3.2481 \(\int \frac{1}{\left (a+b x+c x^2\right )^{7/3}} \, dx\)

Optimal. Leaf size=993 \[ \text{result too large to display} \]

[Out]

(-3*(b + 2*c*x))/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(4/3)) + (15*c*(b + 2*c*x))/
(2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(1/3)) - (15*c^(4/3)*(b + 2*c*x))/(2^(1/3)*
(b^2 - 4*a*c)^2*((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x +
c*x^2)^(1/3))) + (15*3^(1/4)*Sqrt[2 - Sqrt[3]]*c^(4/3)*((b^2 - 4*a*c)^(1/3) + 2^
(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))*Sqrt[((b^2 - 4*a*c)^(2/3) - 2^(2/3)*c^(1/
3)*(b^2 - 4*a*c)^(1/3)*(a + b*x + c*x^2)^(1/3) + 2*2^(1/3)*c^(2/3)*(a + b*x + c*
x^2)^(2/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^
2)^(1/3))^2]*EllipticE[ArcSin[((1 - Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/
3)*(a + b*x + c*x^2)^(1/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)
*(a + b*x + c*x^2)^(1/3))], -7 - 4*Sqrt[3]])/(2*2^(1/3)*(b^2 - 4*a*c)^(5/3)*(b +
2*c*x)*Sqrt[((b^2 - 4*a*c)^(1/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*
x + c*x^2)^(1/3)))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x
+ c*x^2)^(1/3))^2]) - (5*2^(1/6)*3^(3/4)*c^(4/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)
*c^(1/3)*(a + b*x + c*x^2)^(1/3))*Sqrt[((b^2 - 4*a*c)^(2/3) - 2^(2/3)*c^(1/3)*(b
^2 - 4*a*c)^(1/3)*(a + b*x + c*x^2)^(1/3) + 2*2^(1/3)*c^(2/3)*(a + b*x + c*x^2)^
(2/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1
/3))^2]*EllipticF[ArcSin[((1 - Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a
+ b*x + c*x^2)^(1/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a +
b*x + c*x^2)^(1/3))], -7 - 4*Sqrt[3]])/((b^2 - 4*a*c)^(5/3)*(b + 2*c*x)*Sqrt[((
b^2 - 4*a*c)^(1/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3
)))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3)
)^2])

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Rubi [A]  time = 2.186, antiderivative size = 993, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ \frac{15 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right ) \sqrt{\frac{\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a} \sqrt [3]{b^2-4 a c}+2 \sqrt [3]{2} c^{2/3} \left (c x^2+b x+a\right )^{2/3}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right )|-7-4 \sqrt{3}\right ) c^{4/3}}{2 \sqrt [3]{2} \left (b^2-4 a c\right )^{5/3} (b+2 c x) \sqrt{\frac{\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}}}-\frac{5 \sqrt [6]{2} 3^{3/4} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right ) \sqrt{\frac{\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a} \sqrt [3]{b^2-4 a c}+2 \sqrt [3]{2} c^{2/3} \left (c x^2+b x+a\right )^{2/3}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right )|-7-4 \sqrt{3}\right ) c^{4/3}}{\left (b^2-4 a c\right )^{5/3} (b+2 c x) \sqrt{\frac{\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}}}-\frac{15 (b+2 c x) c^{4/3}}{\sqrt [3]{2} \left (b^2-4 a c\right )^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )}+\frac{15 (b+2 c x) c}{2 \left (b^2-4 a c\right )^2 \sqrt [3]{c x^2+b x+a}}-\frac{3 (b+2 c x)}{4 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{4/3}} \]

Warning: Unable to verify antiderivative.

[In]  Int[(a + b*x + c*x^2)^(-7/3),x]

[Out]

(-3*(b + 2*c*x))/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(4/3)) + (15*c*(b + 2*c*x))/
(2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(1/3)) - (15*c^(4/3)*(b + 2*c*x))/(2^(1/3)*
(b^2 - 4*a*c)^2*((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x +
c*x^2)^(1/3))) + (15*3^(1/4)*Sqrt[2 - Sqrt[3]]*c^(4/3)*((b^2 - 4*a*c)^(1/3) + 2^
(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))*Sqrt[((b^2 - 4*a*c)^(2/3) - 2^(2/3)*c^(1/
3)*(b^2 - 4*a*c)^(1/3)*(a + b*x + c*x^2)^(1/3) + 2*2^(1/3)*c^(2/3)*(a + b*x + c*
x^2)^(2/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^
2)^(1/3))^2]*EllipticE[ArcSin[((1 - Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/
3)*(a + b*x + c*x^2)^(1/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)
*(a + b*x + c*x^2)^(1/3))], -7 - 4*Sqrt[3]])/(2*2^(1/3)*(b^2 - 4*a*c)^(5/3)*(b +
2*c*x)*Sqrt[((b^2 - 4*a*c)^(1/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*
x + c*x^2)^(1/3)))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x
+ c*x^2)^(1/3))^2]) - (5*2^(1/6)*3^(3/4)*c^(4/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)
*c^(1/3)*(a + b*x + c*x^2)^(1/3))*Sqrt[((b^2 - 4*a*c)^(2/3) - 2^(2/3)*c^(1/3)*(b
^2 - 4*a*c)^(1/3)*(a + b*x + c*x^2)^(1/3) + 2*2^(1/3)*c^(2/3)*(a + b*x + c*x^2)^
(2/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1
/3))^2]*EllipticF[ArcSin[((1 - Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a
+ b*x + c*x^2)^(1/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a +
b*x + c*x^2)^(1/3))], -7 - 4*Sqrt[3]])/((b^2 - 4*a*c)^(5/3)*(b + 2*c*x)*Sqrt[((
b^2 - 4*a*c)^(1/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3
)))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3)
)^2])

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Rubi in Sympy [A]  time = 111.165, size = 1124, normalized size = 1.13 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15*2**(2/3)*3**(1/4)*c**(4/3)*sqrt((2*2**(1/3)*c**(2/3)*(a + b*x + c*x**2)**(2/3
) - 2**(2/3)*c**(1/3)*(-4*a*c + b**2)**(1/3)*(a + b*x + c*x**2)**(1/3) + (-4*a*c
+ b**2)**(2/3))/(2**(2/3)*c**(1/3)*(a + b*x + c*x**2)**(1/3) + (1 + sqrt(3))*(-
4*a*c + b**2)**(1/3))**2)*sqrt(-sqrt(3) + 2)*(2**(2/3)*c**(1/3)*(a + b*x + c*x**
2)**(1/3) + (-4*a*c + b**2)**(1/3))*sqrt((b + 2*c*x)**2)*elliptic_e(asin((2**(2/
3)*c**(1/3)*(a + b*x + c*x**2)**(1/3) - (-1 + sqrt(3))*(-4*a*c + b**2)**(1/3))/(
2**(2/3)*c**(1/3)*(a + b*x + c*x**2)**(1/3) + (1 + sqrt(3))*(-4*a*c + b**2)**(1/
3))), -7 - 4*sqrt(3))/(4*sqrt((-4*a*c + b**2)**(1/3)*(2**(2/3)*c**(1/3)*(a + b*x
+ c*x**2)**(1/3) + (-4*a*c + b**2)**(1/3))/(2**(2/3)*c**(1/3)*(a + b*x + c*x**2
)**(1/3) + (1 + sqrt(3))*(-4*a*c + b**2)**(1/3))**2)*(b + 2*c*x)*(-4*a*c + b**2)
**(5/3)*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2))) - 5*2**(1/6)*3**(3/4)*
c**(4/3)*sqrt((2*2**(1/3)*c**(2/3)*(a + b*x + c*x**2)**(2/3) - 2**(2/3)*c**(1/3)
*(-4*a*c + b**2)**(1/3)*(a + b*x + c*x**2)**(1/3) + (-4*a*c + b**2)**(2/3))/(2**
(2/3)*c**(1/3)*(a + b*x + c*x**2)**(1/3) + (1 + sqrt(3))*(-4*a*c + b**2)**(1/3))
**2)*(2**(2/3)*c**(1/3)*(a + b*x + c*x**2)**(1/3) + (-4*a*c + b**2)**(1/3))*sqrt
((b + 2*c*x)**2)*elliptic_f(asin((2**(2/3)*c**(1/3)*(a + b*x + c*x**2)**(1/3) -
(-1 + sqrt(3))*(-4*a*c + b**2)**(1/3))/(2**(2/3)*c**(1/3)*(a + b*x + c*x**2)**(1
/3) + (1 + sqrt(3))*(-4*a*c + b**2)**(1/3))), -7 - 4*sqrt(3))/(sqrt((-4*a*c + b*
*2)**(1/3)*(2**(2/3)*c**(1/3)*(a + b*x + c*x**2)**(1/3) + (-4*a*c + b**2)**(1/3)
)/(2**(2/3)*c**(1/3)*(a + b*x + c*x**2)**(1/3) + (1 + sqrt(3))*(-4*a*c + b**2)**
(1/3))**2)*(b + 2*c*x)*(-4*a*c + b**2)**(5/3)*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*
x + 4*c*x**2))) - 15*2**(2/3)*c**(4/3)*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c
*x**2))*sqrt((b + 2*c*x)**2)/(2*(b + 2*c*x)*(-4*a*c + b**2)**2*(2**(2/3)*c**(1/3
)*(a + b*x + c*x**2)**(1/3) + (1 + sqrt(3))*(-4*a*c + b**2)**(1/3))) + 15*c*sqrt
(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2))*sqrt((b + 2*c*x)**2)/(2*(b + 2*c*x)
*(-4*a*c + b**2)**2*(a + b*x + c*x**2)**(1/3)) - 3*sqrt(-4*a*c + b**2 + c*(4*a +
4*b*x + 4*c*x**2))*sqrt((b + 2*c*x)**2)/(4*(b + 2*c*x)*(-4*a*c + b**2)*(a + b*x
+ c*x**2)**(4/3))

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Mathematica [C]  time = 0.457765, size = 173, normalized size = 0.17 \[ \frac{3 \left (-5\ 2^{2/3} c \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [3]{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )-\frac{2 \left (b^2-4 a c\right ) (b+2 c x)}{a+x (b+c x)}+20 c (b+2 c x)\right )}{8 \left (b^2-4 a c\right )^2 \sqrt [3]{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(20*c*(b + 2*c*x) - (2*(b^2 - 4*a*c)*(b + 2*c*x))/(a + x*(b + c*x)) - 5*2^(2/
3)*c*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 -
4*a*c])^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x)
/(2*Sqrt[b^2 - 4*a*c])]))/(8*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(1/3))

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Maple [F]  time = 0.197, size = 0, normalized size = 0. \[ \int \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{7}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x+a)^(7/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x + a)^(-7/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}{\left (c x^{2} + b x + a\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(-7/3),x, algorithm="fricas")

[Out]

integral(1/((c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*(c*x^2 + b
*x + a)^(1/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x + c*x**2)**(-7/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x + a)^(-7/3), x)