3.7.75 \(\int \frac {-3 b+a x^4}{(b-x^3+a x^4) \sqrt [4]{b x+a x^5}} \, dx\)

Optimal. Leaf size=53 \[ -2 \tan ^{-1}\left (\frac {\left (a x^5+b x\right )^{3/4}}{a x^4+b}\right )-2 \tanh ^{-1}\left (\frac {\left (a x^5+b x\right )^{3/4}}{a x^4+b}\right ) \]

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Rubi [F]  time = 2.29, 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 {-3 b+a x^4}{\left (b-x^3+a x^4\right ) \sqrt [4]{b x+a x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-3*b + a*x^4)/((b - x^3 + a*x^4)*(b*x + a*x^5)^(1/4)),x]

[Out]

(4*x*(1 + (a*x^4)/b)^(1/4)*Hypergeometric2F1[3/16, 1/4, 19/16, -((a*x^4)/b)])/(3*(b*x + a*x^5)^(1/4)) - (16*b*
x^(1/4)*(b + a*x^4)^(1/4)*Defer[Subst][Defer[Int][x^2/((b + a*x^16)^(1/4)*(b - x^12 + a*x^16)), x], x, x^(1/4)
])/(b*x + a*x^5)^(1/4) + (4*x^(1/4)*(b + a*x^4)^(1/4)*Defer[Subst][Defer[Int][x^14/((b + a*x^16)^(1/4)*(b - x^
12 + a*x^16)), x], x, x^(1/4)])/(b*x + a*x^5)^(1/4)

Rubi steps

\begin {align*} \int \frac {-3 b+a x^4}{\left (b-x^3+a x^4\right ) \sqrt [4]{b x+a x^5}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^4}\right ) \int \frac {-3 b+a x^4}{\sqrt [4]{x} \sqrt [4]{b+a x^4} \left (b-x^3+a x^4\right )} \, dx}{\sqrt [4]{b x+a x^5}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-3 b+a x^{16}\right )}{\sqrt [4]{b+a x^{16}} \left (b-x^{12}+a x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^5}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {x^2}{\sqrt [4]{b+a x^{16}}}+\frac {x^2 \left (-4 b+x^{12}\right )}{\sqrt [4]{b+a x^{16}} \left (b-x^{12}+a x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^5}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{b+a x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-4 b+x^{12}\right )}{\sqrt [4]{b+a x^{16}} \left (b-x^{12}+a x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^5}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {4 b x^2}{\sqrt [4]{b+a x^{16}} \left (b-x^{12}+a x^{16}\right )}+\frac {x^{14}}{\sqrt [4]{b+a x^{16}} \left (b-x^{12}+a x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+\frac {a x^4}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+\frac {a x^{16}}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^5}}\\ &=\frac {4 x \sqrt [4]{1+\frac {a x^4}{b}} \, _2F_1\left (\frac {3}{16},\frac {1}{4};\frac {19}{16};-\frac {a x^4}{b}\right )}{3 \sqrt [4]{b x+a x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{b+a x^{16}} \left (b-x^{12}+a x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^5}}-\frac {\left (16 b \sqrt [4]{x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{b+a x^{16}} \left (b-x^{12}+a x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^5}}\\ \end {align*}

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

Verification is not applicable to the result.

[In]

Integrate[(-3*b + a*x^4)/((b - x^3 + a*x^4)*(b*x + a*x^5)^(1/4)),x]

[Out]

Integrate[(-3*b + a*x^4)/((b - x^3 + a*x^4)*(b*x + a*x^5)^(1/4)), x]

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IntegrateAlgebraic [A]  time = 1.16, size = 53, normalized size = 1.00 \begin {gather*} -2 \tan ^{-1}\left (\frac {\left (b x+a x^5\right )^{3/4}}{b+a x^4}\right )-2 \tanh ^{-1}\left (\frac {\left (b x+a x^5\right )^{3/4}}{b+a x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-3*b + a*x^4)/((b - x^3 + a*x^4)*(b*x + a*x^5)^(1/4)),x]

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4-3*b)/(a*x^4-x^3+b)/(a*x^5+b*x)^(1/4),x, algorithm="giac")

[Out]

integrate((a*x^4 - 3*b)/((a*x^5 + b*x)^(1/4)*(a*x^4 - x^3 + b)), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{4}-3 b}{\left (a \,x^{4}-x^{3}+b \right ) \left (a \,x^{5}+b x \right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^4-3*b)/(a*x^4-x^3+b)/(a*x^5+b*x)^(1/4),x)

[Out]

int((a*x^4-3*b)/(a*x^4-x^3+b)/(a*x^5+b*x)^(1/4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{4} - 3 \, b}{{\left (a x^{5} + b x\right )}^{\frac {1}{4}} {\left (a x^{4} - x^{3} + b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4-3*b)/(a*x^4-x^3+b)/(a*x^5+b*x)^(1/4),x, algorithm="maxima")

[Out]

integrate((a*x^4 - 3*b)/((a*x^5 + b*x)^(1/4)*(a*x^4 - x^3 + b)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int -\frac {3\,b-a\,x^4}{{\left (a\,x^5+b\,x\right )}^{1/4}\,\left (a\,x^4-x^3+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*b - a*x^4)/((b*x + a*x^5)^(1/4)*(b + a*x^4 - x^3)),x)

[Out]

int(-(3*b - a*x^4)/((b*x + a*x^5)^(1/4)*(b + a*x^4 - x^3)), x)

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sympy [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**4-3*b)/(a*x**4-x**3+b)/(a*x**5+b*x)**(1/4),x)

[Out]

Timed out

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