3.1408 \(\int \frac{\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{20/3}} \, dx\)

Optimal. Leaf size=637 \[ \frac{3\ 3^{3/4} \left (-4 a c+b^2-(b+2 c x)^2\right ) \sqrt [3]{d (b+2 c x)} \left (2 \sqrt [3]{c} d^{2/3}-\frac{\sqrt [3]{2} (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right ) \sqrt{\frac{\frac{2^{2/3} \sqrt [3]{c} d^{2/3} (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}+\frac{(d (b+2 c x))^{4/3}}{\left (a+b x+c x^2\right )^{2/3}}+2 \sqrt [3]{2} c^{2/3} d^{4/3}}{\left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac{\left (1+\sqrt{3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac{2^{2/3} \sqrt [3]{c} d^{2/3}-\frac{\left (1-\sqrt{3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{c x^2+b x+a}}}{2^{2/3} \sqrt [3]{c} d^{2/3}-\frac{\left (1+\sqrt{3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{c x^2+b x+a}}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{7480 c^{10/3} d^{23/3} \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{2/3} \sqrt{-\frac{(d (b+2 c x))^{2/3} \left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac{(d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )}{\sqrt [3]{a+b x+c x^2} \left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac{\left (1+\sqrt{3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )^2}}}+\frac{6 \sqrt [3]{a+b x+c x^2}}{935 c^2 d^5 \left (b^2-4 a c\right ) (d (b+2 c x))^{5/3}}-\frac{3 \sqrt [3]{a+b x+c x^2}}{187 c^2 d^3 (d (b+2 c x))^{11/3}}-\frac{3 \left (a+b x+c x^2\right )^{4/3}}{34 c d (d (b+2 c x))^{17/3}} \]

[Out]

(-3*(a + b*x + c*x^2)^(1/3))/(187*c^2*d^3*(d*(b + 2*c*x))^(11/3)) + (6*(a + b*x
+ c*x^2)^(1/3))/(935*c^2*(b^2 - 4*a*c)*d^5*(d*(b + 2*c*x))^(5/3)) - (3*(a + b*x
+ c*x^2)^(4/3))/(34*c*d*(d*(b + 2*c*x))^(17/3)) + (3*3^(3/4)*(d*(b + 2*c*x))^(1/
3)*(b^2 - 4*a*c - (b + 2*c*x)^2)*(2*c^(1/3)*d^(2/3) - (2^(1/3)*(d*(b + 2*c*x))^(
2/3))/(a + b*x + c*x^2)^(1/3))*Sqrt[(2*2^(1/3)*c^(2/3)*d^(4/3) + (d*(b + 2*c*x))
^(4/3)/(a + b*x + c*x^2)^(2/3) + (2^(2/3)*c^(1/3)*d^(2/3)*(d*(b + 2*c*x))^(2/3))
/(a + b*x + c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*d^(2/3) - ((1 + Sqrt[3])*(d*(b + 2*c*
x))^(2/3))/(a + b*x + c*x^2)^(1/3))^2]*EllipticF[ArcCos[(2^(2/3)*c^(1/3)*d^(2/3)
- ((1 - Sqrt[3])*(d*(b + 2*c*x))^(2/3))/(a + b*x + c*x^2)^(1/3))/(2^(2/3)*c^(1/
3)*d^(2/3) - ((1 + Sqrt[3])*(d*(b + 2*c*x))^(2/3))/(a + b*x + c*x^2)^(1/3))], (2
+ Sqrt[3])/4])/(7480*c^(10/3)*(b^2 - 4*a*c)^2*d^(23/3)*(a + b*x + c*x^2)^(2/3)*
Sqrt[-(((d*(b + 2*c*x))^(2/3)*(2^(2/3)*c^(1/3)*d^(2/3) - (d*(b + 2*c*x))^(2/3)/(
a + b*x + c*x^2)^(1/3)))/((a + b*x + c*x^2)^(1/3)*(2^(2/3)*c^(1/3)*d^(2/3) - ((1
+ Sqrt[3])*(d*(b + 2*c*x))^(2/3))/(a + b*x + c*x^2)^(1/3))^2))])

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Rubi [A]  time = 4.68584, antiderivative size = 637, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{3\ 3^{3/4} \left (-4 a c+b^2-(b+2 c x)^2\right ) \sqrt [3]{d (b+2 c x)} \left (2 \sqrt [3]{c} d^{2/3}-\frac{\sqrt [3]{2} (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right ) \sqrt{\frac{\frac{2^{2/3} \sqrt [3]{c} d^{2/3} (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}+\frac{(d (b+2 c x))^{4/3}}{\left (a+b x+c x^2\right )^{2/3}}+2 \sqrt [3]{2} c^{2/3} d^{4/3}}{\left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac{\left (1+\sqrt{3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac{2^{2/3} \sqrt [3]{c} d^{2/3}-\frac{\left (1-\sqrt{3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{c x^2+b x+a}}}{2^{2/3} \sqrt [3]{c} d^{2/3}-\frac{\left (1+\sqrt{3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{c x^2+b x+a}}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{7480 c^{10/3} d^{23/3} \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{2/3} \sqrt{-\frac{(d (b+2 c x))^{2/3} \left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac{(d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )}{\sqrt [3]{a+b x+c x^2} \left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac{\left (1+\sqrt{3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )^2}}}+\frac{6 \sqrt [3]{a+b x+c x^2}}{935 c^2 d^5 \left (b^2-4 a c\right ) (d (b+2 c x))^{5/3}}-\frac{3 \sqrt [3]{a+b x+c x^2}}{187 c^2 d^3 (d (b+2 c x))^{11/3}}-\frac{3 \left (a+b x+c x^2\right )^{4/3}}{34 c d (d (b+2 c x))^{17/3}} \]

Warning: Unable to verify antiderivative.

[In]  Int[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(20/3),x]

[Out]

(-3*(a + b*x + c*x^2)^(1/3))/(187*c^2*d^3*(d*(b + 2*c*x))^(11/3)) + (6*(a + b*x
+ c*x^2)^(1/3))/(935*c^2*(b^2 - 4*a*c)*d^5*(d*(b + 2*c*x))^(5/3)) - (3*(a + b*x
+ c*x^2)^(4/3))/(34*c*d*(d*(b + 2*c*x))^(17/3)) + (3*3^(3/4)*(d*(b + 2*c*x))^(1/
3)*(b^2 - 4*a*c - (b + 2*c*x)^2)*(2*c^(1/3)*d^(2/3) - (2^(1/3)*(d*(b + 2*c*x))^(
2/3))/(a + b*x + c*x^2)^(1/3))*Sqrt[(2*2^(1/3)*c^(2/3)*d^(4/3) + (d*(b + 2*c*x))
^(4/3)/(a + b*x + c*x^2)^(2/3) + (2^(2/3)*c^(1/3)*d^(2/3)*(d*(b + 2*c*x))^(2/3))
/(a + b*x + c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*d^(2/3) - ((1 + Sqrt[3])*(d*(b + 2*c*
x))^(2/3))/(a + b*x + c*x^2)^(1/3))^2]*EllipticF[ArcCos[(2^(2/3)*c^(1/3)*d^(2/3)
- ((1 - Sqrt[3])*(d*(b + 2*c*x))^(2/3))/(a + b*x + c*x^2)^(1/3))/(2^(2/3)*c^(1/
3)*d^(2/3) - ((1 + Sqrt[3])*(d*(b + 2*c*x))^(2/3))/(a + b*x + c*x^2)^(1/3))], (2
+ Sqrt[3])/4])/(7480*c^(10/3)*(b^2 - 4*a*c)^2*d^(23/3)*(a + b*x + c*x^2)^(2/3)*
Sqrt[-(((d*(b + 2*c*x))^(2/3)*(2^(2/3)*c^(1/3)*d^(2/3) - (d*(b + 2*c*x))^(2/3)/(
a + b*x + c*x^2)^(1/3)))/((a + b*x + c*x^2)^(1/3)*(2^(2/3)*c^(1/3)*d^(2/3) - ((1
+ Sqrt[3])*(d*(b + 2*c*x))^(2/3))/(a + b*x + c*x^2)^(1/3))^2))])

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Rubi in Sympy [A]  time = 130.194, size = 813, normalized size = 1.28 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(4/3)/(2*c*d*x+b*d)**(20/3),x)

[Out]

-3*(a - b**2/(4*c) + (b + 2*c*x)**2/(4*c))**(4/3)/(34*c*d*(b*d + 2*c*d*x)**(17/3
)) - 3*(a - b**2/(4*c) + (b + 2*c*x)**2/(4*c))**(1/3)/(187*c**2*d**3*(b*d + 2*c*
d*x)**(11/3)) + 6*(a - b**2/(4*c) + (b + 2*c*x)**2/(4*c))**(1/3)/(935*c**2*d**5*
(-4*a*c + b**2)*(b*d + 2*c*d*x)**(5/3)) - 3*3**(3/4)*sqrt((4*c**(2/3)*d**(4/3) +
2*2**(1/3)*c**(1/3)*d**(2/3)*(b*d + 2*c*d*x)**(2/3)/(a - b**2/(4*c) + (b + 2*c*
x)**2/(4*c))**(1/3) + 2**(2/3)*(b*d + 2*c*d*x)**(4/3)/(a - b**2/(4*c) + (b + 2*c
*x)**2/(4*c))**(2/3))/(2**(2/3)*c**(1/3)*d**(2/3) + (-sqrt(3) - 1)*(b*d + 2*c*d*
x)**(2/3)/(a - b**2/(4*c) + (b + 2*c*x)**2/(4*c))**(1/3))**2)*(b*d + 2*c*d*x)**(
1/3)*(-2**(2/3)*c**(1/3)*d**(2/3) + (b*d + 2*c*d*x)**(2/3)/(a - b**2/(4*c) + (b
+ 2*c*x)**2/(4*c))**(1/3))*elliptic_f(acos((2**(2/3)*c**(1/3)*d**(2/3) + (-1 + s
qrt(3))*(b*d + 2*c*d*x)**(2/3)/(a - b**2/(4*c) + (b + 2*c*x)**2/(4*c))**(1/3))/(
2**(2/3)*c**(1/3)*d**(2/3) + (-sqrt(3) - 1)*(b*d + 2*c*d*x)**(2/3)/(a - b**2/(4*
c) + (b + 2*c*x)**2/(4*c))**(1/3))), sqrt(3)/4 + 1/2)/(7480*c**(10/3)*d**(23/3)*
sqrt(d**2*(a*c - b**2/4)/(d**2*(b + 2*c*x)**2/4 + d**2*(a*c - b**2/4)))*sqrt(-(b
*d + 2*c*d*x)**(2/3)*(2**(2/3)*c**(1/3)*d**(2/3) - (b*d + 2*c*d*x)**(2/3)/(a - b
**2/(4*c) + (b + 2*c*x)**2/(4*c))**(1/3))/((2**(2/3)*c**(1/3)*d**(2/3) + (-sqrt(
3) - 1)*(b*d + 2*c*d*x)**(2/3)/(a - b**2/(4*c) + (b + 2*c*x)**2/(4*c))**(1/3))**
2*(a - b**2/(4*c) + (b + 2*c*x)**2/(4*c))**(1/3)))*(-4*a*c + b**2)*sqrt(-d**2*(b
+ 2*c*x)**2/(4*(d**2*(b + 2*c*x)**2/4 + d**2*(a*c - b**2/4))) + 1)*(a - b**2/(4
*c) + (b + 2*c*x)**2/(4*c))**(2/3))

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Mathematica [C]  time = 0.55977, size = 170, normalized size = 0.27 \[ \frac{3 \left (24 \sqrt [3]{2} (b+2 c x)^6 \left (-\frac{c (a+x (b+c x))}{b^2-4 a c}\right )^{2/3} \, _2F_1\left (\frac{1}{6},\frac{2}{3};\frac{7}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right )+c (a+x (b+c x)) \left (-95 \left (b^2-4 a c\right ) (b+2 c x)^2+55 \left (b^2-4 a c\right )^2+16 (b+2 c x)^4\right )\right )}{7480 c^3 d \left (b^2-4 a c\right ) (a+x (b+c x))^{2/3} (d (b+2 c x))^{17/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(20/3),x]

[Out]

(3*(c*(a + x*(b + c*x))*(55*(b^2 - 4*a*c)^2 - 95*(b^2 - 4*a*c)*(b + 2*c*x)^2 + 1
6*(b + 2*c*x)^4) + 24*2^(1/3)*(b + 2*c*x)^6*(-((c*(a + x*(b + c*x)))/(b^2 - 4*a*
c)))^(2/3)*HypergeometricPFQ[{1/6, 2/3}, {7/6}, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(
7480*c^3*(b^2 - 4*a*c)*d*(d*(b + 2*c*x))^(17/3)*(a + x*(b + c*x))^(2/3))

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Maple [F]  time = 0.176, size = 0, normalized size = 0. \[ \int{1 \left ( c{x}^{2}+bx+a \right ) ^{{\frac{4}{3}}} \left ( 2\,cdx+bd \right ) ^{-{\frac{20}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(4/3)/(2*c*d*x+b*d)^(20/3),x)

[Out]

int((c*x^2+b*x+a)^(4/3)/(2*c*d*x+b*d)^(20/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{4}{3}}}{{\left (2 \, c d x + b d\right )}^{\frac{20}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(4/3)/(2*c*d*x + b*d)^(20/3),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(4/3)/(2*c*d*x + b*d)^(20/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{2} + b x + a\right )}^{\frac{4}{3}}}{{\left (64 \, c^{6} d^{6} x^{6} + 192 \, b c^{5} d^{6} x^{5} + 240 \, b^{2} c^{4} d^{6} x^{4} + 160 \, b^{3} c^{3} d^{6} x^{3} + 60 \, b^{4} c^{2} d^{6} x^{2} + 12 \, b^{5} c d^{6} x + b^{6} d^{6}\right )}{\left (2 \, c d x + b d\right )}^{\frac{2}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(4/3)/(2*c*d*x + b*d)^(20/3),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^(4/3)/((64*c^6*d^6*x^6 + 192*b*c^5*d^6*x^5 + 240*b^2*
c^4*d^6*x^4 + 160*b^3*c^3*d^6*x^3 + 60*b^4*c^2*d^6*x^2 + 12*b^5*c*d^6*x + b^6*d^
6)*(2*c*d*x + b*d)^(2/3)), 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((c*x**2+b*x+a)**(4/3)/(2*c*d*x+b*d)**(20/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{4}{3}}}{{\left (2 \, c d x + b d\right )}^{\frac{20}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(4/3)/(2*c*d*x + b*d)^(20/3),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(4/3)/(2*c*d*x + b*d)^(20/3), x)