )}{\sqrt {2} c^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(c*x^6*(-4*b + a*x^5))/((b + a*x^5)^(3/4)*(b^2 + 2*a*b*x^5 - c^2*x^8 + a^2*x^10)),x]

[Out]

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

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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(c*x^6*(a*x^5-4*b)/(a*x^5+b)^(3/4)/(a^2*x^10-c^2*x^8+2*a*b*x^5+b^2),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 )} c x^{6}}{{\left (a^{2} x^{10} - c^{2} x^{8} + 2 \, a b x^{5} + b^{2}\right )} {\left (a x^{5} + b\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(c*x^6*(a*x^5-4*b)/(a*x^5+b)^(3/4)/(a^2*x^10-c^2*x^8+2*a*b*x^5+b^2),x, algorithm="giac")

[Out]

integrate((a*x^5 - 4*b)*c*x^6/((a^2*x^10 - c^2*x^8 + 2*a*b*x^5 + b^2)*(a*x^5 + b)^(3/4)), x)

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {c \,x^{6} \left (a \,x^{5}-4 b \right )}{\left (a \,x^{5}+b \right )^{\frac {3}{4}} \left (a^{2} x^{10}-c^{2} x^{8}+2 a b \,x^{5}+b^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(c*x^6*(a*x^5-4*b)/(a*x^5+b)^(3/4)/(a^2*x^10-c^2*x^8+2*a*b*x^5+b^2),x)

[Out]

int(c*x^6*(a*x^5-4*b)/(a*x^5+b)^(3/4)/(a^2*x^10-c^2*x^8+2*a*b*x^5+b^2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} c \int \frac {{\left (a x^{5} - 4 \, b\right )} x^{6}}{{\left (a^{2} x^{10} - c^{2} x^{8} + 2 \, a b x^{5} + b^{2}\right )} {\left (a x^{5} + b\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(c*x^6*(a*x^5-4*b)/(a*x^5+b)^(3/4)/(a^2*x^10-c^2*x^8+2*a*b*x^5+b^2),x, algorithm="maxima")

[Out]

c*integrate((a*x^5 - 4*b)*x^6/((a^2*x^10 - c^2*x^8 + 2*a*b*x^5 + b^2)*(a*x^5 + b)^(3/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {c\,x^6\,\left (4\,b-a\,x^5\right )}{{\left (a\,x^5+b\right )}^{3/4}\,\left (a^2\,x^{10}+2\,a\,b\,x^5+b^2-c^2\,x^8\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(c*x^6*(4*b - a*x^5))/((b + a*x^5)^(3/4)*(b^2 + a^2*x^10 - c^2*x^8 + 2*a*b*x^5)),x)

[Out]

-int((c*x^6*(4*b - a*x^5))/((b + a*x^5)^(3/4)*(b^2 + a^2*x^10 - c^2*x^8 + 2*a*b*x^5)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} c \left (\int \frac {a x^{11}}{a^{2} x^{10} \left (a x^{5} + b\right )^{\frac {3}{4}} + 2 a b x^{5} \left (a x^{5} + b\right )^{\frac {3}{4}} + b^{2} \left (a x^{5} + b\right )^{\frac {3}{4}} - c^{2} x^{8} \left (a x^{5} + b\right )^{\frac {3}{4}}}\, dx + \int \left (- \frac {4 b x^{6}}{a^{2} x^{10} \left (a x^{5} + b\right )^{\frac {3}{4}} + 2 a b x^{5} \left (a x^{5} + b\right )^{\frac {3}{4}} + b^{2} \left (a x^{5} + b\right )^{\frac {3}{4}} - c^{2} x^{8} \left (a x^{5} + b\right )^{\frac {3}{4}}}\right )\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(c*x**6*(a*x**5-4*b)/(a*x**5+b)**(3/4)/(a**2*x**10-c**2*x**8+2*a*b*x**5+b**2),x)

[Out]

c*(Integral(a*x**11/(a**2*x**10*(a*x**5 + b)**(3/4) + 2*a*b*x**5*(a*x**5 + b)**(3/4) + b**2*(a*x**5 + b)**(3/4
) - c**2*x**8*(a*x**5 + b)**(3/4)), x) + Integral(-4*b*x**6/(a**2*x**10*(a*x**5 + b)**(3/4) + 2*a*b*x**5*(a*x*
*5 + b)**(3/4) + b**2*(a*x**5 + b)**(3/4) - c**2*x**8*(a*x**5 + b)**(3/4)), x))

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