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 = 0.644026, 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 \[ -\frac{3 \left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} F_1\left (\frac{17}{3};\frac{7}{3},\frac{7}{3};\frac{20}{3};\frac{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 d-\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{272\ 2^{2/3} e (d+e x) \left (a+b x+c x^2\right )^{7/3}} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^2*(a + b*x + c*x^2)^(7/3)),x]
[Out]
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Rubi in Sympy [A] time = 32.9824, size = 151, normalized size = 0.8 \[ - \frac{3 \sqrt [3]{2} \left (\frac{e \left (b + 2 c x - \sqrt{- 4 a c + b^{2}}\right )}{c \left (d + e x\right )}\right )^{\frac{7}{3}} \left (\frac{e \left (b + 2 c x + \sqrt{- 4 a c + b^{2}}\right )}{c \left (d + e x\right )}\right )^{\frac{7}{3}} \operatorname{appellf_{1}}{\left (\frac{17}{3},\frac{7}{3},\frac{7}{3},\frac{20}{3},\frac{c d - \frac{e \left (b - \sqrt{- 4 a c + b^{2}}\right )}{2}}{c \left (d + e x\right )},\frac{c d - \frac{e \left (b + \sqrt{- 4 a c + b^{2}}\right )}{2}}{c \left (d + e x\right )} \right )}}{544 e \left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac{7}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**2/(c*x**2+b*x+a)**(7/3),x)
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Mathematica [A] time = 2.00677, 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.
[In] Integrate[1/((d + e*x)^2*(a + b*x + c*x^2)^(7/3)),x]
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Maple [F] time = 0.138, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( ex+d \right ) ^{2}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{7}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^2/(c*x^2+b*x+a)^(7/3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{7}{3}}{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x + a)^(7/3)*(e*x + d)^2),x, algorithm="maxima")
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x + a)^(7/3)*(e*x + d)^2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)**2/(c*x**2+b*x+a)**(7/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{7}{3}}{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x + a)^(7/3)*(e*x + d)^2),x, algorithm="giac")
[Out]