3.30.70 \(\int \frac {x^3 (5 b+9 a x^4)}{\sqrt [4]{b x+a x^5} (1+b x^5+a x^9)} \, dx\)

Optimal. Leaf size=371 \[ -\sqrt {2} \tan ^{-1}\left (\frac {x^2 \sqrt [4]{a x^5+b x}-2^{2/3} x \sqrt [4]{a x^5+b x}}{2^{2/3} x \sqrt [4]{a x^5+b x}+x^2 \left (-\sqrt [4]{a x^5+b x}\right )-\sqrt {2} x+2 \sqrt [6]{2}}\right )+\sqrt {2} \tan ^{-1}\left (\frac {x^2 \sqrt [4]{a x^5+b x}-2^{2/3} x \sqrt [4]{a x^5+b x}}{2^{2/3} x \sqrt [4]{a x^5+b x}+x^2 \left (-\sqrt [4]{a x^5+b x}\right )+\sqrt {2} x-2 \sqrt [6]{2}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {-2\ 2^{5/6} x \sqrt [4]{a x^5+b x}-\sqrt {2} x^3 \sqrt [4]{a x^5+b x}+4 \sqrt [6]{2} x^2 \sqrt [4]{a x^5+b x}}{x^4 \left (-\sqrt {a x^5+b x}\right )+2\ 2^{2/3} x^3 \sqrt {a x^5+b x}-2 \sqrt [3]{2} x^2 \sqrt {a x^5+b x}-x^2+2\ 2^{2/3} x-2 \sqrt [3]{2}}\right ) \]

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Rubi [F]  time = 2.31, 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^3 \left (5 b+9 a x^4\right )}{\sqrt [4]{b x+a x^5} \left (1+b x^5+a x^9\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^3*(5*b + 9*a*x^4))/((b*x + a*x^5)^(1/4)*(1 + b*x^5 + a*x^9)),x]

[Out]

(20*b*x^(1/4)*(b + a*x^4)^(1/4)*Defer[Subst][Defer[Int][x^14/((b + a*x^16)^(1/4)*(1 + b*x^20 + a*x^36)), x], x
, x^(1/4)])/(b*x + a*x^5)^(1/4) + (36*a*x^(1/4)*(b + a*x^4)^(1/4)*Defer[Subst][Defer[Int][x^30/((b + a*x^16)^(
1/4)*(1 + b*x^20 + a*x^36)), x], x, x^(1/4)])/(b*x + a*x^5)^(1/4)

Rubi steps

\begin {align*} \int \frac {x^3 \left (5 b+9 a x^4\right )}{\sqrt [4]{b x+a x^5} \left (1+b x^5+a x^9\right )} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^4}\right ) \int \frac {x^{11/4} \left (5 b+9 a x^4\right )}{\sqrt [4]{b+a x^4} \left (1+b x^5+a x^9\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^{14} \left (5 b+9 a x^{16}\right )}{\sqrt [4]{b+a x^{16}} \left (1+b x^{20}+a x^{36}\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 {5 b x^{14}}{\sqrt [4]{b+a x^{16}} \left (1+b x^{20}+a x^{36}\right )}+\frac {9 a x^{30}}{\sqrt [4]{b+a x^{16}} \left (1+b x^{20}+a x^{36}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^5}}\\ &=\frac {\left (36 a \sqrt [4]{x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{30}}{\sqrt [4]{b+a x^{16}} \left (1+b x^{20}+a x^{36}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^5}}+\frac {\left (20 b \sqrt [4]{x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{b+a x^{16}} \left (1+b x^{20}+a x^{36}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^5}}\\ \end {align*}

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Mathematica [F]  time = 0.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3 \left (5 b+9 a x^4\right )}{\sqrt [4]{b x+a x^5} \left (1+b x^5+a x^9\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^3*(5*b + 9*a*x^4))/((b*x + a*x^5)^(1/4)*(1 + b*x^5 + a*x^9)),x]

[Out]

Integrate[(x^3*(5*b + 9*a*x^4))/((b*x + a*x^5)^(1/4)*(1 + b*x^5 + a*x^9)), x]

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IntegrateAlgebraic [A]  time = 16.56, size = 371, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tan ^{-1}\left (\frac {-2^{2/3} x \sqrt [4]{b x+a x^5}+x^2 \sqrt [4]{b x+a x^5}}{2 \sqrt [6]{2}-\sqrt {2} x+2^{2/3} x \sqrt [4]{b x+a x^5}-x^2 \sqrt [4]{b x+a x^5}}\right )+\sqrt {2} \tan ^{-1}\left (\frac {-2^{2/3} x \sqrt [4]{b x+a x^5}+x^2 \sqrt [4]{b x+a x^5}}{-2 \sqrt [6]{2}+\sqrt {2} x+2^{2/3} x \sqrt [4]{b x+a x^5}-x^2 \sqrt [4]{b x+a x^5}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {-2 2^{5/6} x \sqrt [4]{b x+a x^5}+4 \sqrt [6]{2} x^2 \sqrt [4]{b x+a x^5}-\sqrt {2} x^3 \sqrt [4]{b x+a x^5}}{-2 \sqrt [3]{2}+2\ 2^{2/3} x-x^2-2 \sqrt [3]{2} x^2 \sqrt {b x+a x^5}+2\ 2^{2/3} x^3 \sqrt {b x+a x^5}-x^4 \sqrt {b x+a x^5}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^3*(5*b + 9*a*x^4))/((b*x + a*x^5)^(1/4)*(1 + b*x^5 + a*x^9)),x]

[Out]

-(Sqrt[2]*ArcTan[(-(2^(2/3)*x*(b*x + a*x^5)^(1/4)) + x^2*(b*x + a*x^5)^(1/4))/(2*2^(1/6) - Sqrt[2]*x + 2^(2/3)
*x*(b*x + a*x^5)^(1/4) - x^2*(b*x + a*x^5)^(1/4))]) + Sqrt[2]*ArcTan[(-(2^(2/3)*x*(b*x + a*x^5)^(1/4)) + x^2*(
b*x + a*x^5)^(1/4))/(-2*2^(1/6) + Sqrt[2]*x + 2^(2/3)*x*(b*x + a*x^5)^(1/4) - x^2*(b*x + a*x^5)^(1/4))] - Sqrt
[2]*ArcTanh[(-2*2^(5/6)*x*(b*x + a*x^5)^(1/4) + 4*2^(1/6)*x^2*(b*x + a*x^5)^(1/4) - Sqrt[2]*x^3*(b*x + a*x^5)^
(1/4))/(-2*2^(1/3) + 2*2^(2/3)*x - x^2 - 2*2^(1/3)*x^2*Sqrt[b*x + a*x^5] + 2*2^(2/3)*x^3*Sqrt[b*x + a*x^5] - x
^4*Sqrt[b*x + a*x^5])]

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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^3*(9*a*x^4+5*b)/(a*x^5+b*x)^(1/4)/(a*x^9+b*x^5+1),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 (9 \, a x^{4} + 5 \, b\right )} x^{3}}{{\left (a x^{9} + b x^{5} + 1\right )} {\left (a x^{5} + b x\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(9*a*x^4+5*b)/(a*x^5+b*x)^(1/4)/(a*x^9+b*x^5+1),x, algorithm="giac")

[Out]

integrate((9*a*x^4 + 5*b)*x^3/((a*x^9 + b*x^5 + 1)*(a*x^5 + b*x)^(1/4)), x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (9 a \,x^{4}+5 b \right )}{\left (a \,x^{5}+b x \right )^{\frac {1}{4}} \left (a \,x^{9}+b \,x^{5}+1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(9*a*x^4+5*b)/(a*x^5+b*x)^(1/4)/(a*x^9+b*x^5+1),x)

[Out]

int(x^3*(9*a*x^4+5*b)/(a*x^5+b*x)^(1/4)/(a*x^9+b*x^5+1),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (9 \, a x^{4} + 5 \, b\right )} x^{3}}{{\left (a x^{9} + b x^{5} + 1\right )} {\left (a x^{5} + b x\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(9*a*x^4+5*b)/(a*x^5+b*x)^(1/4)/(a*x^9+b*x^5+1),x, algorithm="maxima")

[Out]

integrate((9*a*x^4 + 5*b)*x^3/((a*x^9 + b*x^5 + 1)*(a*x^5 + b*x)^(1/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3\,\left (9\,a\,x^4+5\,b\right )}{{\left (a\,x^5+b\,x\right )}^{1/4}\,\left (a\,x^9+b\,x^5+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(5*b + 9*a*x^4))/((b*x + a*x^5)^(1/4)*(a*x^9 + b*x^5 + 1)),x)

[Out]

int((x^3*(5*b + 9*a*x^4))/((b*x + a*x^5)^(1/4)*(a*x^9 + b*x^5 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \left (9 a x^{4} + 5 b\right )}{\sqrt [4]{x \left (a x^{4} + b\right )} \left (a x^{9} + b x^{5} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(9*a*x**4+5*b)/(a*x**5+b*x)**(1/4)/(a*x**9+b*x**5+1),x)

[Out]

Integral(x**3*(9*a*x**4 + 5*b)/((x*(a*x**4 + b))**(1/4)*(a*x**9 + b*x**5 + 1)), x)

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