3.9.13 \(\int (d+e x)^m (f+g x)^n (a+2 c d x+c e x^2) \, dx\) [813]

Optimal. Leaf size=231 \[ -\frac {c (e f-d g) (2+m) (d+e x)^{1+m} (f+g x)^{1+n}}{e g^2 (2+m+n) (3+m+n)}+\frac {c (d+e x)^{2+m} (f+g x)^{1+n}}{e g (3+m+n)}+\frac {(c (e f-d g) (2+m) (e f (1+m)+d g (1+n))+g (2+m+n) (a e g (3+m+n)-c d (e f (2+m)+d g (1+n)))) (d+e x)^{1+m} (f+g x)^n \left (\frac {e (f+g x)}{e f-d g}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac {g (d+e x)}{e f-d g}\right )}{e^2 g^2 (1+m) (2+m+n) (3+m+n)} \]

[Out]

-c*(-d*g+e*f)*(2+m)*(e*x+d)^(1+m)*(g*x+f)^(1+n)/e/g^2/(2+m+n)/(3+m+n)+c*(e*x+d)^(2+m)*(g*x+f)^(1+n)/e/g/(3+m+n
)+(c*(-d*g+e*f)*(2+m)*(e*f*(1+m)+d*g*(1+n))+g*(2+m+n)*(a*e*g*(3+m+n)-c*d*(e*f*(2+m)+d*g*(1+n))))*(e*x+d)^(1+m)
*(g*x+f)^n*hypergeom([-n, 1+m],[2+m],-g*(e*x+d)/(-d*g+e*f))/e^2/g^2/(1+m)/(2+m+n)/(3+m+n)/((e*(g*x+f)/(-d*g+e*
f))^n)

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Rubi [A]
time = 0.17, antiderivative size = 227, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {965, 81, 72, 71} \begin {gather*} \frac {(d+e x)^{m+1} (f+g x)^n \left (\frac {e (f+g x)}{e f-d g}\right )^{-n} \left (a e g (m+n+3)+\frac {c (m+2) (e f-d g) (d g (n+1)+e f (m+1))}{g (m+n+2)}-c d (d g (n+1)+e f (m+2))\right ) \, _2F_1\left (m+1,-n;m+2;-\frac {g (d+e x)}{e f-d g}\right )}{e^2 g (m+1) (m+n+3)}-\frac {c (m+2) (e f-d g) (d+e x)^{m+1} (f+g x)^{n+1}}{e g^2 (m+n+2) (m+n+3)}+\frac {c (d+e x)^{m+2} (f+g x)^{n+1}}{e g (m+n+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m*(f + g*x)^n*(a + 2*c*d*x + c*e*x^2),x]

[Out]

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

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], 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 72

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 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 965

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

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
time = 0.32, size = 190, normalized size = 0.82 \begin {gather*} \frac {1}{3} (d+e x)^m (f+g x)^n \left (3 c d x^2 \left (1+\frac {e x}{d}\right )^{-m} \left (1+\frac {g x}{f}\right )^{-n} F_1\left (2;-m,-n;3;-\frac {e x}{d},-\frac {g x}{f}\right )+c e x^3 \left (1+\frac {e x}{d}\right )^{-m} \left (1+\frac {g x}{f}\right )^{-n} F_1\left (3;-m,-n;4;-\frac {e x}{d},-\frac {g x}{f}\right )+\frac {3 a \left (\frac {g (d+e x)}{-e f+d g}\right )^{-m} (f+g x) \, _2F_1\left (-m,1+n;2+n;\frac {e (f+g x)}{e f-d g}\right )}{g (1+n)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^m*(f + g*x)^n*(a + 2*c*d*x + c*e*x^2),x]

[Out]

((d + e*x)^m*(f + g*x)^n*((3*c*d*x^2*AppellF1[2, -m, -n, 3, -((e*x)/d), -((g*x)/f)])/((1 + (e*x)/d)^m*(1 + (g*
x)/f)^n) + (c*e*x^3*AppellF1[3, -m, -n, 4, -((e*x)/d), -((g*x)/f)])/((1 + (e*x)/d)^m*(1 + (g*x)/f)^n) + (3*a*(
f + g*x)*Hypergeometric2F1[-m, 1 + n, 2 + n, (e*(f + g*x))/(e*f - d*g)])/(g*(1 + n)*((g*(d + e*x))/(-(e*f) + d
*g))^m)))/3

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{m} \left (g x +f \right )^{n} \left (c e \,x^{2}+2 c d x +a \right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*(g*x+f)^n*(c*e*x^2+2*c*d*x+a),x)

[Out]

int((e*x+d)^m*(g*x+f)^n*(c*e*x^2+2*c*d*x+a),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(g*x+f)^n*(c*e*x^2+2*c*d*x+a),x, algorithm="maxima")

[Out]

integrate((c*x^2*e + 2*c*d*x + a)*(g*x + f)^n*(x*e + d)^m, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(g*x+f)^n*(c*e*x^2+2*c*d*x+a),x, algorithm="fricas")

[Out]

integral((c*x^2*e + 2*c*d*x + a)*(g*x + f)^n*(x*e + d)^m, x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**m*(g*x+f)**n*(c*e*x**2+2*c*d*x+a),x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(g*x+f)^n*(c*e*x^2+2*c*d*x+a),x, algorithm="giac")

[Out]

integrate((c*x^2*e + 2*c*d*x + a)*(g*x + f)^n*(x*e + d)^m, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (f+g\,x\right )}^n\,{\left (d+e\,x\right )}^m\,\left (c\,e\,x^2+2\,c\,d\,x+a\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x)^n*(d + e*x)^m*(a + 2*c*d*x + c*e*x^2),x)

[Out]

int((f + g*x)^n*(d + e*x)^m*(a + 2*c*d*x + c*e*x^2), x)

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