3.2483 \(\int \frac{1}{(d+e x)^2 \left (a+b x+c x^2\right )^{7/3}} \, dx\)

Optimal. Leaf size=189 \[ -\frac{3 \left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} F_1\left (\frac{17}{3};\frac{7}{3},\frac{7}{3};\frac{20}{3};\frac{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 d-\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{272\ 2^{2/3} e (d+e x) \left (a+b x+c x^2\right )^{7/3}} \]

[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[17/3, 7/3, 7/3, 20/3, (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))])/(272*2^(2/3)*e*(d + e*x)*(a + b*x + c*x^2)^(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]

(-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[17/3, 7/3, 7/3, 20/3, (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))])/(272*2^(2/3)*e*(d + e*x)*(a + b*x + c*x^2)^(7/3))

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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)

[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(17/3, 7/3, 7/3, 20/3,
(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)))/(544*e*(d + e*x)*(a + b*x + c*x**2)**(7/3))

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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]

[Out]

(-3*e^3*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(1/3)*((e*(b + Sqrt[
b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(1/3)*AppellF1[17/3, 7/3, 7/3, 20/3, (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)])/(17*2^(2/3)*c^2*(d + e*x)^5*(a + x*(b + c*x))^(1/3))

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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]

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

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

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")

[Out]

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

[Out]

Timed 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]

Timed 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]

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