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

Optimal. Leaf size=125 \[ \frac {\sqrt {2} \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^5+b}}{\sqrt {2} \sqrt [4]{c}}+\frac {\sqrt [4]{c} x^2}{\sqrt {2}}}{x \sqrt [4]{a x^5+b}}\right )}{c^{3/4}}-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{a x^5+b}}{\sqrt {a x^5+b}-\sqrt {c} x^2}\right )}{c^{3/4}} \]

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Rubi [F]  time = 2.00, 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 {x^2 \left (-4 b+a x^5\right )}{\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[(x^2*(-4*b + a*x^5))/((b + a*x^5)^(3/4)*(b + c*x^4 + a*x^5)),x]

[Out]

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

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Mathematica [F]  time = 0.41, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (-4 b+a x^5\right )}{\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[(x^2*(-4*b + a*x^5))/((b + a*x^5)^(3/4)*(b + c*x^4 + a*x^5)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 12.77, size = 125, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2} \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 )}{c^{3/4}}+\frac {\sqrt {2} \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 )}{c^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[2]*ArcTan[(Sqrt[2]*c^(1/4)*x*(b + a*x^5)^(1/4))/(-(Sqrt[c]*x^2) + Sqrt[b + a*x^5])])/c^(3/4)) + (Sqrt[
2]*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))])/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(x^2*(a*x^5-4*b)/(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} - 4 \, b\right )} x^{2}}{{\left (a x^{5} + c x^{4} + b\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(x^2*(a*x^5-4*b)/(a*x^5+b)^(3/4)/(a*x^5+c*x^4+b),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (a \,x^{5}-4 b \right )}{\left (a \,x^{5}+b \right )^{\frac {3}{4}} \left (a \,x^{5}+c \,x^{4}+b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

[Out]

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

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