Optimal. Leaf size=1389 \[ \frac {3 (a+b x)^2 (c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{20 d^2}+\frac {9 (b c-a d) (c+d x)^{2/3} (23 b c-39 a d-16 b d x) (b c+a d+2 b d x)^{2/3}}{560 d^4}-\frac {81 (b c-a d)^3 \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}}{112 b^{2/3} d^6 \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 {81 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (b c-a d)^{11/3} \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 )}{224 b^{2/3} d^4 \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 {27\ 3^{3/4} (b c-a d)^{11/3} \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 )}{56 \sqrt {2} b^{2/3} d^4 \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}}} \]
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Rubi [A]
time = 1.70, antiderivative size = 1389, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {102, 152, 64,
637, 309, 224, 1891} \begin {gather*} \frac {81 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \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 ) (b c-a d)^{11/3}}{224 b^{2/3} d^4 \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 {27\ 3^{3/4} \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 ) (b c-a d)^{11/3}}{56 \sqrt {2} b^{2/3} d^4 \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 {81 \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 c-a d)^3}{112 b^{2/3} d^6 \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 {9 (c+d x)^{2/3} (23 b c-39 a d-16 b d x) (b c+a d+2 b d x)^{2/3} (b c-a d)}{560 d^4}+\frac {3 (a+b x)^2 (c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{20 d^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 64
Rule 102
Rule 152
Rule 224
Rule 309
Rule 637
Rule 1891
Rubi steps
\begin {align*} \int \frac {(a+b x)^3}{\sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx &=\frac {3 (a+b x)^2 (c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{20 d^2}+\frac {3 \int \frac {(a+b x) \left (-2 b (b c-a d) (b c+3 a d)-8 b^2 d (b c-a d) x\right )}{\sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx}{20 b d^2}\\ &=\frac {3 (a+b x)^2 (c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{20 d^2}+\frac {9 (b c-a d) (c+d x)^{2/3} (23 b c-39 a d-16 b d x) (b c+a d+2 b d x)^{2/3}}{560 d^4}-\frac {\left (27 (b c-a d)^3\right ) \int \frac {1}{\sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx}{56 d^3}\\ &=\frac {3 (a+b x)^2 (c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{20 d^2}+\frac {9 (b c-a d) (c+d x)^{2/3} (23 b c-39 a d-16 b d x) (b c+a d+2 b d x)^{2/3}}{560 d^4}-\frac {\left (27 (b c-a d)^3 \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}{56 d^3 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}}\\ &=\frac {3 (a+b x)^2 (c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{20 d^2}+\frac {9 (b c-a d) (c+d x)^{2/3} (23 b c-39 a d-16 b d x) (b c+a d+2 b d x)^{2/3}}{560 d^4}-\frac {\left (81 (b c-a d)^3 \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 )}{56 d^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 {3 (a+b x)^2 (c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{20 d^2}+\frac {9 (b c-a d) (c+d x)^{2/3} (23 b c-39 a d-16 b d x) (b c+a d+2 b d x)^{2/3}}{560 d^4}-\frac {\left (81 (b c-a d)^3 \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 )}{112 \sqrt [3]{b} d^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 {\left (81 (b c-a d)^{11/3} \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 )}{56 \sqrt {2 \left (2+\sqrt {3}\right )} \sqrt [3]{b} d^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 {3 (a+b x)^2 (c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{20 d^2}+\frac {9 (b c-a d) (c+d x)^{2/3} (23 b c-39 a d-16 b d x) (b c+a d+2 b d x)^{2/3}}{560 d^4}-\frac {81 (b c-a d)^3 \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}}{112 b^{2/3} d^6 \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 {81 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (b c-a d)^{11/3} \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 )}{224 b^{2/3} d^4 \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 {27\ 3^{3/4} (b c-a d)^{11/3} \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 )}{56 \sqrt {2} b^{2/3} d^4 \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}}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.36, size = 160, normalized size = 0.12 \begin {gather*} -\frac {3 (a d+b (c+2 d x))^{2/3} \left (-2 b (c+d x) \left (145 a^2 d^2+2 a b d (-93 c+52 d x)+b^2 \left (69 c^2-48 c d x+28 d^2 x^2\right )\right )+135 \sqrt [3]{2} (b c-a d)^3 \sqrt [3]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};\frac {a d+b (c+2 d x)}{-b c+a d}\right )\right )}{1120 b d^4 \sqrt [3]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{3}}{\left (d x +c \right )^{\frac {1}{3}} \left (2 b d x +a d +b c \right )^{\frac {1}{3}}}\, dx\]
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]
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right )^{3}}{\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 {{\left (a+b\,x\right )}^3}{{\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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