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

Optimal. Leaf size=52 \[ \frac {(d+e x)^{-p-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{c d (p+1)} \]

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Rubi [A]  time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {648} \begin {gather*} \frac {(d+e x)^{-p-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{c d (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 648

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 41, normalized size = 0.79 \begin {gather*} \frac {(d+e x)^{-p-1} ((d+e x) (a e+c d x))^{p+1}}{c d (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.11, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^{-p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 55, normalized size = 1.06 \begin {gather*} \frac {{\left (c d x + a e\right )} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{{\left (c d p + c d\right )} {\left (e x + d\right )}^{p}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.23, size = 85, normalized size = 1.63 \begin {gather*} \frac {\frac {c d x e^{\left (p \log \left (c d x + a e\right ) + p \log \left (x e + d\right )\right )}}{{\left (x e + d\right )}^{p}} + \frac {a e^{\left (p \log \left (c d x + a e\right ) + p \log \left (x e + d\right ) + 1\right )}}{{\left (x e + d\right )}^{p}}}{c d p + c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 56, normalized size = 1.08 \begin {gather*} \frac {\left (c d x +a e \right ) \left (e x +d \right )^{-p} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p}}{\left (p +1\right ) c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.07, size = 30, normalized size = 0.58 \begin {gather*} \frac {{\left (c d x + a e\right )} {\left (c d x + a e\right )}^{p}}{c d {\left (p + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.75, size = 56, normalized size = 1.08 \begin {gather*} \frac {\left (a\,e+c\,d\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^p}{c\,d\,\left (p+1\right )\,{\left (d+e\,x\right )}^p} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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