3.2472 \(\int (d+e x)^2 \left (a+b x+c x^2\right )^{4/3} \, dx\)

Optimal. Leaf size=638 \[ \frac{\sqrt [3]{2} 3^{3/4} \sqrt{2+\sqrt{3}} \left (b^2-4 a c\right )^2 \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt{\frac{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+\left (b^2-4 a c\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} \left (-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right )|-7-4 \sqrt{3}\right )}{935 c^{13/3} (b+2 c x) \sqrt{\frac{\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [3]{a+b x+c x^2} \left (-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right )}{935 c^4}+\frac{3 (b+2 c x) \left (a+b x+c x^2\right )^{4/3} \left (-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right )}{374 c^3}+\frac{15 e \left (a+b x+c x^2\right )^{7/3} (2 c d-b e)}{119 c^2}+\frac{3 e (d+e x) \left (a+b x+c x^2\right )^{7/3}}{17 c} \]

[Out]

(-3*(b^2 - 4*a*c)*(17*c^2*d^2 + 5*b^2*e^2 - c*e*(17*b*d + 3*a*e))*(b + 2*c*x)*(a
+ b*x + c*x^2)^(1/3))/(935*c^4) + (3*(17*c^2*d^2 + 5*b^2*e^2 - c*e*(17*b*d + 3*
a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(4/3))/(374*c^3) + (15*e*(2*c*d - b*e)*(a +
b*x + c*x^2)^(7/3))/(119*c^2) + (3*e*(d + e*x)*(a + b*x + c*x^2)^(7/3))/(17*c) +
(2^(1/3)*3^(3/4)*Sqrt[2 + Sqrt[3]]*(b^2 - 4*a*c)^2*(17*c^2*d^2 + 5*b^2*e^2 - c*
e*(17*b*d + 3*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 + S
qrt[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]])/(935*c^(13/3)*(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 = 2.27773, antiderivative size = 638, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\sqrt [3]{2} 3^{3/4} \sqrt{2+\sqrt{3}} \left (b^2-4 a c\right )^2 \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt{\frac{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+\left (b^2-4 a c\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} \left (-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right )|-7-4 \sqrt{3}\right )}{935 c^{13/3} (b+2 c x) \sqrt{\frac{\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [3]{a+b x+c x^2} \left (-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right )}{935 c^4}+\frac{3 (b+2 c x) \left (a+b x+c x^2\right )^{4/3} \left (-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right )}{374 c^3}+\frac{15 e \left (a+b x+c x^2\right )^{7/3} (2 c d-b e)}{119 c^2}+\frac{3 e (d+e x) \left (a+b x+c x^2\right )^{7/3}}{17 c} \]

Warning: Unable to verify antiderivative.

[In]  Int[(d + e*x)^2*(a + b*x + c*x^2)^(4/3),x]

[Out]

(-3*(b^2 - 4*a*c)*(17*c^2*d^2 + 5*b^2*e^2 - c*e*(17*b*d + 3*a*e))*(b + 2*c*x)*(a
+ b*x + c*x^2)^(1/3))/(935*c^4) + (3*(17*c^2*d^2 + 5*b^2*e^2 - c*e*(17*b*d + 3*
a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(4/3))/(374*c^3) + (15*e*(2*c*d - b*e)*(a +
b*x + c*x^2)^(7/3))/(119*c^2) + (3*e*(d + e*x)*(a + b*x + c*x^2)^(7/3))/(17*c) +
(2^(1/3)*3^(3/4)*Sqrt[2 + Sqrt[3]]*(b^2 - 4*a*c)^2*(17*c^2*d^2 + 5*b^2*e^2 - c*
e*(17*b*d + 3*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 + S
qrt[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]])/(935*c^(13/3)*(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 = 106.51, size = 734, normalized size = 1.15 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**2*(c*x**2+b*x+a)**(4/3),x)

[Out]

3*e*(d + e*x)*(a + b*x + c*x**2)**(7/3)/(17*c) - 15*e*(b*e - 2*c*d)*(a + b*x + c
*x**2)**(7/3)/(119*c**2) + 3*(a + b*x + c*x**2)**(4/3)*sqrt(-4*a*c + b**2 + c*(4
*a + 4*b*x + 4*c*x**2))*(-3*a*c*e**2 + 5*b**2*e**2 - 17*b*c*d*e + 17*c**2*d**2)*
sqrt((b + 2*c*x)**2)/(374*c**3*(b + 2*c*x)) - 3*(-4*a*c + b**2)*(a + b*x + c*x**
2)**(1/3)*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2))*(-3*a*c*e**2 + 5*b**2
*e**2 - 17*b*c*d*e + 17*c**2*d**2)*sqrt((b + 2*c*x)**2)/(935*c**4*(b + 2*c*x)) +
2**(1/3)*3**(3/4)*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)*(-4*a*c + b**2)**2*(2**(2/3)*c**(1/3)*(a + b*x
+ c*x**2)**(1/3) + (-4*a*c + b**2)**(1/3))*(-3*a*c*e**2 + 5*b**2*e**2 - 17*b*c*d
*e + 17*c**2*d**2)*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))/(935*c**(13/3)*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)*sqrt(-4*a*c + b**2 +
c*(4*a + 4*b*x + 4*c*x**2)))

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Mathematica [C]  time = 1.44207, size = 413, normalized size = 0.65 \[ -\frac{3 \left (2 c (a+x (b+c x)) \left (-b c^2 \left (-823 a^2 e^2+a c \left (1547 d^2+646 d e x+125 e^2 x^2\right )+15 c^2 x^2 \left (119 d^2+170 d e x+66 e^2 x^2\right )\right )-2 c^3 \left (a^2 e (935 d+112 e x)+a c x \left (1547 d^2+1870 d e x+665 e^2 x^2\right )+5 c^2 x^3 \left (119 d^2+187 d e x+77 e^2 x^2\right )\right )+b^3 c \left (c \left (238 d^2+119 d e x+25 e^2 x^2\right )-497 a e^2\right )+b^2 c^2 \left (a e (1547 d+211 e x)-c x \left (119 d^2+85 d e x+20 e^2 x^2\right )\right )+70 b^5 e^2-7 b^4 c e (34 d+5 e x)\right )-7 \sqrt [3]{2} \left (b^2-4 a c\right )^2 \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{2/3} \left (-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )\right )}{26180 c^5 (a+x (b+c x))^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^2*(a + b*x + c*x^2)^(4/3),x]

[Out]

(-3*(2*c*(a + x*(b + c*x))*(70*b^5*e^2 - 7*b^4*c*e*(34*d + 5*e*x) + b^2*c^2*(a*e
*(1547*d + 211*e*x) - c*x*(119*d^2 + 85*d*e*x + 20*e^2*x^2)) + b^3*c*(-497*a*e^2
+ c*(238*d^2 + 119*d*e*x + 25*e^2*x^2)) - b*c^2*(-823*a^2*e^2 + 15*c^2*x^2*(119
*d^2 + 170*d*e*x + 66*e^2*x^2) + a*c*(1547*d^2 + 646*d*e*x + 125*e^2*x^2)) - 2*c
^3*(a^2*e*(935*d + 112*e*x) + 5*c^2*x^3*(119*d^2 + 187*d*e*x + 77*e^2*x^2) + a*c
*x*(1547*d^2 + 1870*d*e*x + 665*e^2*x^2))) - 7*2^(1/3)*(b^2 - 4*a*c)^2*(17*c^2*d
^2 + 5*b^2*e^2 - c*e*(17*b*d + 3*a*e))*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*((b + Sqr
t[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c])^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3
, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])]))/(26180*c^5*(a + x*(b
+ c*x))^(2/3))

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Maple [F]  time = 0.171, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{4}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2*(c*x^2+b*x+a)^(4/3),x)

[Out]

int((e*x+d)^2*(c*x^2+b*x+a)^(4/3),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(4/3)*(e*x + d)^2,x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(4/3)*(e*x + d)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(4/3)*(e*x + d)^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**2*(c*x**2+b*x+a)**(4/3),x)

[Out]

Integral((d + e*x)**2*(a + b*x + c*x**2)**(4/3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{4}{3}}{\left (e x + d\right )}^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(4/3)*(e*x + d)^2,x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(4/3)*(e*x + d)^2, x)