3.139 \(\int (d+e x)^m (-c d^2+b d e+b e^2 x+c e^2 x^2)^p ((-c d+b e) f+(c e f-c d g+b e g) x+c e g x^2) \, dx\)

Optimal. Leaf size=222 \[ \frac {g (d+e x)^{m-1} \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^{p+2}}{c e^2 (m+2 p+3)}-\frac {(d+e x)^m (-b e+c d-c e x)^2 \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^p \left (\frac {c (d+e x)}{2 c d-b e}\right )^{-m-p} (b e g (m+p+1)+c (d g (1-m)-e f (m+2 p+3))) \, _2F_1\left (-m-p,p+2;p+3;\frac {c d-b e-c e x}{2 c d-b e}\right )}{c^2 e^2 (p+2) (m+2 p+3)} \]

[Out]

g*(e*x+d)^(-1+m)*(-d*(-b*e+c*d)+b*e^2*x+c*e^2*x^2)^(2+p)/c/e^2/(3+m+2*p)-(b*e*g*(1+m+p)+c*(d*g*(1-m)-e*f*(3+m+
2*p)))*(e*x+d)^m*(c*(e*x+d)/(-b*e+2*c*d))^(-m-p)*(-c*e*x-b*e+c*d)^2*(-d*(-b*e+c*d)+b*e^2*x+c*e^2*x^2)^p*hyperg
eom([-m-p, 2+p],[3+p],(-c*e*x-b*e+c*d)/(-b*e+2*c*d))/c^2/e^2/(2+p)/(3+m+2*p)

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Rubi [A]  time = 0.41, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 70, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {1632, 794, 679, 677, 70, 69} \[ \frac {g (d+e x)^{m-1} \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^{p+2}}{c e^2 (m+2 p+3)}-\frac {(d+e x)^m (-b e+c d-c e x)^2 \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^p \left (\frac {c (d+e x)}{2 c d-b e}\right )^{-m-p} (b e g (m+p+1)+c (d g (1-m)-e f (m+2 p+3))) \, _2F_1\left (-m-p,p+2;p+3;\frac {c d-b e-c e x}{2 c d-b e}\right )}{c^2 e^2 (p+2) (m+2 p+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g*(d + e*x)^(-1 + m)*(-(d*(c*d - b*e)) + b*e^2*x + c*e^2*x^2)^(2 + p))/(c*e^2*(3 + m + 2*p)) - ((b*e*g*(1 + m
+ p) + c*(d*g*(1 - m) - e*f*(3 + m + 2*p)))*(d + e*x)^m*((c*(d + e*x))/(2*c*d - b*e))^(-m - p)*(c*d - b*e - c
*e*x)^2*(-(d*(c*d - b*e)) + b*e^2*x + c*e^2*x^2)^p*Hypergeometric2F1[-m - p, 2 + p, 3 + p, (c*d - b*e - c*e*x)
/(2*c*d - b*e)])/(c^2*e^2*(2 + p)*(3 + m + 2*p))

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*Simp[(b*c)/(b*c - a*d) + (b*d*x)/(b*c -
a*d), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 677

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(d^m*(a + b*x + c*x^2
)^FracPart[p])/((1 + (e*x)/d)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p]), Int[(1 + (e*x)/d)^(m + p)*(a/d + (c*x)
/e)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !Int
egerQ[p] && (IntegerQ[m] || GtQ[d, 0]) &&  !(IGtQ[m, 0] && (IntegerQ[3*p] || IntegerQ[4*p]))

Rule 679

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

Rule 794

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] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g
, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 0])

Rule 1632

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

Rubi steps

\begin {align*} \int (d+e x)^m \left (-c d^2+b d e+b e^2 x+c e^2 x^2\right )^p \left (-(c d-b e) f+(c e f-c d g+b e g) x+c e g x^2\right ) \, dx &=(d e) \int (d+e x)^{-1+m} \left (\frac {f}{d e}+\frac {g x}{d e}\right ) \left (-c d^2+b d e+b e^2 x+c e^2 x^2\right )^{1+p} \, dx\\ &=\frac {g (d+e x)^{-1+m} \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^{2+p}}{c e^2 (3+m+2 p)}+\frac {\left (d \left (\left (\frac {c e^2 f}{d}+\frac {\left (c d e^2-b e^3\right ) g}{d e}\right ) (-1+m)+e \left (\frac {2 c e f}{d}-\frac {b e g}{d}\right ) (2+p)\right )\right ) \int (d+e x)^{-1+m} \left (-c d^2+b d e+b e^2 x+c e^2 x^2\right )^{1+p} \, dx}{c e^2 (1+m+2 (1+p))}\\ &=\frac {g (d+e x)^{-1+m} \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^{2+p}}{c e^2 (3+m+2 p)}+\frac {\left (\left (\left (\frac {c e^2 f}{d}+\frac {\left (c d e^2-b e^3\right ) g}{d e}\right ) (-1+m)+e \left (\frac {2 c e f}{d}-\frac {b e g}{d}\right ) (2+p)\right ) (d+e x)^m \left (1+\frac {e x}{d}\right )^{-m}\right ) \int \left (1+\frac {e x}{d}\right )^{-1+m} \left (-c d^2+b d e+b e^2 x+c e^2 x^2\right )^{1+p} \, dx}{c e^2 (1+m+2 (1+p))}\\ &=\frac {g (d+e x)^{-1+m} \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^{2+p}}{c e^2 (3+m+2 p)}+\frac {\left (\left (\left (\frac {c e^2 f}{d}+\frac {\left (c d e^2-b e^3\right ) g}{d e}\right ) (-1+m)+e \left (\frac {2 c e f}{d}-\frac {b e g}{d}\right ) (2+p)\right ) (d+e x)^m \left (1+\frac {e x}{d}\right )^{-m-p} \left (-c d^2+b d e+c d e x\right )^{-p} \left (-c d^2+b d e+b e^2 x+c e^2 x^2\right )^p\right ) \int \left (1+\frac {e x}{d}\right )^{m+p} \left (-c d^2+b d e+c d e x\right )^{1+p} \, dx}{c e^2 (1+m+2 (1+p))}\\ &=\frac {g (d+e x)^{-1+m} \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^{2+p}}{c e^2 (3+m+2 p)}+\frac {\left (\left (\left (\frac {c e^2 f}{d}+\frac {\left (c d e^2-b e^3\right ) g}{d e}\right ) (-1+m)+e \left (\frac {2 c e f}{d}-\frac {b e g}{d}\right ) (2+p)\right ) (d+e x)^m \left (\frac {c d e \left (1+\frac {e x}{d}\right )}{c d e-\frac {e \left (-c d^2+b d e\right )}{d}}\right )^{-m-p} \left (-c d^2+b d e+c d e x\right )^{-p} \left (-c d^2+b d e+b e^2 x+c e^2 x^2\right )^p\right ) \int \left (-c d^2+b d e+c d e x\right )^{1+p} \left (\frac {c d}{2 c d-b e}+\frac {c e x}{2 c d-b e}\right )^{m+p} \, dx}{c e^2 (1+m+2 (1+p))}\\ &=\frac {g (d+e x)^{-1+m} \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^{2+p}}{c e^2 (3+m+2 p)}-\frac {(c d g (1-m)+b e g (1+m+p)-c e f (3+m+2 p)) (d+e x)^m \left (\frac {c (d+e x)}{2 c d-b e}\right )^{-m-p} (c d-b e-c e x)^2 \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^p \, _2F_1\left (-m-p,2+p;3+p;\frac {c d-b e-c e x}{2 c d-b e}\right )}{c^2 e^2 (2+p) (3+m+2 p)}\\ \end {align*}

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Mathematica [A]  time = 0.32, size = 165, normalized size = 0.74 \[ \frac {(d+e x)^m (b e-c d+c e x)^2 (-((d+e x) (c (d-e x)-b e)))^p \left (\frac {e \left (\frac {c (d+e x)}{2 c d-b e}\right )^{-m-p} (-b e g (m+p+1)+c d g (m-1)+c e f (m+2 p+3)) \, _2F_1\left (-m-p,p+2;p+3;\frac {-c d+b e+c e x}{b e-2 c d}\right )}{p+2}+c e g (d+e x)\right )}{c^2 e^3 (m+2 p+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^m*(-(c*d) + b*e + c*e*x)^2*(-((d + e*x)*(-(b*e) + c*(d - e*x))))^p*(c*e*g*(d + e*x) + (e*(c*d*g*(-1
+ m) - b*e*g*(1 + m + p) + c*e*f*(3 + m + 2*p))*((c*(d + e*x))/(2*c*d - b*e))^(-m - p)*Hypergeometric2F1[-m -
p, 2 + p, 3 + p, (-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)])/(2 + p)))/(c^2*e^3*(3 + m + 2*p))

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fricas [F]  time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c e g x^{2} - {\left (c d - b e\right )} f + {\left (c e f - {\left (c d - b e\right )} g\right )} x\right )} {\left (c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e\right )}^{p} {\left (e x + d\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*e^2*x^2+b*e^2*x+b*d*e-c*d^2)^p*(-(-b*e+c*d)*f+(b*e*g-c*d*g+c*e*f)*x+c*e*g*x^2),x, algor
ithm="fricas")

[Out]

integral((c*e*g*x^2 - (c*d - b*e)*f + (c*e*f - (c*d - b*e)*g)*x)*(c*e^2*x^2 + b*e^2*x - c*d^2 + b*d*e)^p*(e*x
+ d)^m, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c e g x^{2} - {\left (c d - b e\right )} f + {\left (c e f - c d g + b e g\right )} x\right )} {\left (c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e\right )}^{p} {\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*e^2*x^2+b*e^2*x+b*d*e-c*d^2)^p*(-(-b*e+c*d)*f+(b*e*g-c*d*g+c*e*f)*x+c*e*g*x^2),x, algor
ithm="giac")

[Out]

integrate((c*e*g*x^2 - (c*d - b*e)*f + (c*e*f - c*d*g + b*e*g)*x)*(c*e^2*x^2 + b*e^2*x - c*d^2 + b*d*e)^p*(e*x
+ d)^m, x)

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maple [F]  time = 0.18, size = 0, normalized size = 0.00 \[ \int \left (c e g \,x^{2}-\left (-b e +c d \right ) f +\left (b e g -c d g +c e f \right ) x \right ) \left (e x +d \right )^{m} \left (c \,e^{2} x^{2}+b \,e^{2} x +b d e -c \,d^{2}\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c e g x^{2} - {\left (c d - b e\right )} f + {\left (c e f - c d g + b e g\right )} x\right )} {\left (c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e\right )}^{p} {\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*e^2*x^2+b*e^2*x+b*d*e-c*d^2)^p*(-(-b*e+c*d)*f+(b*e*g-c*d*g+c*e*f)*x+c*e*g*x^2),x, algor
ithm="maxima")

[Out]

integrate((c*e*g*x^2 - (c*d - b*e)*f + (c*e*f - c*d*g + b*e*g)*x)*(c*e^2*x^2 + b*e^2*x - c*d^2 + b*d*e)^p*(e*x
+ d)^m, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (d+e\,x\right )}^m\,\left (c\,e\,g\,x^2+\left (b\,e\,g-c\,d\,g+c\,e\,f\right )\,x+f\,\left (b\,e-c\,d\right )\right )\,{\left (-c\,d^2+b\,d\,e+c\,e^2\,x^2+b\,e^2\,x\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: HeuristicGCDFailed

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