Optimal. Leaf size=1510 \[ -\frac {(c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{(b c-a d)^2 (a+b x)}+\frac {\sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\left (d (3 b c+a d)+4 b d^2 x\right )^2}}{b^{2/3} d (b c-a d)^2 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}+\frac {\sqrt {3} d \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 b^{2/3} (c+d x)^{2/3}}{\sqrt {3} \sqrt [3]{b c-a d} \sqrt [3]{b c+a d+2 b d x}}\right )}{2 b^{2/3} (b c-a d)^{5/3}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} d \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (d (3 b c+a d)+4 b d^2 x\right )^2} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2 \sqrt [3]{b} (b c-a d)^{2/3} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}+4 b^{2/3} ((c+d x) (a d+b (c+2 d x)))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}\right )|-7-4 \sqrt {3}\right )}{2 b^{2/3} (b c-a d)^{4/3} \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}}}+\frac {\sqrt {2} d \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (d (3 b c+a d)+4 b d^2 x\right )^2} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2 \sqrt [3]{b} (b c-a d)^{2/3} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}+4 b^{2/3} ((c+d x) (a d+b (c+2 d x)))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} (b c-a d)^{4/3} \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}}}+\frac {d \log (a+b x)}{2 b^{2/3} (b c-a d)^{5/3}}-\frac {3 d \log \left (\frac {b^{2/3} (c+d x)^{2/3}}{\sqrt [3]{b c-a d}}-\sqrt [3]{b c+a d+2 b d x}\right )}{4 b^{2/3} (b c-a d)^{5/3}} \]
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Rubi [A]
time = 1.26, antiderivative size = 1510, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {125, 163, 64,
637, 309, 224, 1891, 124} \begin {gather*} \frac {\sqrt {3} d \text {ArcTan}\left (\frac {2 b^{2/3} (c+d x)^{2/3}}{\sqrt {3} \sqrt [3]{b c-a d} \sqrt [3]{b c+a d+2 b d x}}+\frac {1}{\sqrt {3}}\right )}{2 b^{2/3} (b c-a d)^{5/3}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} d \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (4 b x d^2+(3 b c+a d) d\right )^2} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))} (b c-a d)^{2/3}+4 b^{2/3} ((c+d x) (a d+b (c+2 d x)))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}} E\left (\text {ArcSin}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}\right )|-7-4 \sqrt {3}\right )}{2 b^{2/3} (b c-a d)^{4/3} \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}}}+\frac {\sqrt {2} d \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (4 b x d^2+(3 b c+a d) d\right )^2} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))} (b c-a d)^{2/3}+4 b^{2/3} ((c+d x) (a d+b (c+2 d x)))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}} F\left (\text {ArcSin}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} (b c-a d)^{4/3} \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}}}+\frac {d \log (a+b x)}{2 b^{2/3} (b c-a d)^{5/3}}-\frac {3 d \log \left (\frac {b^{2/3} (c+d x)^{2/3}}{\sqrt [3]{b c-a d}}-\sqrt [3]{b c+a d+2 b d x}\right )}{4 b^{2/3} (b c-a d)^{5/3}}-\frac {(c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{(b c-a d)^2 (a+b x)}+\frac {\sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\left (4 b x d^2+(3 b c+a d) d\right )^2}}{b^{2/3} d (b c-a d)^2 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 64
Rule 124
Rule 125
Rule 163
Rule 224
Rule 309
Rule 637
Rule 1891
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^2 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx &=-\frac {(c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{(b c-a d)^2 (a+b x)}-\frac {d \int \frac {3 b c-5 a d-2 b d x}{(a+b x) \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx}{3 (b c-a d)^2}\\ &=-\frac {(c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{(b c-a d)^2 (a+b x)}+\frac {\left (2 d^2\right ) \int \frac {1}{\sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx}{3 (b c-a d)^2}-\frac {d \int \frac {1}{(a+b x) \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx}{b c-a d}\\ &=-\frac {(c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{(b c-a d)^2 (a+b x)}+\frac {\sqrt {3} d \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 b^{2/3} (c+d x)^{2/3}}{\sqrt {3} \sqrt [3]{b c-a d} \sqrt [3]{b c+a d+2 b d x}}\right )}{2 b^{2/3} (b c-a d)^{5/3}}+\frac {d \log (a+b x)}{2 b^{2/3} (b c-a d)^{5/3}}-\frac {3 d \log \left (\frac {b^{2/3} (c+d x)^{2/3}}{\sqrt [3]{b c-a d}}-\sqrt [3]{b c+a d+2 b d x}\right )}{4 b^{2/3} (b c-a d)^{5/3}}+\frac {\left (2 d^2 \sqrt [3]{(c+d x) (b c+a d+2 b d x)}\right ) \int \frac {1}{\sqrt [3]{c (b c+a d)+(2 b c d+d (b c+a d)) x+2 b d^2 x^2}} \, dx}{3 (b c-a d)^2 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}}\\ &=-\frac {(c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{(b c-a d)^2 (a+b x)}+\frac {\sqrt {3} d \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 b^{2/3} (c+d x)^{2/3}}{\sqrt {3} \sqrt [3]{b c-a d} \sqrt [3]{b c+a d+2 b d x}}\right )}{2 b^{2/3} (b c-a d)^{5/3}}+\frac {d \log (a+b x)}{2 b^{2/3} (b c-a d)^{5/3}}-\frac {3 d \log \left (\frac {b^{2/3} (c+d x)^{2/3}}{\sqrt [3]{b c-a d}}-\sqrt [3]{b c+a d+2 b d x}\right )}{4 b^{2/3} (b c-a d)^{5/3}}+\frac {\left (2 d^2 \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (2 b c d+d (b c+a d)+4 b d^2 x\right )^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-8 b c d^2 (b c+a d)+(2 b c d+d (b c+a d))^2+8 b d^2 x^3}} \, dx,x,\sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{(b c-a d)^2 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} \left (2 b c d+d (b c+a d)+4 b d^2 x\right )}\\ &=-\frac {(c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{(b c-a d)^2 (a+b x)}+\frac {\sqrt {3} d \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 b^{2/3} (c+d x)^{2/3}}{\sqrt {3} \sqrt [3]{b c-a d} \sqrt [3]{b c+a d+2 b d x}}\right )}{2 b^{2/3} (b c-a d)^{5/3}}+\frac {d \log (a+b x)}{2 b^{2/3} (b c-a d)^{5/3}}-\frac {3 d \log \left (\frac {b^{2/3} (c+d x)^{2/3}}{\sqrt [3]{b c-a d}}-\sqrt [3]{b c+a d+2 b d x}\right )}{4 b^{2/3} (b c-a d)^{5/3}}+\frac {\left (d^2 \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (2 b c d+d (b c+a d)+4 b d^2 x\right )^2}\right ) \text {Subst}\left (\int \frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} x}{\sqrt {-8 b c d^2 (b c+a d)+(2 b c d+d (b c+a d))^2+8 b d^2 x^3}} \, dx,x,\sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\sqrt [3]{b} (b c-a d)^2 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} \left (2 b c d+d (b c+a d)+4 b d^2 x\right )}+\frac {\left (\sqrt {\frac {2}{2+\sqrt {3}}} d^2 \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (2 b c d+d (b c+a d)+4 b d^2 x\right )^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-8 b c d^2 (b c+a d)+(2 b c d+d (b c+a d))^2+8 b d^2 x^3}} \, dx,x,\sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\sqrt [3]{b} (b c-a d)^{4/3} \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} \left (2 b c d+d (b c+a d)+4 b d^2 x\right )}\\ &=-\frac {(c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{(b c-a d)^2 (a+b x)}+\frac {\sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\left (d (3 b c+a d)+4 b d^2 x\right )^2}}{b^{2/3} d (b c-a d)^2 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}+\frac {\sqrt {3} d \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 b^{2/3} (c+d x)^{2/3}}{\sqrt {3} \sqrt [3]{b c-a d} \sqrt [3]{b c+a d+2 b d x}}\right )}{2 b^{2/3} (b c-a d)^{5/3}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} d \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (d (3 b c+a d)+4 b d^2 x\right )^2} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2 \sqrt [3]{b} (b c-a d)^{2/3} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}+4 b^{2/3} ((c+d x) (a d+b (c+2 d x)))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}\right )|-7-4 \sqrt {3}\right )}{2 b^{2/3} (b c-a d)^{4/3} \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}}}+\frac {\sqrt {2} d \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (d (3 b c+a d)+4 b d^2 x\right )^2} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2 \sqrt [3]{b} (b c-a d)^{2/3} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}+4 b^{2/3} ((c+d x) (a d+b (c+2 d x)))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} (b c-a d)^{4/3} \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}}}+\frac {d \log (a+b x)}{2 b^{2/3} (b c-a d)^{5/3}}-\frac {3 d \log \left (\frac {b^{2/3} (c+d x)^{2/3}}{\sqrt [3]{b c-a d}}-\sqrt [3]{b c+a d+2 b d x}\right )}{4 b^{2/3} (b c-a d)^{5/3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 20.88, size = 288, normalized size = 0.19 \begin {gather*} \frac {(c+d x)^{2/3} \left (-\frac {10 (a d+b (c+2 d x))}{a+b x}+\frac {d \left (10 b (c+d x) (a d+b (c+2 d x))+10\ 2^{2/3} b (b c-a d) (c+d x) \sqrt [3]{\frac {b c+a d+2 b d x}{b c+b d x}} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};\frac {b c-a d}{2 b c+2 b d x},\frac {b c-a d}{b c+b d x}\right )-2^{2/3} (b c-a d)^2 \sqrt [3]{\frac {b c+a d+2 b d x}{b c+b d x}} F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};\frac {b c-a d}{2 b c+2 b d x},\frac {b c-a d}{b c+b d x}\right )\right )}{b^2 (c+d x)^2}\right )}{10 (b c-a d)^2 \sqrt [3]{a d+b (c+2 d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b x +a \right )^{2} \left (d x +c \right )^{\frac {1}{3}} \left (2 b d x +a d +b c \right )^{\frac {1}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x\right )^{2} \sqrt [3]{c + d x} \sqrt [3]{a d + b c + 2 b d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^{1/3}\,{\left (a\,d+b\,c+2\,b\,d\,x\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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