\(\int \frac {x^2 (4 b+a x^5)}{(-b+a x^5)^{3/4} (-b+c x^4+a x^5)} \, dx\) [1918]

Optimal result

Integrand size = 42, antiderivative size = 133 \[ \int \frac {x^2 \left (4 b+a x^5\right )}{\left (-b+a x^5\right )^{3/4} \left (-b+c x^4+a x^5\right )} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^5}}{-\sqrt {c} x^2+\sqrt {-b+a x^5}}\right )}{c^{3/4}}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {-b+a x^5}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{-b+a x^5}}\right )}{c^{3/4}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(c^2*x*(1 - (a*x^5)/b)^(3/4)*Hypergeometric2F1[1/5, 3/4, 6/5, (a*x^5)/b])/(a^2*(-b + a*x^5)^(3/4)) - (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)) + (x^3*(1 - (a*x^5)/
b)^(3/4)*Hypergeometric2F1[3/5, 3/4, 8/5, (a*x^5)/b])/(3*(-b + a*x^5)^(3/4)) - (b*c^2*Defer[Int][1/((b - c*x^4
- a*x^5)*(-b + a*x^5)^(3/4)), x])/a^2 - (b*c*Defer[Int][x/((-b + a*x^5)^(3/4)*(-b + c*x^4 + a*x^5)), x])/a +
5*b*Defer[Int][x^2/((-b + a*x^5)^(3/4)*(-b + c*x^4 + a*x^5)), x] - (c^3*Defer[Int][x^4/((-b + a*x^5)^(3/4)*(-b
+ c*x^4 + a*x^5)), x])/a^2

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

Mathematica [A] (verified)

Time = 7.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.87 \[ \int \frac {x^2 \left (4 b+a x^5\right )}{\left (-b+a x^5\right )^{3/4} \left (-b+c x^4+a x^5\right )} \, dx=\frac {\sqrt {2} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^5}}{\sqrt {c} x^2-\sqrt {-b+a x^5}}\right )+\text {arctanh}\left (\frac {\sqrt {c} x^2+\sqrt {-b+a x^5}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^5}}\right )\right )}{c^{3/4}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.15 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.16

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {\left (a \,x^{5}-b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}+\sqrt {a \,x^{5}-b}}{\sqrt {a \,x^{5}-b}-\left (a \,x^{5}-b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{5}-b \right )^{\frac {1}{4}}+c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{5}-b \right )^{\frac {1}{4}}-c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right )\right )}{2 c^{\frac {3}{4}}}\)	\(154\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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