3.1448 \(\int \frac{b+c x}{(a+2 b x+c x^2)^{3/7}} \, dx\)

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

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Rubi [A]  time = 0.0064562, 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, Rules used = {629} \[ \frac{7}{8} \left (a+2 b x+c x^2\right )^{4/7} \]

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

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{b+c x}{\left (a+2 b x+c x^2\right )^{3/7}} \, dx &=\frac{7}{8} \left (a+2 b x+c x^2\right )^{4/7}\\ \end{align*}

Mathematica [A]  time = 0.011354, 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

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Maple [A]  time = 0.044, size = 16, normalized size = 0.8 \begin{align*}{\frac{7}{8} \left ( c{x}^{2}+2\,bx+a \right ) ^{{\frac{4}{7}}}} \end{align*}

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)

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Maxima [A]  time = 1.28466, size = 20, normalized size = 1.05 \begin{align*} \frac{7}{8} \,{\left (c x^{2} + 2 \, b x + a\right )}^{\frac{4}{7}} \end{align*}

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)

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Fricas [A]  time = 2.00528, size = 42, normalized size = 2.21 \begin{align*} \frac{7}{8} \,{\left (c x^{2} + 2 \, b x + a\right )}^{\frac{4}{7}} \end{align*}

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)

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Sympy [A]  time = 0.371145, size = 17, normalized size = 0.89 \begin{align*} \frac{7 \left (a + 2 b x + c x^{2}\right )^{\frac{4}{7}}}{8} \end{align*}

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

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Giac [A]  time = 1.13929, size = 20, normalized size = 1.05 \begin{align*} \frac{7}{8} \,{\left (c x^{2} + 2 \, b x + a\right )}^{\frac{4}{7}} \end{align*}

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)