3.14.1 \(\int (-a e+c d x) (d+e x)^{-3-2 p} (a+c x^2)^p \, dx\)

Optimal. Leaf size=31 \[ \frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{2 (p+1)} \]

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Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {803} \begin {gather*} \frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{2 (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-(a*e) + c*d*x)*(d + e*x)^(-3 - 2*p)*(a + c*x^2)^p,x]

[Out]

(a + c*x^2)^(1 + p)/(2*(1 + p)*(d + e*x)^(2*(1 + p)))

Rule 803

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] /; FreeQ[{a, c, d, e, f, g, m, p}, x]
&& NeQ[c*d^2 + a*e^2, 0] && EqQ[Simplify[m + 2*p + 3], 0] && EqQ[c*d*f + a*e*g, 0]

Rubi steps

\begin {align*} \int (-a e+c d x) (d+e x)^{-3-2 p} \left (a+c x^2\right )^p \, dx &=\frac {(d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{2 (1+p)}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 31, normalized size = 1.00 \begin {gather*} \frac {\left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-2}}{2 (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-(a*e) + c*d*x)*(d + e*x)^(-3 - 2*p)*(a + c*x^2)^p,x]

[Out]

((d + e*x)^(-2 - 2*p)*(a + c*x^2)^(1 + p))/(2*(1 + p))

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IntegrateAlgebraic [F]  time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int (-a e+c d x) (d+e x)^{-3-2 p} \left (a+c x^2\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(-(a*e) + c*d*x)*(d + e*x)^(-3 - 2*p)*(a + c*x^2)^p,x]

[Out]

Defer[IntegrateAlgebraic][(-(a*e) + c*d*x)*(d + e*x)^(-3 - 2*p)*(a + c*x^2)^p, x]

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fricas [A]  time = 0.45, size = 47, normalized size = 1.52 \begin {gather*} \frac {{\left (c e x^{3} + c d x^{2} + a e x + a d\right )} {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}}{2 \, {\left (p + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x-a*e)*(e*x+d)^(-3-2*p)*(c*x^2+a)^p,x, algorithm="fricas")

[Out]

1/2*(c*e*x^3 + c*d*x^2 + a*e*x + a*d)*(c*x^2 + a)^p*(e*x + d)^(-2*p - 3)/(p + 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (c d x - a e\right )} {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x-a*e)*(e*x+d)^(-3-2*p)*(c*x^2+a)^p,x, algorithm="giac")

[Out]

integrate((c*d*x - a*e)*(c*x^2 + a)^p*(e*x + d)^(-2*p - 3), x)

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maple [A]  time = 0.05, size = 30, normalized size = 0.97 \begin {gather*} \frac {\left (c \,x^{2}+a \right )^{p +1} \left (e x +d \right )^{-2 p -2}}{2 p +2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*x-a*e)*(e*x+d)^(-2*p-3)*(c*x^2+a)^p,x)

[Out]

1/2*(c*x^2+a)^(p+1)*(e*x+d)^(-2*p-2)/(p+1)

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maxima [A]  time = 1.28, size = 58, normalized size = 1.87 \begin {gather*} \frac {{\left (c x^{2} + a\right )} e^{\left (p \log \left (c x^{2} + a\right ) - 2 \, p \log \left (e x + d\right )\right )}}{2 \, {\left (e^{2} {\left (p + 1\right )} x^{2} + 2 \, d e {\left (p + 1\right )} x + d^{2} {\left (p + 1\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x-a*e)*(e*x+d)^(-3-2*p)*(c*x^2+a)^p,x, algorithm="maxima")

[Out]

1/2*(c*x^2 + a)*e^(p*log(c*x^2 + a) - 2*p*log(e*x + d))/(e^2*(p + 1)*x^2 + 2*d*e*(p + 1)*x + d^2*(p + 1))

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mupad [B]  time = 1.90, size = 98, normalized size = 3.16 \begin {gather*} \frac {\frac {a\,d\,{\left (c\,x^2+a\right )}^p}{2\,p+2}+\frac {a\,e\,x\,{\left (c\,x^2+a\right )}^p}{2\,p+2}+\frac {c\,d\,x^2\,{\left (c\,x^2+a\right )}^p}{2\,p+2}+\frac {c\,e\,x^3\,{\left (c\,x^2+a\right )}^p}{2\,p+2}}{{\left (d+e\,x\right )}^{2\,p+3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((a*e - c*d*x)*(a + c*x^2)^p)/(d + e*x)^(2*p + 3),x)

[Out]

((a*d*(a + c*x^2)^p)/(2*p + 2) + (a*e*x*(a + c*x^2)^p)/(2*p + 2) + (c*d*x^2*(a + c*x^2)^p)/(2*p + 2) + (c*e*x^
3*(a + c*x^2)^p)/(2*p + 2))/(d + e*x)^(2*p + 3)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x-a*e)*(e*x+d)**(-3-2*p)*(c*x**2+a)**p,x)

[Out]

Timed out

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