\(\int \frac {x^3 (-5 b+9 a x^4)}{\sqrt [4]{-b x+a x^5} (-2-b x^5+a x^9)} \, dx\) [751]

Optimal result

Integrand size = 43, antiderivative size = 58 \[ \int \frac {x^3 \left (-5 b+9 a x^4\right )}{\sqrt [4]{-b x+a x^5} \left (-2-b x^5+a x^9\right )} \, dx=2^{3/4} \arctan \left (\frac {x \sqrt [4]{-b x+a x^5}}{\sqrt [4]{2}}\right )-2^{3/4} \text {arctanh}\left (\frac {x \sqrt [4]{-b x+a x^5}}{\sqrt [4]{2}}\right ) \]

[Out]

2^(3/4)*arctan(1/2*x*(a*x^5-b*x)^(1/4)*2^(3/4))-2^(3/4)*arctanh(1/2*x*(a*x^5-b*x)^(1/4)*2^(3/4))

Rubi [F]

\[ \int \frac {x^3 \left (-5 b+9 a x^4\right )}{\sqrt [4]{-b x+a x^5} \left (-2-b x^5+a x^9\right )} \, dx=\int \frac {x^3 \left (-5 b+9 a x^4\right )}{\sqrt [4]{-b x+a x^5} \left (-2-b x^5+a x^9\right )} \, dx \]

[In]

Int[(x^3*(-5*b + 9*a*x^4))/((-(b*x) + a*x^5)^(1/4)*(-2 - 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)*(2 + 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)*(-2 - b*x^20 + a*x^36)), x], x, x^(1/4)])/(-(b*x) + a*x^5)^(1/4)

Rubi steps \begin{align*} \text {integral}& = \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 (-2-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 ) \text {Subst}\left (\int \frac {x^{14} \left (-5 b+9 a x^{16}\right )}{\sqrt [4]{-b+a x^{16}} \left (-2-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 ) \text {Subst}\left (\int \left (\frac {5 b x^{14}}{\sqrt [4]{-b+a x^{16}} \left (2+b x^{20}-a x^{36}\right )}+\frac {9 a x^{30}}{\sqrt [4]{-b+a x^{16}} \left (-2-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 ) \text {Subst}\left (\int \frac {x^{30}}{\sqrt [4]{-b+a x^{16}} \left (-2-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 ) \text {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{-b+a x^{16}} \left (2+b x^{20}-a x^{36}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {x^3 \left (-5 b+9 a x^4\right )}{\sqrt [4]{-b x+a x^5} \left (-2-b x^5+a x^9\right )} \, dx=\int \frac {x^3 \left (-5 b+9 a x^4\right )}{\sqrt [4]{-b x+a x^5} \left (-2-b x^5+a x^9\right )} \, dx \]

[In]

Integrate[(x^3*(-5*b + 9*a*x^4))/((-(b*x) + a*x^5)^(1/4)*(-2 - b*x^5 + a*x^9)),x]

[Out]

Integrate[(x^3*(-5*b + 9*a*x^4))/((-(b*x) + a*x^5)^(1/4)*(-2 - b*x^5 + a*x^9)), x]

Maple [F]

\[\int \frac {x^{3} \left (9 a \,x^{4}-5 b \right )}{\left (a \,x^{5}-b x \right )^{\frac {1}{4}} \left (a \,x^{9}-b \,x^{5}-2\right )}d x\]

[In]

int(x^3*(9*a*x^4-5*b)/(a*x^5-b*x)^(1/4)/(a*x^9-b*x^5-2),x)

[Out]

int(x^3*(9*a*x^4-5*b)/(a*x^5-b*x)^(1/4)/(a*x^9-b*x^5-2),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {x^3 \left (-5 b+9 a x^4\right )}{\sqrt [4]{-b x+a x^5} \left (-2-b x^5+a x^9\right )} \, dx=\text {Timed out} \]

[In]

integrate(x^3*(9*a*x^4-5*b)/(a*x^5-b*x)^(1/4)/(a*x^9-b*x^5-2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {x^3 \left (-5 b+9 a x^4\right )}{\sqrt [4]{-b x+a x^5} \left (-2-b x^5+a x^9\right )} \, dx=\int \frac {x^{3} \cdot \left (9 a x^{4} - 5 b\right )}{\sqrt [4]{x \left (a x^{4} - b\right )} \left (a x^{9} - b x^{5} - 2\right )}\, dx \]

[In]

integrate(x**3*(9*a*x**4-5*b)/(a*x**5-b*x)**(1/4)/(a*x**9-b*x**5-2),x)

[Out]

Integral(x**3*(9*a*x**4 - 5*b)/((x*(a*x**4 - b))**(1/4)*(a*x**9 - b*x**5 - 2)), x)

Maxima [F]

\[ \int \frac {x^3 \left (-5 b+9 a x^4\right )}{\sqrt [4]{-b x+a x^5} \left (-2-b x^5+a x^9\right )} \, dx=\int { \frac {{\left (9 \, a x^{4} - 5 \, b\right )} x^{3}}{{\left (a x^{9} - b x^{5} - 2\right )} {\left (a x^{5} - b x\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(x^3*(9*a*x^4-5*b)/(a*x^5-b*x)^(1/4)/(a*x^9-b*x^5-2),x, algorithm="maxima")

[Out]

integrate((9*a*x^4 - 5*b)*x^3/((a*x^9 - b*x^5 - 2)*(a*x^5 - b*x)^(1/4)), x)

Giac [F]

\[ \int \frac {x^3 \left (-5 b+9 a x^4\right )}{\sqrt [4]{-b x+a x^5} \left (-2-b x^5+a x^9\right )} \, dx=\int { \frac {{\left (9 \, a x^{4} - 5 \, b\right )} x^{3}}{{\left (a x^{9} - b x^{5} - 2\right )} {\left (a x^{5} - b x\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(x^3*(9*a*x^4-5*b)/(a*x^5-b*x)^(1/4)/(a*x^9-b*x^5-2),x, algorithm="giac")

[Out]

integrate((9*a*x^4 - 5*b)*x^3/((a*x^9 - b*x^5 - 2)*(a*x^5 - b*x)^(1/4)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \left (-5 b+9 a x^4\right )}{\sqrt [4]{-b x+a x^5} \left (-2-b x^5+a x^9\right )} \, dx=\int \frac {x^3\,\left (5\,b-9\,a\,x^4\right )}{{\left (a\,x^5-b\,x\right )}^{1/4}\,\left (-a\,x^9+b\,x^5+2\right )} \,d x \]

[In]

int((x^3*(5*b - 9*a*x^4))/((a*x^5 - b*x)^(1/4)*(b*x^5 - a*x^9 + 2)),x)

[Out]

int((x^3*(5*b - 9*a*x^4))/((a*x^5 - b*x)^(1/4)*(b*x^5 - a*x^9 + 2)), x)