3.31.66 \(\int \frac {x^3 (-5 b+6 a x)}{\sqrt [4]{-b x+a x^2} (c-b x^5+a x^6)} \, dx\)

Optimal. Leaf size=477 \[ -\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt [3]{c} x^2 \sqrt [4]{a x^2-b x}-2^{2/3} x \sqrt [4]{a x^2-b x}}{-\sqrt [3]{c} x^2 \sqrt [4]{a x^2-b x}+2^{2/3} x \sqrt [4]{a x^2-b x}-\sqrt {2} c^{7/12} x+2 \sqrt [6]{2} \sqrt [4]{c}}\right )}{\sqrt [4]{c}}+\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt [3]{c} x^2 \sqrt [4]{a x^2-b x}-2^{2/3} x \sqrt [4]{a x^2-b x}}{-\sqrt [3]{c} x^2 \sqrt [4]{a x^2-b x}+2^{2/3} x \sqrt [4]{a x^2-b x}+\sqrt {2} c^{7/12} x-2 \sqrt [6]{2} \sqrt [4]{c}}\right )}{\sqrt [4]{c}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {-4 \sqrt [6]{2} c^{7/12} x^2 \sqrt [4]{a x^2-b x}+\sqrt {2} c^{11/12} x^3 \sqrt [4]{a x^2-b x}+2\ 2^{5/6} \sqrt [4]{c} x \sqrt [4]{a x^2-b x}}{c^{2/3} x^4 \sqrt {a x^2-b x}-2\ 2^{2/3} \sqrt [3]{c} x^3 \sqrt {a x^2-b x}+2 \sqrt [3]{2} x^2 \sqrt {a x^2-b x}+c^{7/6} x^2-2\ 2^{2/3} c^{5/6} x+2 \sqrt [3]{2} \sqrt {c}}\right )}{\sqrt [4]{c}} \]

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Rubi [F]  time = 3.00, 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^3 (-5 b+6 a x)}{\sqrt [4]{-b x+a x^2} \left (c-b x^5+a x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^3*(-5*b + 6*a*x))/((-(b*x) + a*x^2)^(1/4)*(c - b*x^5 + a*x^6)),x]

[Out]

(20*b*x^(1/4)*(-b + a*x)^(1/4)*Defer[Subst][Defer[Int][x^14/((-b + a*x^4)^(1/4)*(-c + b*x^20 - a*x^24)), x], x
, x^(1/4)])/(-(b*x) + a*x^2)^(1/4) + (24*a*x^(1/4)*(-b + a*x)^(1/4)*Defer[Subst][Defer[Int][x^18/((-b + a*x^4)
^(1/4)*(c - b*x^20 + a*x^24)), x], x, x^(1/4)])/(-(b*x) + a*x^2)^(1/4)

Rubi steps

\begin {align*} \int \frac {x^3 (-5 b+6 a x)}{\sqrt [4]{-b x+a x^2} \left (c-b x^5+a x^6\right )} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x}\right ) \int \frac {x^{11/4} (-5 b+6 a x)}{\sqrt [4]{-b+a x} \left (c-b x^5+a x^6\right )} \, dx}{\sqrt [4]{-b x+a x^2}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {x^{14} \left (-5 b+6 a x^4\right )}{\sqrt [4]{-b+a x^4} \left (c-b x^{20}+a x^{24}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^2}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \left (\frac {5 b x^{14}}{\sqrt [4]{-b+a x^4} \left (-c+b x^{20}-a x^{24}\right )}+\frac {6 a x^{18}}{\sqrt [4]{-b+a x^4} \left (c-b x^{20}+a x^{24}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^2}}\\ &=\frac {\left (24 a \sqrt [4]{x} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {x^{18}}{\sqrt [4]{-b+a x^4} \left (c-b x^{20}+a x^{24}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^2}}+\frac {\left (20 b \sqrt [4]{x} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{-b+a x^4} \left (-c+b x^{20}-a x^{24}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^2}}\\ \end {align*}

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Mathematica [F]  time = 2.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3 (-5 b+6 a x)}{\sqrt [4]{-b x+a x^2} \left (c-b x^5+a x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^3*(-5*b + 6*a*x))/((-(b*x) + a*x^2)^(1/4)*(c - b*x^5 + a*x^6)),x]

[Out]

Integrate[(x^3*(-5*b + 6*a*x))/((-(b*x) + a*x^2)^(1/4)*(c - b*x^5 + a*x^6)), x]

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IntegrateAlgebraic [A]  time = 8.06, size = 477, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2} \tan ^{-1}\left (\frac {-2^{2/3} x \sqrt [4]{-b x+a x^2}+\sqrt [3]{c} x^2 \sqrt [4]{-b x+a x^2}}{2 \sqrt [6]{2} \sqrt [4]{c}-\sqrt {2} c^{7/12} x+2^{2/3} x \sqrt [4]{-b x+a x^2}-\sqrt [3]{c} x^2 \sqrt [4]{-b x+a x^2}}\right )}{\sqrt [4]{c}}+\frac {\sqrt {2} \tan ^{-1}\left (\frac {-2^{2/3} x \sqrt [4]{-b x+a x^2}+\sqrt [3]{c} x^2 \sqrt [4]{-b x+a x^2}}{-2 \sqrt [6]{2} \sqrt [4]{c}+\sqrt {2} c^{7/12} x+2^{2/3} x \sqrt [4]{-b x+a x^2}-\sqrt [3]{c} x^2 \sqrt [4]{-b x+a x^2}}\right )}{\sqrt [4]{c}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {2\ 2^{5/6} \sqrt [4]{c} x \sqrt [4]{-b x+a x^2}-4 \sqrt [6]{2} c^{7/12} x^2 \sqrt [4]{-b x+a x^2}+\sqrt {2} c^{11/12} x^3 \sqrt [4]{-b x+a x^2}}{2 \sqrt [3]{2} \sqrt {c}-2\ 2^{2/3} c^{5/6} x+c^{7/6} x^2+2 \sqrt [3]{2} x^2 \sqrt {-b x+a x^2}-2\ 2^{2/3} \sqrt [3]{c} x^3 \sqrt {-b x+a x^2}+c^{2/3} x^4 \sqrt {-b x+a x^2}}\right )}{\sqrt [4]{c}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^3*(-5*b + 6*a*x))/((-(b*x) + a*x^2)^(1/4)*(c - b*x^5 + a*x^6)),x]

[Out]

-((Sqrt[2]*ArcTan[(-(2^(2/3)*x*(-(b*x) + a*x^2)^(1/4)) + c^(1/3)*x^2*(-(b*x) + a*x^2)^(1/4))/(2*2^(1/6)*c^(1/4
) - Sqrt[2]*c^(7/12)*x + 2^(2/3)*x*(-(b*x) + a*x^2)^(1/4) - c^(1/3)*x^2*(-(b*x) + a*x^2)^(1/4))])/c^(1/4)) + (
Sqrt[2]*ArcTan[(-(2^(2/3)*x*(-(b*x) + a*x^2)^(1/4)) + c^(1/3)*x^2*(-(b*x) + a*x^2)^(1/4))/(-2*2^(1/6)*c^(1/4)
+ Sqrt[2]*c^(7/12)*x + 2^(2/3)*x*(-(b*x) + a*x^2)^(1/4) - c^(1/3)*x^2*(-(b*x) + a*x^2)^(1/4))])/c^(1/4) - (Sqr
t[2]*ArcTanh[(2*2^(5/6)*c^(1/4)*x*(-(b*x) + a*x^2)^(1/4) - 4*2^(1/6)*c^(7/12)*x^2*(-(b*x) + a*x^2)^(1/4) + Sqr
t[2]*c^(11/12)*x^3*(-(b*x) + a*x^2)^(1/4))/(2*2^(1/3)*Sqrt[c] - 2*2^(2/3)*c^(5/6)*x + c^(7/6)*x^2 + 2*2^(1/3)*
x^2*Sqrt[-(b*x) + a*x^2] - 2*2^(2/3)*c^(1/3)*x^3*Sqrt[-(b*x) + a*x^2] + c^(2/3)*x^4*Sqrt[-(b*x) + a*x^2])])/c^
(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(x^3*(6*a*x-5*b)/(a*x^2-b*x)^(1/4)/(a*x^6-b*x^5+c),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 (6 \, a x - 5 \, b\right )} x^{3}}{{\left (a x^{6} - b x^{5} + c\right )} {\left (a x^{2} - b x\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(6*a*x-5*b)/(a*x^2-b*x)^(1/4)/(a*x^6-b*x^5+c),x, algorithm="giac")

[Out]

integrate((6*a*x - 5*b)*x^3/((a*x^6 - b*x^5 + c)*(a*x^2 - b*x)^(1/4)), x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (6 a x -5 b \right )}{\left (a \,x^{2}-b x \right )^{\frac {1}{4}} \left (a \,x^{6}-b \,x^{5}+c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(6*a*x-5*b)/(a*x^2-b*x)^(1/4)/(a*x^6-b*x^5+c),x)

[Out]

int(x^3*(6*a*x-5*b)/(a*x^2-b*x)^(1/4)/(a*x^6-b*x^5+c),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (6 \, a x - 5 \, b\right )} x^{3}}{{\left (a x^{6} - b x^{5} + c\right )} {\left (a x^{2} - b x\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(6*a*x-5*b)/(a*x^2-b*x)^(1/4)/(a*x^6-b*x^5+c),x, algorithm="maxima")

[Out]

integrate((6*a*x - 5*b)*x^3/((a*x^6 - b*x^5 + c)*(a*x^2 - 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 (5\,b-6\,a\,x\right )}{{\left (a\,x^2-b\,x\right )}^{1/4}\,\left (a\,x^6-b\,x^5+c\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^3*(5*b - 6*a*x))/((a*x^2 - b*x)^(1/4)*(c + a*x^6 - b*x^5)),x)

[Out]

int(-(x^3*(5*b - 6*a*x))/((a*x^2 - b*x)^(1/4)*(c + a*x^6 - b*x^5)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(6*a*x-5*b)/(a*x**2-b*x)**(1/4)/(a*x**6-b*x**5+c),x)

[Out]

Integral(x**3*(6*a*x - 5*b)/((x*(a*x - b))**(1/4)*(a*x**6 - b*x**5 + c)), x)

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