Optimal. Leaf size=96 \[ 2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{k x^3+(-k-1) x^2+x}}\right )-2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{k x^3+(-k-1) x^2+x}}\right )+\frac {4 \sqrt [4]{k x^3+(-k-1) x^2+x}}{x} \]
________________________________________________________________________________________
Rubi [F] time = 16.65, 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 {(-1+x) (-1+k x) \left (3-2 (1+k) x+k x^2\right )}{x ((1-x) x (1-k x))^{3/4} \left (-1+(1+k) x-k x^2+d x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {(-1+x) (-1+k x) \left (3-2 (1+k) x+k x^2\right )}{x ((1-x) x (1-k x))^{3/4} \left (-1+(1+k) x-k x^2+d x^3\right )} \, dx &=\frac {\left ((1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \int \frac {(-1+x) (-1+k x) \left (3-2 (1+k) x+k x^2\right )}{(1-x)^{3/4} x^{7/4} (1-k x)^{3/4} \left (-1+(1+k) x-k x^2+d x^3\right )} \, dx}{((1-x) x (1-k x))^{3/4}}\\ &=-\frac {\left ((1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \int \frac {\sqrt [4]{1-x} (-1+k x) \left (3-2 (1+k) x+k x^2\right )}{x^{7/4} (1-k x)^{3/4} \left (-1+(1+k) x-k x^2+d x^3\right )} \, dx}{((1-x) x (1-k x))^{3/4}}\\ &=\frac {\left ((1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \int \frac {\sqrt [4]{1-x} \sqrt [4]{1-k x} \left (3-2 (1+k) x+k x^2\right )}{x^{7/4} \left (-1+(1+k) x-k x^2+d x^3\right )} \, dx}{((1-x) x (1-k x))^{3/4}}\\ &=\frac {\left (4 (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{1-x^4} \sqrt [4]{1-k x^4} \left (3-2 (1+k) x^4+k x^8\right )}{x^4 \left (-1+(1+k) x^4-k x^8+d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}}\\ &=\frac {\left (4 (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (-\frac {3 \sqrt [4]{1-x^4} \sqrt [4]{1-k x^4}}{x^4}+\frac {\sqrt [4]{1-x^4} \sqrt [4]{1-k x^4} \left (-1-k+2 k x^4-3 d x^8\right )}{1-(1+k) x^4+k x^8-d x^{12}}\right ) \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}}\\ &=\frac {\left (4 (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{1-x^4} \sqrt [4]{1-k x^4} \left (-1-k+2 k x^4-3 d x^8\right )}{1-(1+k) x^4+k x^8-d x^{12}} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}}-\frac {\left (12 (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{1-x^4} \sqrt [4]{1-k x^4}}{x^4} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}}\\ &=\frac {4 (1-x)^{3/4} (1-k x)^{3/4} F_1\left (-\frac {3}{4};-\frac {1}{4},-\frac {1}{4};\frac {1}{4};x,k x\right )}{((1-x) x (1-k x))^{3/4}}+\frac {\left (4 (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {2 k x^4 \sqrt [4]{1-x^4} \sqrt [4]{1-k x^4}}{1-(1+k) x^4+k x^8-d x^{12}}+\frac {(1+k) \sqrt [4]{1-x^4} \sqrt [4]{1-k x^4}}{-1+(1+k) x^4-k x^8+d x^{12}}+\frac {3 d x^8 \sqrt [4]{1-x^4} \sqrt [4]{1-k x^4}}{-1+(1+k) x^4-k x^8+d x^{12}}\right ) \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}}\\ &=\frac {4 (1-x)^{3/4} (1-k x)^{3/4} F_1\left (-\frac {3}{4};-\frac {1}{4},-\frac {1}{4};\frac {1}{4};x,k x\right )}{((1-x) x (1-k x))^{3/4}}+\frac {\left (12 d (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8 \sqrt [4]{1-x^4} \sqrt [4]{1-k x^4}}{-1+(1+k) x^4-k x^8+d x^{12}} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}}+\frac {\left (8 k (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt [4]{1-x^4} \sqrt [4]{1-k x^4}}{1-(1+k) x^4+k x^8-d x^{12}} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}}+\frac {\left (4 (1+k) (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{1-x^4} \sqrt [4]{1-k x^4}}{-1+(1+k) x^4-k x^8+d x^{12}} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 1.70, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-1+x) (-1+k x) \left (3-2 (1+k) x+k x^2\right )}{x ((1-x) x (1-k x))^{3/4} \left (-1+(1+k) x-k x^2+d x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 4.25, size = 96, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{x+(-1-k) x^2+k x^3}}{x}+2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{x+(-1-k) x^2+k x^3}}\right )-2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{x+(-1-k) x^2+k x^3}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (k x^{2} - 2 \, {\left (k + 1\right )} x + 3\right )} {\left (k x - 1\right )} {\left (x - 1\right )}}{{\left (d x^{3} - k x^{2} + {\left (k + 1\right )} x - 1\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {3}{4}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (-1+x \right ) \left (k x -1\right ) \left (3-2 \left (1+k \right ) x +k \,x^{2}\right )}{x \left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {3}{4}} \left (-1+\left (1+k \right ) x -k \,x^{2}+d \,x^{3}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (k x^{2} - 2 \, {\left (k + 1\right )} x + 3\right )} {\left (k x - 1\right )} {\left (x - 1\right )}}{{\left (d x^{3} - k x^{2} + {\left (k + 1\right )} x - 1\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {3}{4}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (k\,x-1\right )\,\left (x-1\right )\,\left (k\,x^2-2\,x\,\left (k+1\right )+3\right )}{x\,{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{3/4}\,\left (d\,x^3-k\,x^2+\left (k+1\right )\,x-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [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]
[Out]
________________________________________________________________________________________