3.1405 \(\int \frac{\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{2/3}} \, dx\)

Optimal. Leaf size=597 \[ \frac{\left (b^2-4 a c\right ) \left (-4 a c+b^2-(b+2 c x)^2\right ) \sqrt [3]{d (b+2 c x)} \left (2 \sqrt [3]{c} d^{2/3}-\frac{\sqrt [3]{2} (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right ) \sqrt{\frac{\frac{2^{2/3} \sqrt [3]{c} d^{2/3} (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}+\frac{(d (b+2 c x))^{4/3}}{\left (a+b x+c x^2\right )^{2/3}}+2 \sqrt [3]{2} c^{2/3} d^{4/3}}{\left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac{\left (1+\sqrt{3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac{2^{2/3} \sqrt [3]{c} d^{2/3}-\frac{\left (1-\sqrt{3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{c x^2+b x+a}}}{2^{2/3} \sqrt [3]{c} d^{2/3}-\frac{\left (1+\sqrt{3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{c x^2+b x+a}}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{72 \sqrt [4]{3} c^{10/3} d^{5/3} \left (a+b x+c x^2\right )^{2/3} \sqrt{-\frac{(d (b+2 c x))^{2/3} \left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac{(d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )}{\sqrt [3]{a+b x+c x^2} \left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac{\left (1+\sqrt{3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )^2}}}-\frac{\left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2} \sqrt [3]{d (b+2 c x)}}{9 c^2 d}+\frac{\left (a+b x+c x^2\right )^{4/3} \sqrt [3]{d (b+2 c x)}}{6 c d} \]

[Out]

-((b^2 - 4*a*c)*(d*(b + 2*c*x))^(1/3)*(a + b*x + c*x^2)^(1/3))/(9*c^2*d) + ((d*(
b + 2*c*x))^(1/3)*(a + b*x + c*x^2)^(4/3))/(6*c*d) + ((b^2 - 4*a*c)*(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])/(72*3^(1/4)*c^(10/3)*d^(5/3)*(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 + S
qrt[3])*(d*(b + 2*c*x))^(2/3))/(a + b*x + c*x^2)^(1/3))^2))])

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Rubi [A]  time = 4.8462, antiderivative size = 597, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{\left (b^2-4 a c\right ) \left (-4 a c+b^2-(b+2 c x)^2\right ) \sqrt [3]{d (b+2 c x)} \left (2 \sqrt [3]{c} d^{2/3}-\frac{\sqrt [3]{2} (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right ) \sqrt{\frac{\frac{2^{2/3} \sqrt [3]{c} d^{2/3} (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}+\frac{(d (b+2 c x))^{4/3}}{\left (a+b x+c x^2\right )^{2/3}}+2 \sqrt [3]{2} c^{2/3} d^{4/3}}{\left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac{\left (1+\sqrt{3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac{2^{2/3} \sqrt [3]{c} d^{2/3}-\frac{\left (1-\sqrt{3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{c x^2+b x+a}}}{2^{2/3} \sqrt [3]{c} d^{2/3}-\frac{\left (1+\sqrt{3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{c x^2+b x+a}}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{72 \sqrt [4]{3} c^{10/3} d^{5/3} \left (a+b x+c x^2\right )^{2/3} \sqrt{-\frac{(d (b+2 c x))^{2/3} \left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac{(d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )}{\sqrt [3]{a+b x+c x^2} \left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac{\left (1+\sqrt{3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )^2}}}-\frac{\left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2} \sqrt [3]{d (b+2 c x)}}{9 c^2 d}+\frac{\left (a+b x+c x^2\right )^{4/3} \sqrt [3]{d (b+2 c x)}}{6 c d} \]

Warning: Unable to verify antiderivative.

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

[Out]

-((b^2 - 4*a*c)*(d*(b + 2*c*x))^(1/3)*(a + b*x + c*x^2)^(1/3))/(9*c^2*d) + ((d*(
b + 2*c*x))^(1/3)*(a + b*x + c*x^2)^(4/3))/(6*c*d) + ((b^2 - 4*a*c)*(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])/(72*3^(1/4)*c^(10/3)*d^(5/3)*(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 + S
qrt[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 = 105.395, size = 760, normalized size = 1.27 \[ \text{result too large to display} \]

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)**(2/3),x)

[Out]

(b*d + 2*c*d*x)**(1/3)*(a - b**2/(4*c) + (b + 2*c*x)**2/(4*c))**(4/3)/(6*c*d) -
(-4*a*c + b**2)*(b*d + 2*c*d*x)**(1/3)*(a - b**2/(4*c) + (b + 2*c*x)**2/(4*c))**
(1/3)/(9*c**2*d) - 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)*(-4*a*c + b**2)**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 + sqrt(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)/(216*c**(10/3)*d**(5/3)*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)))*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 = 0.560527, size = 171, normalized size = 0.29 \[ \frac{\sqrt [3]{d (b+2 c x)} \left (2 \sqrt [3]{2} \left (b^2-4 a c\right )^2 \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{2/3} \, _2F_1\left (\frac{1}{6},\frac{2}{3};\frac{7}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right )+c \left (11 a^2 c-2 a \left (b^2-7 b c x-7 c^2 x^2\right )+x \left (-2 b^3+b^2 c x+6 b c^2 x^2+3 c^3 x^3\right )\right )\right )}{18 c^3 d (a+x (b+c x))^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

((d*(b + 2*c*x))^(1/3)*(c*(11*a^2*c - 2*a*(b^2 - 7*b*c*x - 7*c^2*x^2) + x*(-2*b^
3 + b^2*c*x + 6*b*c^2*x^2 + 3*c^3*x^3)) + 2*2^(1/3)*(b^2 - 4*a*c)^2*((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)]))/(18*c^3*d*(a + x*(b + c*x))^(2/3))

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \[ \int{1 \left ( c{x}^{2}+bx+a \right ) ^{{\frac{4}{3}}} \left ( 2\,cdx+bd \right ) ^{-{\frac{2}{3}}}}\, dx \]

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)^(2/3),x)

[Out]

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

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

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)^(2/3),x, algorithm="maxima")

[Out]

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

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

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)^(2/3),x, algorithm="fricas")

[Out]

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

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

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)**(2/3),x)

[Out]

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

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

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)^(2/3),x, algorithm="giac")

[Out]

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