3.21.55 \(\int (d+e x)^m (c d m-b e (1+m+p)-c e (2+m+2 p) x) (c d^2-b d e-b e^2 x-c e^2 x^2)^p \, dx\)

Optimal. Leaf size=42 \[ \frac {(d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{p+1}}{e} \]

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Rubi [A]  time = 0.05, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {786} \begin {gather*} \frac {(d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{p+1}}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m*(c*d*m - b*e*(1 + m + p) - c*e*(2 + m + 2*p)*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^p,x]

[Out]

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

Rule 786

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

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 34, normalized size = 0.81 \begin {gather*} \frac {(d+e x)^m ((d+e x) (c (d-e x)-b e))^{p+1}}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^m*(c*d*m - b*e*(1 + m + p) - c*e*(2 + m + 2*p)*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^p,
x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.50, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^m (c d m-b e (1+m+p)-c e (2+m+2 p) x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^m*(c*d*m - b*e*(1 + m + p) - c*e*(2 + m + 2*p)*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^
2*x^2)^p,x]

[Out]

Defer[IntegrateAlgebraic][(d + e*x)^m*(c*d*m - b*e*(1 + m + p) - c*e*(2 + m + 2*p)*x)*(c*d^2 - b*d*e - b*e^2*x
- c*e^2*x^2)^p, x]

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fricas [A]  time = 0.44, size = 66, normalized size = 1.57 \begin {gather*} -\frac {{\left (c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e\right )} {\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{p} {\left (e x + d\right )}^{m}}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*d*m-b*e*(1+m+p)-c*e*(2+m+2*p)*x)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^p,x, algorithm="frica
s")

[Out]

-(c*e^2*x^2 + b*e^2*x - c*d^2 + b*d*e)*(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^p*(e*x + d)^m/e

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giac [B]  time = 0.46, size = 173, normalized size = 4.12 \begin {gather*} -{\left ({\left (x e + d\right )}^{m} c x^{2} e^{\left (p \log \left (-c x e + c d - b e\right ) + p \log \left (x e + d\right ) + 2\right )} - {\left (x e + d\right )}^{m} c d^{2} e^{\left (p \log \left (-c x e + c d - b e\right ) + p \log \left (x e + d\right )\right )} + {\left (x e + d\right )}^{m} b x e^{\left (p \log \left (-c x e + c d - b e\right ) + p \log \left (x e + d\right ) + 2\right )} + {\left (x e + d\right )}^{m} b d e^{\left (p \log \left (-c x e + c d - b e\right ) + p \log \left (x e + d\right ) + 1\right )}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*d*m-b*e*(1+m+p)-c*e*(2+m+2*p)*x)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^p,x, algorithm="giac"
)

[Out]

-((x*e + d)^m*c*x^2*e^(p*log(-c*x*e + c*d - b*e) + p*log(x*e + d) + 2) - (x*e + d)^m*c*d^2*e^(p*log(-c*x*e + c
*d - b*e) + p*log(x*e + d)) + (x*e + d)^m*b*x*e^(p*log(-c*x*e + c*d - b*e) + p*log(x*e + d) + 2) + (x*e + d)^m
*b*d*e^(p*log(-c*x*e + c*d - b*e) + p*log(x*e + d) + 1))*e^(-1)

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maple [A]  time = 0.06, size = 56, normalized size = 1.33 \begin {gather*} -\frac {\left (c e x +b e -c d \right ) \left (e x +d \right )^{m +1} \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{p}}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*(c*d*m-b*e*(m+p+1)-c*e*(m+2*p+2)*x)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^p,x)

[Out]

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

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maxima [A]  time = 0.80, size = 64, normalized size = 1.52 \begin {gather*} -\frac {{\left (c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e\right )} e^{\left (p \log \left (-c e x + c d - b e\right ) + m \log \left (e x + d\right ) + p \log \left (e x + d\right )\right )}}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*d*m-b*e*(1+m+p)-c*e*(2+m+2*p)*x)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^p,x, algorithm="maxim
a")

[Out]

-(c*e^2*x^2 + b*e^2*x - c*d^2 + b*d*e)*e^(p*log(-c*e*x + c*d - b*e) + m*log(e*x + d) + p*log(e*x + d))/e

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mupad [B]  time = 2.58, size = 79, normalized size = 1.88 \begin {gather*} -\left (b\,e\,x\,{\left (d+e\,x\right )}^m-\frac {\left (c\,d^2-b\,d\,e\right )\,{\left (d+e\,x\right )}^m}{e}+c\,e\,x^2\,{\left (d+e\,x\right )}^m\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^p \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(d + e*x)^m*(b*e*(m + p + 1) - c*d*m + c*e*x*(m + 2*p + 2))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^p,x)

[Out]

-(b*e*x*(d + e*x)^m - ((c*d^2 - b*d*e)*(d + e*x)^m)/e + c*e*x^2*(d + e*x)^m)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^
2*x)^p

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sympy [A]  time = 16.86, size = 173, normalized size = 4.12 \begin {gather*} \begin {cases} - b d \left (d + e x\right )^{m} \left (- b d e - b e^{2} x + c d^{2} - c e^{2} x^{2}\right )^{p} - b e x \left (d + e x\right )^{m} \left (- b d e - b e^{2} x + c d^{2} - c e^{2} x^{2}\right )^{p} + \frac {c d^{2} \left (d + e x\right )^{m} \left (- b d e - b e^{2} x + c d^{2} - c e^{2} x^{2}\right )^{p}}{e} - c e x^{2} \left (d + e x\right )^{m} \left (- b d e - b e^{2} x + c d^{2} - c e^{2} x^{2}\right )^{p} & \text {for}\: e \neq 0 \\c d d^{m} m x \left (c d^{2}\right )^{p} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**m*(c*d*m-b*e*(1+m+p)-c*e*(2+m+2*p)*x)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**p,x)

[Out]

Piecewise((-b*d*(d + e*x)**m*(-b*d*e - b*e**2*x + c*d**2 - c*e**2*x**2)**p - b*e*x*(d + e*x)**m*(-b*d*e - b*e*
*2*x + c*d**2 - c*e**2*x**2)**p + c*d**2*(d + e*x)**m*(-b*d*e - b*e**2*x + c*d**2 - c*e**2*x**2)**p/e - c*e*x*
*2*(d + e*x)**m*(-b*d*e - b*e**2*x + c*d**2 - c*e**2*x**2)**p, Ne(e, 0)), (c*d*d**m*m*x*(c*d**2)**p, True))

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