Optimal. Leaf size=638 \[ -\frac {3 \left (b^2-4 a c\right ) \left (17 c^2 d^2+5 b^2 e^2-c e (17 b d+3 a e)\right ) (b+2 c x) \sqrt [3]{a+b x+c x^2}}{935 c^4}+\frac {3 \left (17 c^2 d^2+5 b^2 e^2-c e (17 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{374 c^3}+\frac {15 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/3}}{119 c^2}+\frac {3 e (d+e x) \left (a+b x+c x^2\right )^{7/3}}{17 c}+\frac {\sqrt [3]{2} 3^{3/4} \sqrt {2+\sqrt {3}} \left (b^2-4 a c\right )^2 \left (17 c^2 d^2+5 b^2 e^2-c e (17 b d+3 a e)\right ) \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}\right )|-7-4 \sqrt {3}\right )}{935 c^{13/3} (b+2 c x) \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}} \]
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Rubi [A]
time = 0.72, antiderivative size = 638, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {756, 654, 637,
327, 224} \begin {gather*} \frac {\sqrt [3]{2} 3^{3/4} \sqrt {2+\sqrt {3}} \left (b^2-4 a c\right )^2 \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt {\frac {-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+\left (b^2-4 a c\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} \left (-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right ) F\left (\text {ArcSin}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right )|-7-4 \sqrt {3}\right )}{935 c^{13/3} (b+2 c x) \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}}-\frac {3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [3]{a+b x+c x^2} \left (-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right )}{935 c^4}+\frac {3 (b+2 c x) \left (a+b x+c x^2\right )^{4/3} \left (-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right )}{374 c^3}+\frac {15 e \left (a+b x+c x^2\right )^{7/3} (2 c d-b e)}{119 c^2}+\frac {3 e (d+e x) \left (a+b x+c x^2\right )^{7/3}}{17 c} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 224
Rule 327
Rule 637
Rule 654
Rule 756
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+b x+c x^2\right )^{4/3} \, dx &=\frac {3 e (d+e x) \left (a+b x+c x^2\right )^{7/3}}{17 c}+\frac {3 \int \left (\frac {1}{3} \left (17 c d^2-3 e \left (\frac {7 b d}{3}+a e\right )\right )+\frac {10}{3} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{4/3} \, dx}{17 c}\\ &=\frac {15 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/3}}{119 c^2}+\frac {3 e (d+e x) \left (a+b x+c x^2\right )^{7/3}}{17 c}+\frac {\left (3 \left (-\frac {10}{3} b e (2 c d-b e)+\frac {2}{3} c \left (17 c d^2-3 e \left (\frac {7 b d}{3}+a e\right )\right )\right )\right ) \int \left (a+b x+c x^2\right )^{4/3} \, dx}{34 c^2}\\ &=\frac {15 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/3}}{119 c^2}+\frac {3 e (d+e x) \left (a+b x+c x^2\right )^{7/3}}{17 c}+\frac {\left (9 \left (-\frac {10}{3} b e (2 c d-b e)+\frac {2}{3} c \left (17 c d^2-3 e \left (\frac {7 b d}{3}+a e\right )\right )\right ) \sqrt {(b+2 c x)^2}\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt {b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{34 c^2 (b+2 c x)}\\ &=\frac {3 \left (17 c^2 d^2+5 b^2 e^2-c e (17 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{374 c^3}+\frac {15 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/3}}{119 c^2}+\frac {3 e (d+e x) \left (a+b x+c x^2\right )^{7/3}}{17 c}-\frac {\left (9 \left (b^2-4 a c\right ) \left (-\frac {10}{3} b e (2 c d-b e)+\frac {2}{3} c \left (17 c d^2-3 e \left (\frac {7 b d}{3}+a e\right )\right )\right ) \sqrt {(b+2 c x)^2}\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{187 c^3 (b+2 c x)}\\ &=-\frac {3 \left (b^2-4 a c\right ) \left (17 c^2 d^2+5 b^2 e^2-c e (17 b d+3 a e)\right ) (b+2 c x) \sqrt [3]{a+b x+c x^2}}{935 c^4}+\frac {3 \left (17 c^2 d^2+5 b^2 e^2-c e (17 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{374 c^3}+\frac {15 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/3}}{119 c^2}+\frac {3 e (d+e x) \left (a+b x+c x^2\right )^{7/3}}{17 c}+\frac {\left (9 \left (b^2-4 a c\right )^2 \left (-\frac {10}{3} b e (2 c d-b e)+\frac {2}{3} c \left (17 c d^2-3 e \left (\frac {7 b d}{3}+a e\right )\right )\right ) \sqrt {(b+2 c x)^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{1870 c^4 (b+2 c x)}\\ &=-\frac {3 \left (b^2-4 a c\right ) \left (17 c^2 d^2+5 b^2 e^2-c e (17 b d+3 a e)\right ) (b+2 c x) \sqrt [3]{a+b x+c x^2}}{935 c^4}+\frac {3 \left (17 c^2 d^2+5 b^2 e^2-c e (17 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{374 c^3}+\frac {15 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/3}}{119 c^2}+\frac {3 e (d+e x) \left (a+b x+c x^2\right )^{7/3}}{17 c}+\frac {\sqrt [3]{2} 3^{3/4} \sqrt {2+\sqrt {3}} \left (b^2-4 a c\right )^2 \left (17 c^2 d^2+5 b^2 e^2-c e (17 b d+3 a e)\right ) \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}\right )|-7-4 \sqrt {3}\right )}{935 c^{13/3} (b+2 c x) \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\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 = 10.26, size = 164, normalized size = 0.26 \begin {gather*} \frac {3 (a+x (b+c x))^{4/3} \left (-\frac {160 e (-2 c d+b e) (a+x (b+c x))}{c}+224 e (d+e x) (a+x (b+c x))+\frac {14 \sqrt [3]{2} \left (17 c^2 d^2+5 b^2 e^2-c e (17 b d+3 a e)\right ) (b+2 c x) \, _2F_1\left (-\frac {4}{3},\frac {1}{2};\frac {3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{3 c^2 \left (-\frac {c (a+x (b+c x))}{b^2-4 a c}\right )^{4/3}}\right )}{3808 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{2} \left (c \,x^{2}+b x +a \right )^{\frac {4}{3}}\, dx\]
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 \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac {4}{3}}\, 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 {\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{4/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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