3.1437 \(\int \frac{b+c x}{\left (a+2 b x+c x^2\right )^{3/7}} \, dx\)

Optimal. Leaf size=19 \[ \frac{7}{8} \left (a+2 b x+c x^2\right )^{4/7} \]

[Out]

(7*(a + 2*b*x + c*x^2)^(4/7))/8

_______________________________________________________________________________________

Rubi [A]  time = 0.0139967, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{7}{8} \left (a+2 b x+c x^2\right )^{4/7} \]

Antiderivative was successfully verified.

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

[Out]

(7*(a + 2*b*x + c*x^2)^(4/7))/8

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.05616, size = 17, normalized size = 0.89 \[ \frac{7 \left (a + 2 b x + c x^{2}\right )^{\frac{4}{7}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*(a + 2*b*x + c*x**2)**(4/7)/8

_______________________________________________________________________________________

Mathematica [A]  time = 0.0290945, size = 19, normalized size = 1. \[ \frac{7}{8} (a+x (2 b+c x))^{4/7} \]

Antiderivative was successfully verified.

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

[Out]

(7*(a + x*(2*b + c*x))^(4/7))/8

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 16, normalized size = 0.8 \[{\frac{7}{8} \left ( c{x}^{2}+2\,bx+a \right ) ^{{\frac{4}{7}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/8*(c*x^2+2*b*x+a)^(4/7)

_______________________________________________________________________________________

Maxima [A]  time = 0.683786, size = 20, normalized size = 1.05 \[ \frac{7}{8} \,{\left (c x^{2} + 2 \, b x + a\right )}^{\frac{4}{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/8*(c*x^2 + 2*b*x + a)^(4/7)

_______________________________________________________________________________________

Fricas [A]  time = 0.206775, size = 20, normalized size = 1.05 \[ \frac{7}{8} \,{\left (c x^{2} + 2 \, b x + a\right )}^{\frac{4}{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/8*(c*x^2 + 2*b*x + a)^(4/7)

_______________________________________________________________________________________

Sympy [A]  time = 0.976939, size = 17, normalized size = 0.89 \[ \frac{7 \left (a + 2 b x + c x^{2}\right )^{\frac{4}{7}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*(a + 2*b*x + c*x**2)**(4/7)/8

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.208863, size = 20, normalized size = 1.05 \[ \frac{7}{8} \,{\left (c x^{2} + 2 \, b x + a\right )}^{\frac{4}{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/8*(c*x^2 + 2*b*x + a)^(4/7)