Optimal. Leaf size=1043 \[ \frac {15 (2 c d-b e) (b+2 c x)}{4 \left (b^2-4 a c\right )^2 \sqrt [3]{c x^2+b x+a}}-\frac {15 \sqrt [3]{c} (2 c d-b e) (b+2 c x)}{2 \sqrt [3]{2} \left (b^2-4 a c\right )^2 \left (\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 )}-\frac {3 (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{4/3}}+\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} (2 c d-b e) \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a} \sqrt [3]{b^2-4 a c}+2 \sqrt [3]{2} c^{2/3} \left (c x^2+b x+a\right )^{2/3}}{\left (\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 )^2}} E\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]{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 )}{4 \sqrt [3]{2} \left (b^2-4 a c\right )^{5/3} \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]{c x^2+b x+a}\right )}{\left (\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 )^2}} (b+2 c x)}-\frac {5\ 3^{3/4} \sqrt [3]{c} (2 c d-b e) \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a} \sqrt [3]{b^2-4 a c}+2 \sqrt [3]{2} c^{2/3} \left (c x^2+b x+a\right )^{2/3}}{\left (\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 )^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]{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 )}{2^{5/6} \left (b^2-4 a c\right )^{5/3} \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]{c x^2+b x+a}\right )}{\left (\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 )^2}} (b+2 c x)} \]
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Rubi [A] time = 0.99, antiderivative size = 1043, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {638, 623, 325, 303, 218, 1877} \[ \frac {15 (2 c d-b e) (b+2 c x)}{4 \left (b^2-4 a c\right )^2 \sqrt [3]{c x^2+b x+a}}-\frac {15 \sqrt [3]{c} (2 c d-b e) (b+2 c x)}{2 \sqrt [3]{2} \left (b^2-4 a c\right )^2 \left (\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 )}-\frac {3 (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{4/3}}+\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} (2 c d-b e) \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a} \sqrt [3]{b^2-4 a c}+2 \sqrt [3]{2} c^{2/3} \left (c x^2+b x+a\right )^{2/3}}{\left (\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 )^2}} E\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]{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 )}{4 \sqrt [3]{2} \left (b^2-4 a c\right )^{5/3} \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]{c x^2+b x+a}\right )}{\left (\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 )^2}} (b+2 c x)}-\frac {5\ 3^{3/4} \sqrt [3]{c} (2 c d-b e) \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a} \sqrt [3]{b^2-4 a c}+2 \sqrt [3]{2} c^{2/3} \left (c x^2+b x+a\right )^{2/3}}{\left (\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 )^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]{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 )}{2^{5/6} \left (b^2-4 a c\right )^{5/3} \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]{c x^2+b x+a}\right )}{\left (\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 )^2}} (b+2 c x)} \]
Antiderivative was successfully verified.
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Rule 218
Rule 303
Rule 325
Rule 623
Rule 638
Rule 1877
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a+b x+c x^2\right )^{7/3}} \, dx &=-\frac {3 (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}-\frac {(5 (2 c d-b e)) \int \frac {1}{\left (a+b x+c x^2\right )^{4/3}} \, dx}{4 \left (b^2-4 a c\right )}\\ &=-\frac {3 (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}-\frac {\left (15 (2 c d-b e) \sqrt {(b+2 c x)^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{4 \left (b^2-4 a c\right ) (b+2 c x)}\\ &=-\frac {3 (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}+\frac {15 (2 c d-b e) (b+2 c x)}{4 \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac {\left (15 c (2 c d-b e) \sqrt {(b+2 c x)^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{2 \left (b^2-4 a c\right )^2 (b+2 c x)}\\ &=-\frac {3 (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}+\frac {15 (2 c d-b e) (b+2 c x)}{4 \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac {\left (15 c^{2/3} (2 c d-b e) \sqrt {(b+2 c x)^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} x}{\sqrt {b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{2\ 2^{2/3} \left (b^2-4 a c\right )^2 (b+2 c x)}-\frac {\left (15 c^{2/3} (2 c d-b e) \sqrt {(b+2 c x)^2}\right ) \operatorname {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 )}{2 \sqrt [6]{2} \sqrt {2+\sqrt {3}} \left (b^2-4 a c\right )^{5/3} (b+2 c x)}\\ &=-\frac {3 (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}+\frac {15 (2 c d-b e) (b+2 c x)}{4 \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac {15 \sqrt [3]{c} (2 c d-b e) (b+2 c x)}{2 \sqrt [3]{2} \left (b^2-4 a c\right )^2 \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 )}+\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} (2 c d-b e) \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}} E\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 )}{4 \sqrt [3]{2} \left (b^2-4 a c\right )^{5/3} (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 {5\ 3^{3/4} \sqrt [3]{c} (2 c d-b e) \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 )}{2^{5/6} \left (b^2-4 a c\right )^{5/3} (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] time = 0.52, size = 200, normalized size = 0.19 \[ \frac {3 \left (5\ 2^{2/3} \left (-\sqrt {b^2-4 a c}+b+2 c x\right ) \sqrt [3]{\frac {\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}}} (b e-2 c d) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};\frac {-b-2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )+\frac {4 \left (b^2-4 a c\right ) (2 a e-b d+b e x-2 c d x)}{a+x (b+c x)}+20 (b+2 c x) (2 c d-b e)\right )}{16 \left (b^2-4 a c\right )^2 \sqrt [3]{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2} + b x + a\right )}^{\frac {2}{3}} {\left (e x + d\right )}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x + {\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} x^{2}}, x\right ) \]
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 \[ \int \frac {e x + d}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.45, size = 0, normalized size = 0.00 \[ \int \frac {e x +d}{\left (c \,x^{2}+b x +a \right )^{\frac {7}{3}}}\, dx \]
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 \[ \int \frac {e x + d}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}}}\,{d x} \]
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 \[ \int \frac {d+e\,x}{{\left (c\,x^2+b\,x+a\right )}^{7/3}} \,d x \]
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 \[ \int \frac {d + e x}{\left (a + b x + c x^{2}\right )^{\frac {7}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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