\(\int \frac {c x^6 (-4 b+a x^5)}{(b+a x^5)^{3/4} (b^2+2 a b x^5-c^2 x^8+a^2 x^{10})} \, dx\) [2312]

Optimal result

Integrand size = 53, antiderivative size = 177 \[ \int \frac {c x^6 \left (-4 b+a x^5\right )}{\left (b+a x^5\right )^{3/4} \left (b^2+2 a b x^5-c^2 x^8+a^2 x^{10}\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b+a x^5}}\right )}{c^{3/4}}-\frac {\arctan \left (\frac {-\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {b+a x^5}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{b+a x^5}}\right )}{\sqrt {2} c^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b+a x^5}}\right )}{c^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^5}}{\sqrt {c} x^2+\sqrt {b+a x^5}}\right )}{\sqrt {2} c^{3/4}} \]

[Out]

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

Rubi [F]

\[ \int \frac {c x^6 \left (-4 b+a x^5\right )}{\left (b+a x^5\right )^{3/4} \left (b^2+2 a b x^5-c^2 x^8+a^2 x^{10}\right )} \, dx=\int \frac {c x^6 \left (-4 b+a x^5\right )}{\left (b+a x^5\right )^{3/4} \left (b^2+2 a b x^5-c^2 x^8+a^2 x^{10}\right )} \, dx \]

[In]

Int[(c*x^6*(-4*b + a*x^5))/((b + a*x^5)^(3/4)*(b^2 + 2*a*b*x^5 - c^2*x^8 + a^2*x^10)),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = c \int \frac {x^6 \left (-4 b+a x^5\right )}{\left (b+a x^5\right )^{3/4} \left (b^2+2 a b x^5-c^2 x^8+a^2 x^{10}\right )} \, dx \\ & = c \int \left (\frac {x}{a \left (b+a x^5\right )^{3/4}}+\frac {x \left (-b^2-6 a b x^5+c^2 x^8\right )}{a \left (b+a x^5\right )^{3/4} \left (b^2+2 a b x^5-c^2 x^8+a^2 x^{10}\right )}\right ) \, dx \\ & = \frac {c \int \frac {x}{\left (b+a x^5\right )^{3/4}} \, dx}{a}+\frac {c \int \frac {x \left (-b^2-6 a b x^5+c^2 x^8\right )}{\left (b+a x^5\right )^{3/4} \left (b^2+2 a b x^5-c^2 x^8+a^2 x^{10}\right )} \, dx}{a} \\ & = \frac {c \int \left (\frac {-b c^2-a b c x-5 a^2 b x^2+c^3 x^4}{2 a c \left (b+a x^5\right )^{3/4} \left (b-c x^4+a x^5\right )}+\frac {b c^2-a b c x+5 a^2 b x^2+c^3 x^4}{2 a c \left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )}\right ) \, dx}{a}+\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 {c x^2 \left (1+\frac {a x^5}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\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 {\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}{2 a^2}+\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}{2 a^2} \\ & = \frac {c x^2 \left (1+\frac {a x^5}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\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 {\int \left (-\frac {c^3 x^4}{\left (-b+c x^4-a x^5\right ) \left (b+a x^5\right )^{3/4}}-\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 )}\right ) \, dx}{2 a^2}+\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}{2 a^2} \\ & = \frac {c x^2 \left (1+\frac {a x^5}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\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 {1}{2} (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 {1}{2} (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}{2 a}-\frac {(b c) \int \frac {x}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )} \, dx}{2 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}{2 a^2}+\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}{2 a^2}-\frac {c^3 \int \frac {x^4}{\left (-b+c x^4-a x^5\right ) \left (b+a x^5\right )^{3/4}} \, dx}{2 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}{2 a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.62 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.90 \[ \int \frac {c x^6 \left (-4 b+a x^5\right )}{\left (b+a x^5\right )^{3/4} \left (b^2+2 a b x^5-c^2 x^8+a^2 x^{10}\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b+a x^5}}\right )+\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^5}}{-\sqrt {c} x^2+\sqrt {b+a x^5}}\right )-2 \text {arctanh}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b+a x^5}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {c} x^2+\sqrt {b+a x^5}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^5}}\right )}{2 c^{3/4}} \]

[In]

Integrate[(c*x^6*(-4*b + a*x^5))/((b + a*x^5)^(3/4)*(b^2 + 2*a*b*x^5 - c^2*x^8 + a^2*x^10)),x]

[Out]

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

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.19

method	result	size
pseudoelliptic	\(\frac {-2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{5}+b \right )^{\frac {1}{4}}+c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right ) \sqrt {2}-2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{5}+b \right )^{\frac {1}{4}}-c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right ) \sqrt {2}-\ln \left (\frac {\left (a \,x^{5}+b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}+\sqrt {a \,x^{5}+b}}{\sqrt {a \,x^{5}+b}-\left (a \,x^{5}+b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}}\right ) \sqrt {2}-4 \arctan \left (\frac {\left (a \,x^{5}+b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x}\right )-2 \ln \left (\frac {-c^{\frac {1}{4}} x -\left (a \,x^{5}+b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x -\left (a \,x^{5}+b \right )^{\frac {1}{4}}}\right )}{4 c^{\frac {3}{4}}}\)	\(210\)

[In]

int(c*x^6*(a*x^5-4*b)/(a*x^5+b)^(3/4)/(a^2*x^10-c^2*x^8+2*a*b*x^5+b^2),x,method=_RETURNVERBOSE)

[Out]

1/4*(-2*arctan((2^(1/2)*(a*x^5+b)^(1/4)+c^(1/4)*x)/c^(1/4)/x)*2^(1/2)-2*arctan((2^(1/2)*(a*x^5+b)^(1/4)-c^(1/4
)*x)/c^(1/4)/x)*2^(1/2)-ln(((a*x^5+b)^(1/4)*x*c^(1/4)*2^(1/2)+c^(1/2)*x^2+(a*x^5+b)^(1/2))/((a*x^5+b)^(1/2)-(a
*x^5+b)^(1/4)*x*c^(1/4)*2^(1/2)+c^(1/2)*x^2))*2^(1/2)-4*arctan(1/c^(1/4)/x*(a*x^5+b)^(1/4))-2*ln((-c^(1/4)*x-(
a*x^5+b)^(1/4))/(c^(1/4)*x-(a*x^5+b)^(1/4))))/c^(3/4)

Fricas [F(-1)]

Timed out. \[ \int \frac {c x^6 \left (-4 b+a x^5\right )}{\left (b+a x^5\right )^{3/4} \left (b^2+2 a b x^5-c^2 x^8+a^2 x^{10}\right )} \, dx=\text {Timed out} \]

[In]

integrate(c*x^6*(a*x^5-4*b)/(a*x^5+b)^(3/4)/(a^2*x^10-c^2*x^8+2*a*b*x^5+b^2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {c x^6 \left (-4 b+a x^5\right )}{\left (b+a x^5\right )^{3/4} \left (b^2+2 a b x^5-c^2 x^8+a^2 x^{10}\right )} \, dx=c \left (\int \frac {a x^{11}}{a^{2} x^{10} \left (a x^{5} + b\right )^{\frac {3}{4}} + 2 a b x^{5} \left (a x^{5} + b\right )^{\frac {3}{4}} + b^{2} \left (a x^{5} + b\right )^{\frac {3}{4}} - c^{2} x^{8} \left (a x^{5} + b\right )^{\frac {3}{4}}}\, dx + \int \left (- \frac {4 b x^{6}}{a^{2} x^{10} \left (a x^{5} + b\right )^{\frac {3}{4}} + 2 a b x^{5} \left (a x^{5} + b\right )^{\frac {3}{4}} + b^{2} \left (a x^{5} + b\right )^{\frac {3}{4}} - c^{2} x^{8} \left (a x^{5} + b\right )^{\frac {3}{4}}}\right )\, dx\right ) \]

[In]

integrate(c*x**6*(a*x**5-4*b)/(a*x**5+b)**(3/4)/(a**2*x**10-c**2*x**8+2*a*b*x**5+b**2),x)

[Out]

c*(Integral(a*x**11/(a**2*x**10*(a*x**5 + b)**(3/4) + 2*a*b*x**5*(a*x**5 + b)**(3/4) + b**2*(a*x**5 + b)**(3/4
) - c**2*x**8*(a*x**5 + b)**(3/4)), x) + Integral(-4*b*x**6/(a**2*x**10*(a*x**5 + b)**(3/4) + 2*a*b*x**5*(a*x*
*5 + b)**(3/4) + b**2*(a*x**5 + b)**(3/4) - c**2*x**8*(a*x**5 + b)**(3/4)), x))

Maxima [F]

\[ \int \frac {c x^6 \left (-4 b+a x^5\right )}{\left (b+a x^5\right )^{3/4} \left (b^2+2 a b x^5-c^2 x^8+a^2 x^{10}\right )} \, dx=\int { \frac {{\left (a x^{5} - 4 \, b\right )} c x^{6}}{{\left (a^{2} x^{10} - c^{2} x^{8} + 2 \, a b x^{5} + b^{2}\right )} {\left (a x^{5} + b\right )}^{\frac {3}{4}}} \,d x } \]

[In]

integrate(c*x^6*(a*x^5-4*b)/(a*x^5+b)^(3/4)/(a^2*x^10-c^2*x^8+2*a*b*x^5+b^2),x, algorithm="maxima")

[Out]

c*integrate((a*x^5 - 4*b)*x^6/((a^2*x^10 - c^2*x^8 + 2*a*b*x^5 + b^2)*(a*x^5 + b)^(3/4)), x)

Giac [F]

\[ \int \frac {c x^6 \left (-4 b+a x^5\right )}{\left (b+a x^5\right )^{3/4} \left (b^2+2 a b x^5-c^2 x^8+a^2 x^{10}\right )} \, dx=\int { \frac {{\left (a x^{5} - 4 \, b\right )} c x^{6}}{{\left (a^{2} x^{10} - c^{2} x^{8} + 2 \, a b x^{5} + b^{2}\right )} {\left (a x^{5} + b\right )}^{\frac {3}{4}}} \,d x } \]

[In]

integrate(c*x^6*(a*x^5-4*b)/(a*x^5+b)^(3/4)/(a^2*x^10-c^2*x^8+2*a*b*x^5+b^2),x, algorithm="giac")

[Out]

integrate((a*x^5 - 4*b)*c*x^6/((a^2*x^10 - c^2*x^8 + 2*a*b*x^5 + b^2)*(a*x^5 + b)^(3/4)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {c x^6 \left (-4 b+a x^5\right )}{\left (b+a x^5\right )^{3/4} \left (b^2+2 a b x^5-c^2 x^8+a^2 x^{10}\right )} \, dx=-\int \frac {c\,x^6\,\left (4\,b-a\,x^5\right )}{{\left (a\,x^5+b\right )}^{3/4}\,\left (a^2\,x^{10}+2\,a\,b\,x^5+b^2-c^2\,x^8\right )} \,d x \]

[In]

int(-(c*x^6*(4*b - a*x^5))/((b + a*x^5)^(3/4)*(b^2 + a^2*x^10 - c^2*x^8 + 2*a*b*x^5)),x)

[Out]

-int((c*x^6*(4*b - a*x^5))/((b + a*x^5)^(3/4)*(b^2 + a^2*x^10 - c^2*x^8 + 2*a*b*x^5)), x)