Optimal. Leaf size=189 \[ -\frac{3 \left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} F_1\left (\frac{17}{3};\frac{7}{3},\frac{7}{3};\frac{20}{3};\frac{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 d-\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{272\ 2^{2/3} e (d+e x) \left (a+b x+c x^2\right )^{7/3}} \]
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Rubi [A] time = 0.0913116, antiderivative size = 189, normalized size of antiderivative = 1., 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} \[ -\frac{3 \left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} F_1\left (\frac{17}{3};\frac{7}{3},\frac{7}{3};\frac{20}{3};\frac{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 d-\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{272\ 2^{2/3} e (d+e x) \left (a+b x+c x^2\right )^{7/3}} \]
Antiderivative was successfully verified.
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Rule 758
Rule 133
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^2 \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^{14/3}}{\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{17}{3};\frac{7}{3},\frac{7}{3};\frac{20}{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 )}{272\ 2^{2/3} e (d+e x) \left (a+b x+c x^2\right )^{7/3}}\\ \end{align*}
Mathematica [A] time = 1.0946, size = 190, normalized size = 1.01 \[ -\frac{3 e^3 \sqrt [3]{\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}} \sqrt [3]{\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}} F_1\left (\frac{17}{3};\frac{7}{3},\frac{7}{3};\frac{20}{3};\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right )}{17\ 2^{2/3} c^2 (d+e x)^5 \sqrt [3]{a+x (b+c x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.274, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( ex+d \right ) ^{2}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{7}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{7}{3}}{\left (e x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{7}{3}}{\left (e x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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