Optimal. Leaf size=1707 \[ \frac {(b-x)^{2/3} \sqrt [3]{-a+x} \left (-\frac {\sqrt {3} b \left (-1+\sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{-2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{5/6}}-\frac {\sqrt {3} a c \left (-1+\sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{-2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{5/6}}+\frac {\sqrt {3} \left (-b+a \sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{-2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{5/6}}+\frac {\sqrt {3} c \left (-b+a \sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{-2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{5/6}}-\frac {\sqrt {3} b \left (1+\sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{5/6}}-\frac {\sqrt {3} a c \left (1+\sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{5/6}}+\frac {\sqrt {3} \left (b+a \sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{5/6}}+\frac {\sqrt {3} c \left (b+a \sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{5/6}}-\frac {b \left (1+\sqrt {d}\right ) \log \left (\sqrt [3]{b-x}-\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{5/6}}-\frac {a c \left (1+\sqrt {d}\right ) \log \left (\sqrt [3]{b-x}-\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{5/6}}+\frac {\left (b+a \sqrt {d}\right ) \log \left (\sqrt [3]{b-x}-\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{5/6}}+\frac {c \left (b+a \sqrt {d}\right ) \log \left (\sqrt [3]{b-x}-\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{5/6}}-\frac {b \left (-1+\sqrt {d}\right ) \log \left (\sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{5/6}}-\frac {a c \left (-1+\sqrt {d}\right ) \log \left (\sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{5/6}}+\frac {\left (-b+a \sqrt {d}\right ) \log \left (\sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{5/6}}+\frac {c \left (-b+a \sqrt {d}\right ) \log \left (\sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{5/6}}+\frac {b \left (-1+\sqrt {d}\right ) \log \left ((b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}+\frac {a c \left (-1+\sqrt {d}\right ) \log \left ((b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}+\frac {\left (b-a \sqrt {d}\right ) \log \left ((b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}-\frac {c \left (-b+a \sqrt {d}\right ) \log \left ((b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}+\frac {b \left (1+\sqrt {d}\right ) \log \left ((b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}+\frac {a c \left (1+\sqrt {d}\right ) \log \left ((b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}+\frac {\left (-b-a \sqrt {d}\right ) \log \left ((b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}-\frac {c \left (b+a \sqrt {d}\right ) \log \left ((b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}\right )}{\sqrt [3]{(b-x)^2 (-a+x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.41, antiderivative size = 561, normalized size of antiderivative = 0.33, number of steps
used = 5, number of rules used = 3, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {6851, 6860, 93}
\begin {gather*} -\frac {\sqrt {3} \sqrt [3]{x-a} (x-b)^{2/3} \left (c-\sqrt {d}\right ) \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{x-b}}\right )}{2 d^{5/6} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt {3} \sqrt [3]{x-a} (x-b)^{2/3} \left (c+\sqrt {d}\right ) \text {ArcTan}\left (\frac {2 \sqrt [6]{d} \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{x-b}}+\frac {1}{\sqrt {3}}\right )}{2 d^{5/6} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {\sqrt [3]{x-a} (x-b)^{2/3} \left (c+\sqrt {d}\right ) \log \left (2 \left (\sqrt {d}+1\right ) \left (b-a \sqrt {d}\right )-2 (1-d) x\right )}{4 d^{5/6} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt [3]{x-a} (x-b)^{2/3} \left (c-\sqrt {d}\right ) \log \left (2 \left (1-\sqrt {d}\right ) \left (a \sqrt {d}+b\right )-2 (1-d) x\right )}{4 d^{5/6} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 \sqrt [3]{x-a} (x-b)^{2/3} \left (c-\sqrt {d}\right ) \log \left (-\sqrt [6]{d} \sqrt [3]{x-a}-\sqrt [3]{x-b}\right )}{4 d^{5/6} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {3 \sqrt [3]{x-a} (x-b)^{2/3} \left (c+\sqrt {d}\right ) \log \left (\sqrt [6]{d} \sqrt [3]{x-a}-\sqrt [3]{x-b}\right )}{4 d^{5/6} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 93
Rule 6851
Rule 6860
Rubi steps
\begin {align*} \int \frac {-b-a c+(1+c) x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx &=\frac {\left (\sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {-b-a c+(1+c) x}{\sqrt [3]{-a+x} (-b+x)^{2/3} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \left (\frac {1+c+\frac {c+d}{\sqrt {d}}}{\sqrt [3]{-a+x} (-b+x)^{2/3} \left (-2 (a-b) \sqrt {d}+2 (b-a d)+2 (-1+d) x\right )}+\frac {1+c-\frac {c+d}{\sqrt {d}}}{\sqrt [3]{-a+x} (-b+x)^{2/3} \left (2 (a-b) \sqrt {d}+2 (b-a d)+2 (-1+d) x\right )}\right ) \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\left (1+c-\frac {c+d}{\sqrt {d}}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{-a+x} (-b+x)^{2/3} \left (2 (a-b) \sqrt {d}+2 (b-a d)+2 (-1+d) x\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}+\frac {\left (\left (1+c+\frac {c+d}{\sqrt {d}}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{-a+x} (-b+x)^{2/3} \left (-2 (a-b) \sqrt {d}+2 (b-a d)+2 (-1+d) x\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=-\frac {\sqrt {3} \left (c-\sqrt {d}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt {3} \sqrt [3]{-b+x}}\right )}{2 (a-b) d^{5/6} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt {3} \left (c+\sqrt {d}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt {3} \sqrt [3]{-b+x}}\right )}{2 (a-b) d^{5/6} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {\left (c+\sqrt {d}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (2 \left (1+\sqrt {d}\right ) \left (b-a \sqrt {d}\right )-2 (1-d) x\right )}{4 (a-b) d^{5/6} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\left (c-\sqrt {d}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (2 \left (1-\sqrt {d}\right ) \left (b+a \sqrt {d}\right )-2 (1-d) x\right )}{4 (a-b) d^{5/6} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 \left (c-\sqrt {d}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (-\sqrt [6]{d} \sqrt [3]{-a+x}-\sqrt [3]{-b+x}\right )}{4 (a-b) d^{5/6} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {3 \left (c+\sqrt {d}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (\sqrt [6]{d} \sqrt [3]{-a+x}-\sqrt [3]{-b+x}\right )}{4 (a-b) d^{5/6} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.42, size = 495, normalized size = 0.29 \begin {gather*} -\frac {(b-x)^{2/3} \sqrt [3]{-a+x} \left (2 \sqrt {3} \left (c+\sqrt {d}\right ) \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{b-x}}{\sqrt [6]{d} \sqrt [3]{-a+x}}}{\sqrt {3}}\right )-2 \sqrt {3} \left (c-\sqrt {d}\right ) \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{b-x}}{\sqrt [6]{d} \sqrt [3]{-a+x}}}{\sqrt {3}}\right )+2 c \log \left (\sqrt [6]{d}-\frac {\sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 \sqrt {d} \log \left (\sqrt [6]{d}-\frac {\sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 c \log \left (\sqrt [6]{d}+\frac {\sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 \sqrt {d} \log \left (\sqrt [6]{d}+\frac {\sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )+c \log \left (\sqrt [3]{d}+\frac {(b-x)^{2/3}}{(-a+x)^{2/3}}-\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )+\sqrt {d} \log \left (\sqrt [3]{d}+\frac {(b-x)^{2/3}}{(-a+x)^{2/3}}-\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-c \log \left (\sqrt [3]{d}+\frac {(b-x)^{2/3}}{(-a+x)^{2/3}}+\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )+\sqrt {d} \log \left (\sqrt [3]{d}+\frac {(b-x)^{2/3}}{(-a+x)^{2/3}}+\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )\right )}{4 (a-b) d^{5/6} \sqrt [3]{(b-x)^2 (-a+x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {-b -a c +\left (1+c \right ) x}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{3}} \left (-b^{2}+a^{2} d +2 \left (-a d +b \right ) x +\left (-1+d \right ) x^{2}\right )}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 11788 vs.
\(2 (1297) = 2594\).
time = 1.09, size = 11788, normalized size = 6.91 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {b+a\,c-x\,\left (c+1\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\left (a^2\,d+2\,x\,\left (b-a\,d\right )-b^2+x^2\,\left (d-1\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________