3.2478 \(\int \frac{(d+e x)^3}{\left (a+b x+c x^2\right )^{7/3}} \, dx\)

Optimal. Leaf size=1224 \[ \text{result too large to display} \]

[Out]

(-3*(d + e*x)^2*(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)) + (3*(10*b*c*d*(c*d^2 + 3*a*e^2) - 8*a*c*e*(2*c*d^2 + 3*a*e^2) - b^2*
(11*c*d^2*e - a*e^3) + (2*c*d - b*e)*(10*c^2*d^2 - b^2*e^2 - 2*c*e*(5*b*d - 7*a*
e))*x))/(4*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(1/3)) - (3*(2*c*d - b*e)*(5*c^2*
d^2 - b^2*e^2 - c*e*(5*b*d - 9*a*e))*(b + 2*c*x))/(2*2^(1/3)*c^(5/3)*(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
))) + (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*(2*c*d - b*e)*(5*c^2*d^2 - b^2*e^2 - c*e*(5*b
*d - 9*a*e))*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))*Sqr
t[((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]
])/(4*2^(1/3)*c^(5/3)*(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]) - (3^(3/4)*(2
*c*d - b*e)*(5*c^2*d^2 - b^2*e^2 - c*e*(5*b*d - 9*a*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]*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]])/(2^(5/6)*c^(5/3)*(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])

_______________________________________________________________________________________

Rubi [A]  time = 3.58935, antiderivative size = 1224, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{3 (b d-2 a e+(2 c d-b e) x) (d+e x)^2}{4 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{4/3}}+\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right ) \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} c^{5/3} \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{3^{3/4} (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right ) \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} c^{5/3} \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{3 \left (-\left (11 c d^2 e-a e^3\right ) b^2+10 c d \left (c d^2+3 a e^2\right ) b-8 a c e \left (2 c d^2+3 a e^2\right )+(2 c d-b e) \left (10 c^2 d^2-b^2 e^2-2 c e (5 b d-7 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \sqrt [3]{c x^2+b x+a}}-\frac{3 (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right ) (b+2 c x)}{2 \sqrt [3]{2} c^{5/3} \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 )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-3*(d + e*x)^2*(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)) + (3*(10*b*c*d*(c*d^2 + 3*a*e^2) - 8*a*c*e*(2*c*d^2 + 3*a*e^2) - b^2*
(11*c*d^2*e - a*e^3) + (2*c*d - b*e)*(10*c^2*d^2 - b^2*e^2 - 2*c*e*(5*b*d - 7*a*
e))*x))/(4*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(1/3)) - (3*(2*c*d - b*e)*(5*c^2*
d^2 - b^2*e^2 - c*e*(5*b*d - 9*a*e))*(b + 2*c*x))/(2*2^(1/3)*c^(5/3)*(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
))) + (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*(2*c*d - b*e)*(5*c^2*d^2 - b^2*e^2 - c*e*(5*b
*d - 9*a*e))*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))*Sqr
t[((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]
])/(4*2^(1/3)*c^(5/3)*(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]) - (3^(3/4)*(2
*c*d - b*e)*(5*c^2*d^2 - b^2*e^2 - c*e*(5*b*d - 9*a*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]*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]])/(2^(5/6)*c^(5/3)*(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])

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 1.3475, size = 400, normalized size = 0.33 \[ -\frac{3 \left (4 \left (b^2-4 a c\right ) \left (2 c \left (a^2 e^3-3 a c d e (d+e x)+c^2 d^3 x\right )+b^2 e^2 (3 c d x-a e)+b c \left (3 a e^2 (d+e x)+c d^2 (d-3 e x)\right )-b^3 e^3 x\right )-4 (a+x (b+c x)) \left (4 c^2 \left (-8 a^2 e^3+9 a c d e^2 x+5 c^2 d^3 x\right )+b^2 c e \left (7 a e^2+3 c d (2 e x-5 d)\right )+2 b c^2 \left (9 a e^2 (d-e x)+5 c d^2 (d-3 e x)\right )+b^4 \left (-e^3\right )+b^3 c e^2 (3 d+2 e x)\right )+2^{2/3} \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [3]{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}} (a+x (b+c x)) (b e-2 c d) \left (c e (5 b d-9 a e)+b^2 e^2-5 c^2 d^2\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )\right )}{16 c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(4*(b^2 - 4*a*c)*(-(b^3*e^3*x) + b^2*e^2*(-(a*e) + 3*c*d*x) + 2*c*(a^2*e^3 +
c^2*d^3*x - 3*a*c*d*e*(d + e*x)) + b*c*(c*d^2*(d - 3*e*x) + 3*a*e^2*(d + e*x)))
- 4*(a + x*(b + c*x))*(-(b^4*e^3) + b^3*c*e^2*(3*d + 2*e*x) + 4*c^2*(-8*a^2*e^3
+ 5*c^2*d^3*x + 9*a*c*d*e^2*x) + 2*b*c^2*(5*c*d^2*(d - 3*e*x) + 9*a*e^2*(d - e*
x)) + b^2*c*e*(7*a*e^2 + 3*c*d*(-5*d + 2*e*x))) + 2^(2/3)*(-2*c*d + b*e)*(-5*c^2
*d^2 + b^2*e^2 + c*e*(5*b*d - 9*a*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)*(a + x*(b + c*x))*Hypergeometric
2F1[1/3, 2/3, 5/3, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])]))/(16
*c^2*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(4/3))

_______________________________________________________________________________________

Maple [F]  time = 0.136, size = 0, normalized size = 0. \[ \int{ \left ( ex+d \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{7}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)/((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)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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