3.1402 \(\int \frac{\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{17/3}} \, dx\)

Optimal. Leaf size=44 \[ \frac{3 \left (a+b x+c x^2\right )^{7/3}}{7 d \left (b^2-4 a c\right ) (b d+2 c d x)^{14/3}} \]

[Out]

(3*(a + b*x + c*x^2)^(7/3))/(7*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(14/3))

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Rubi [A]  time = 0.0609568, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{3 \left (a+b x+c x^2\right )^{7/3}}{7 d \left (b^2-4 a c\right ) (b d+2 c d x)^{14/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(17/3),x]

[Out]

(3*(a + b*x + c*x^2)^(7/3))/(7*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(14/3))

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Rubi in Sympy [A]  time = 13.7267, size = 39, normalized size = 0.89 \[ \frac{3 \left (a + b x + c x^{2}\right )^{\frac{7}{3}}}{7 d \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{14}{3}}} \]

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)**(17/3),x)

[Out]

3*(a + b*x + c*x**2)**(7/3)/(7*d*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(14/3))

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Mathematica [A]  time = 0.161812, size = 50, normalized size = 1.14 \[ \frac{3 (a+x (b+c x))^{7/3} \sqrt [3]{d (b+2 c x)}}{7 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(17/3),x]

[Out]

(3*(d*(b + 2*c*x))^(1/3)*(a + x*(b + c*x))^(7/3))/(7*(b^2 - 4*a*c)*d^6*(b + 2*c*
x)^5)

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Maple [A]  time = 0.007, size = 44, normalized size = 1. \[ -{\frac{6\,cx+3\,b}{28\,ac-7\,{b}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{7}{3}}} \left ( 2\,cdx+bd \right ) ^{-{\frac{17}{3}}}} \]

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)^(17/3),x)

[Out]

-3/7*(c*x^2+b*x+a)^(7/3)*(2*c*x+b)/(4*a*c-b^2)/(2*c*d*x+b*d)^(17/3)

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Maxima [A]  time = 1.12145, size = 247, normalized size = 5.61 \[ \frac{3 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}{\left (c x^{2} + b x + a\right )}^{\frac{1}{3}}}{7 \,{\left (b^{6} d^{5} - 4 \, a b^{4} c d^{5} + 16 \,{\left (b^{2} c^{4} d^{5} - 4 \, a c^{5} d^{5}\right )} x^{4} + 32 \,{\left (b^{3} c^{3} d^{5} - 4 \, a b c^{4} d^{5}\right )} x^{3} + 24 \,{\left (b^{4} c^{2} d^{5} - 4 \, a b^{2} c^{3} d^{5}\right )} x^{2} + 8 \,{\left (b^{5} c d^{5} - 4 \, a b^{3} c^{2} d^{5}\right )} x\right )}{\left (2 \, c x + b\right )}^{\frac{2}{3}} d^{\frac{2}{3}}} \]

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)^(17/3),x, algorithm="maxima")

[Out]

3/7*(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)/((b^6*d^5 - 4*a*b^4*c*d^5 + 16*(b^2*c^4*d^5 - 4*a*c^5*d^5)*x^4 + 32*(b^3*c
^3*d^5 - 4*a*b*c^4*d^5)*x^3 + 24*(b^4*c^2*d^5 - 4*a*b^2*c^3*d^5)*x^2 + 8*(b^5*c*
d^5 - 4*a*b^3*c^2*d^5)*x)*(2*c*x + b)^(2/3)*d^(2/3))

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Fricas [A]  time = 0.266333, size = 262, normalized size = 5.95 \[ \frac{3 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}{\left (2 \, c d x + b d\right )}^{\frac{1}{3}}{\left (c x^{2} + b x + a\right )}^{\frac{1}{3}}}{7 \,{\left (32 \,{\left (b^{2} c^{5} - 4 \, a c^{6}\right )} d^{6} x^{5} + 80 \,{\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} d^{6} x^{4} + 80 \,{\left (b^{4} c^{3} - 4 \, a b^{2} c^{4}\right )} d^{6} x^{3} + 40 \,{\left (b^{5} c^{2} - 4 \, a b^{3} c^{3}\right )} d^{6} x^{2} + 10 \,{\left (b^{6} c - 4 \, a b^{4} c^{2}\right )} d^{6} x +{\left (b^{7} - 4 \, a b^{5} c\right )} d^{6}\right )}} \]

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)^(17/3),x, algorithm="fricas")

[Out]

3/7*(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*(2*c*d*x + b*d)^(1
/3)*(c*x^2 + b*x + a)^(1/3)/(32*(b^2*c^5 - 4*a*c^6)*d^6*x^5 + 80*(b^3*c^4 - 4*a*
b*c^5)*d^6*x^4 + 80*(b^4*c^3 - 4*a*b^2*c^4)*d^6*x^3 + 40*(b^5*c^2 - 4*a*b^3*c^3)
*d^6*x^2 + 10*(b^6*c - 4*a*b^4*c^2)*d^6*x + (b^7 - 4*a*b^5*c)*d^6)

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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)**(17/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{4}{3}}}{{\left (2 \, c d x + b d\right )}^{\frac{17}{3}}}\,{d x} \]

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)^(17/3),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(4/3)/(2*c*d*x + b*d)^(17/3), x)