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

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

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {7}{8} \left (a+2 b x+c x^2\right )^{4/7} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} \frac {7}{8} (a+x (2 b+c x))^{4/7} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.02, size = 19, normalized size = 1.00 \begin {gather*} \frac {7}{8} \left (a+2 b x+c x^2\right )^{4/7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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

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fricas [A]  time = 0.42, size = 15, normalized size = 0.79 \begin {gather*} \frac {7}{8} \, {\left (c x^{2} + 2 \, b x + a\right )}^{\frac {4}{7}} \end {gather*}

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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giac [A]  time = 0.16, size = 15, normalized size = 0.79 \begin {gather*} \frac {7}{8} \, {\left (c x^{2} + 2 \, b x + a\right )}^{\frac {4}{7}} \end {gather*}

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)

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maple [A]  time = 0.05, size = 16, normalized size = 0.84 \begin {gather*} \frac {7 \left (c \,x^{2}+2 b x +a \right )^{\frac {4}{7}}}{8} \end {gather*}

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 = 1.35, size = 15, normalized size = 0.79 \begin {gather*} \frac {7}{8} \, {\left (c x^{2} + 2 \, b x + a\right )}^{\frac {4}{7}} \end {gather*}

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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mupad [B]  time = 0.60, size = 15, normalized size = 0.79 \begin {gather*} \frac {7\,{\left (c\,x^2+2\,b\,x+a\right )}^{4/7}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.33, size = 17, normalized size = 0.89 \begin {gather*} \frac {7 \left (a + 2 b x + c x^{2}\right )^{\frac {4}{7}}}{8} \end {gather*}

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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