3.2482 \(\int \frac{1}{(d+e x) \left (a+b x+c x^2\right )^{7/3}} \, dx\)

Optimal. Leaf size=182 \[ -\frac{3 \left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} F_1\left (\frac{14}{3};\frac{7}{3},\frac{7}{3};\frac{17}{3};\frac{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 d-\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{224\ 2^{2/3} e \left (a+b x+c x^2\right )^{7/3}} \]

[Out]

(-3*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(7/3)*((e*(b + Sqrt[b^2
- 4*a*c] + 2*c*x))/(c*(d + e*x)))^(7/3)*AppellF1[14/3, 7/3, 7/3, 17/3, (2*c*d -
(b - Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c
)/(2*(d + e*x))])/(224*2^(2/3)*e*(a + b*x + c*x^2)^(7/3))

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Rubi [A]  time = 0.676529, antiderivative size = 182, 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 \[ -\frac{3 \left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} F_1\left (\frac{14}{3};\frac{7}{3},\frac{7}{3};\frac{17}{3};\frac{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 d-\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{224\ 2^{2/3} e \left (a+b x+c x^2\right )^{7/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)*(a + b*x + c*x^2)^(7/3)),x]

[Out]

(-3*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(7/3)*((e*(b + Sqrt[b^2
- 4*a*c] + 2*c*x))/(c*(d + e*x)))^(7/3)*AppellF1[14/3, 7/3, 7/3, 17/3, (2*c*d -
(b - Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c
)/(2*(d + e*x))])/(224*2^(2/3)*e*(a + b*x + c*x^2)^(7/3))

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Rubi in Sympy [A]  time = 33.427, size = 146, normalized size = 0.8 \[ - \frac{3 \sqrt [3]{2} \left (\frac{e \left (b + 2 c x - \sqrt{- 4 a c + b^{2}}\right )}{c \left (d + e x\right )}\right )^{\frac{7}{3}} \left (\frac{e \left (b + 2 c x + \sqrt{- 4 a c + b^{2}}\right )}{c \left (d + e x\right )}\right )^{\frac{7}{3}} \operatorname{appellf_{1}}{\left (\frac{14}{3},\frac{7}{3},\frac{7}{3},\frac{17}{3},\frac{c d - \frac{e \left (b - \sqrt{- 4 a c + b^{2}}\right )}{2}}{c \left (d + e x\right )},\frac{c d - \frac{e \left (b + \sqrt{- 4 a c + b^{2}}\right )}{2}}{c \left (d + e x\right )} \right )}}{448 e \left (a + b x + c x^{2}\right )^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/(c*x**2+b*x+a)**(7/3),x)

[Out]

-3*2**(1/3)*(e*(b + 2*c*x - sqrt(-4*a*c + b**2))/(c*(d + e*x)))**(7/3)*(e*(b + 2
*c*x + sqrt(-4*a*c + b**2))/(c*(d + e*x)))**(7/3)*appellf1(14/3, 7/3, 7/3, 17/3,
(c*d - e*(b - sqrt(-4*a*c + b**2))/2)/(c*(d + e*x)), (c*d - e*(b + sqrt(-4*a*c
+ b**2))/2)/(c*(d + e*x)))/(448*e*(a + b*x + c*x**2)**(7/3))

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Mathematica [A]  time = 1.18191, size = 180, normalized size = 0.99 \[ -\frac{3 \left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} F_1\left (\frac{14}{3};\frac{7}{3},\frac{7}{3};\frac{17}{3};\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right )}{224\ 2^{2/3} e (a+x (b+c x))^{7/3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((d + e*x)*(a + b*x + c*x^2)^(7/3)),x]

[Out]

(-3*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(7/3)*((e*(b + Sqrt[b^2
- 4*a*c] + 2*c*x))/(c*(d + e*x)))^(7/3)*AppellF1[14/3, 7/3, 7/3, 17/3, (2*c*d -
(b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/
(2*c*d + 2*c*e*x)])/(224*2^(2/3)*e*(a + x*(b + c*x))^(7/3))

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Maple [F]  time = 0.132, size = 0, normalized size = 0. \[ \int{\frac{1}{ex+d} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{7}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(c*x^2+b*x+a)^(7/3),x)

[Out]

int(1/(e*x+d)/(c*x^2+b*x+a)^(7/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{7}{3}}{\left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^(7/3)*(e*x + d)),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + b*x + a)^(7/3)*(e*x + d)), x)

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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)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac{7}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)/(c*x**2+b*x+a)**(7/3),x)

[Out]

Integral(1/((d + e*x)*(a + b*x + c*x**2)**(7/3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{7}{3}}{\left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^(7/3)*(e*x + d)),x, algorithm="giac")

[Out]

integrate(1/((c*x^2 + b*x + a)^(7/3)*(e*x + d)), x)