Optimal. Leaf size=189 \[ -\frac {3 \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} F_1\left (\frac {20}{3};\frac {7}{3},\frac {7}{3};\frac {23}{3};\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{320\ 2^{2/3} e (d+e x)^2 \left (a+b x+c x^2\right )^{7/3}} \]
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Rubi [A] time = 0.09, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {758, 133} \[ -\frac {3 \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} F_1\left (\frac {20}{3};\frac {7}{3},\frac {7}{3};\frac {23}{3};\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{320\ 2^{2/3} e (d+e x)^2 \left (a+b x+c x^2\right )^{7/3}} \]
Antiderivative was successfully verified.
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Rule 133
Rule 758
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{7/3}} \, dx &=-\frac {\left (\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{7/3}\right ) \operatorname {Subst}\left (\int \frac {x^{17/3}}{\left (1-\frac {1}{2} \left (2 d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{c}\right ) x\right )^{7/3} \left (1-\frac {1}{2} \left (2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}\right ) x\right )^{7/3}} \, dx,x,\frac {1}{d+e x}\right )}{16\ 2^{2/3} e \left (\frac {1}{d+e x}\right )^{14/3} \left (a+b x+c x^2\right )^{7/3}}\\ &=-\frac {3 \left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{7/3} F_1\left (\frac {20}{3};\frac {7}{3},\frac {7}{3};\frac {23}{3};\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{320\ 2^{2/3} e (d+e x)^2 \left (a+b x+c x^2\right )^{7/3}}\\ \end {align*}
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Mathematica [A] time = 1.85, size = 190, normalized size = 1.01 \[ -\frac {3 e^3 \sqrt [3]{\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}} \sqrt [3]{\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}} F_1\left (\frac {20}{3};\frac {7}{3},\frac {7}{3};\frac {23}{3};\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 c d-b e+\sqrt {b^2-4 a c} e}{2 c d+2 c e x}\right )}{20\ 2^{2/3} c^2 (d+e x)^6 \sqrt [3]{a+x (b+c x)}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}} {\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.82, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e x +d \right )^{3} \left (c \,x^{2}+b x +a \right )^{\frac {7}{3}}}\, dx \]
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 \[ \int \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}} {\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (d+e\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{7/3}} \,d x \]
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 \[ \int \frac {1}{\left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac {7}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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