3.2480 \(\int \frac{d+e x}{\left (a+b x+c x^2\right )^{7/3}} \, dx\)

Optimal. Leaf size=1043 \[ \text{result too large to display} \]

[Out]

(-3*(b*d - 2*a*e + (2*c*d - b*e)*x))/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(4/3)) +
(15*(2*c*d - b*e)*(b + 2*c*x))/(4*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(1/3)) - (1
5*c^(1/3)*(2*c*d - b*e)*(b + 2*c*x))/(2*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)*Sqr
t[2 - Sqrt[3]]*c^(1/3)*(2*c*d - b*e)*((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]*Ellip
ticE[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]])/(4*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*3^(3/4)*c^(1/3)*(2*c*d - b*e)*((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]*Elli
pticF[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^(5/6)*(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 = 2.21401, antiderivative size = 1043, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{15 (2 c d-b e) (b+2 c x)}{4 \left (b^2-4 a c\right )^2 \sqrt [3]{c x^2+b x+a}}-\frac{15 \sqrt [3]{c} (2 c d-b e) (b+2 c x)}{2 \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{3 (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{4/3}}+\frac{15 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{c} (2 c d-b e) \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 )}{4 \sqrt [3]{2} \left (b^2-4 a c\right )^{5/3} \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}} (b+2 c x)}-\frac{5\ 3^{3/4} \sqrt [3]{c} (2 c d-b e) \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 )}{2^{5/6} \left (b^2-4 a c\right )^{5/3} \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}} (b+2 c x)} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-3*(b*d - 2*a*e + (2*c*d - b*e)*x))/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(4/3)) +
(15*(2*c*d - b*e)*(b + 2*c*x))/(4*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(1/3)) - (1
5*c^(1/3)*(2*c*d - b*e)*(b + 2*c*x))/(2*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)*Sqr
t[2 - Sqrt[3]]*c^(1/3)*(2*c*d - b*e)*((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]*Ellip
ticE[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]])/(4*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*3^(3/4)*c^(1/3)*(2*c*d - b*e)*((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]*Elli
pticF[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^(5/6)*(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 = 121.535, size = 1130, normalized size = 1.08 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*2**(2/3)*3**(1/4)*c**(1/3)*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)*(b*e - 2*c*d)*(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))/(8*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**(1/3)*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)*(b*e - 2*c*d)*(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))/(2*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)*sqr
t(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2))) + 15*2**(2/3)*c**(1/3)*(b*e - 2*c
*d)*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)**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))) + 3*(2*a*e - b*d + x*(b*e - 2*c*d))/(4*(-4*a*c
+ b**2)*(a + b*x + c*x**2)**(4/3)) - 15*(b*e - 2*c*d)*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)**2*(
a + b*x + c*x**2)**(1/3))

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Mathematica [C]  time = 0.904071, size = 200, normalized size = 0.19 \[ \frac{3 \left (5\ 2^{2/3} \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [3]{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}} (b e-2 c d) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )+\frac{4 \left (b^2-4 a c\right ) (2 a e-b d+b e x-2 c d x)}{a+x (b+c x)}+20 (b+2 c x) (2 c d-b e)\right )}{16 \left (b^2-4 a c\right )^2 \sqrt [3]{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(20*(2*c*d - b*e)*(b + 2*c*x) + (4*(b^2 - 4*a*c)*(-(b*d) + 2*a*e - 2*c*d*x +
b*e*x))/(a + x*(b + c*x)) + 5*2^(2/3)*(-2*c*d + b*e)*(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])]))/(16*(b
^2 - 4*a*c)^2*(a + x*(b + c*x))^(1/3))

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Maple [F]  time = 0.118, size = 0, normalized size = 0. \[ \int{(ex+d) \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{7}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((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. \[ \int \frac{e x + d}{{\left (c x^{2} + b x + a\right )}^{\frac{7}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x + d)/((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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)/(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. \[ \int \frac{e x + d}{{\left (c x^{2} + b x + a\right )}^{\frac{7}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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