3.124 \(\int (a+b x)^m (c+d x)^n (e+f x) (g+h x) \, dx\)

Optimal. Leaf size=266 \[ \frac{(a+b x)^{m+1} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{d (a+b x)}{b c-a d}\right ) \left (a^2 d^2 f h (n+1) (n+2)+a b d (n+1) (2 c f h (m+1)-d (m+n+3) (e h+f g))+b^2 \left (c^2 f h (m+1) (m+2)-c d (m+1) (m+n+3) (e h+f g)+d^2 e g (m+n+2) (m+n+3)\right )\right )}{b^3 d^2 (m+1) (m+n+2) (m+n+3)}-\frac{(a+b x)^{m+1} (c+d x)^{n+1} (a d f h (n+2)+b c f h (m+2)-b d (m+n+3) (e h+f g)-b d f h x (m+n+2))}{b^2 d^2 (m+n+2) (m+n+3)} \]

[Out]

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

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Rubi [A]  time = 0.165118, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {147, 70, 69} \[ \frac{(a+b x)^{m+1} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{d (a+b x)}{b c-a d}\right ) \left (a^2 d^2 f h (n+1) (n+2)+a b d (n+1) (2 c f h (m+1)-d (m+n+3) (e h+f g))+b^2 \left (c^2 f h (m+1) (m+2)-c d (m+1) (m+n+3) (e h+f g)+d^2 e g (m+n+2) (m+n+3)\right )\right )}{b^3 d^2 (m+1) (m+n+2) (m+n+3)}-\frac{(a+b x)^{m+1} (c+d x)^{n+1} (a d f h (n+2)+b c f h (m+2)-b d (m+n+3) (e h+f g)-b d f h x (m+n+2))}{b^2 d^2 (m+n+2) (m+n+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 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 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]))

Rubi steps

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

Mathematica [A]  time = 0.235778, size = 195, normalized size = 0.73 \[ \frac{(a+b x)^{m+1} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (b \left (b (d e-c f) (d g-c h) \, _2F_1\left (m+1,-n;m+2;\frac{d (a+b x)}{a d-b c}\right )-(b c-a d) (2 c f h-d (e h+f g)) \, _2F_1\left (m+1,-n-1;m+2;\frac{d (a+b x)}{a d-b c}\right )\right )+f h (b c-a d)^2 \, _2F_1\left (m+1,-n-2;m+2;\frac{d (a+b x)}{a d-b c}\right )\right )}{b^3 d^2 (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{n} \left ( fx+e \right ) \left ( hx+g \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}{\left (h x + g\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (f h x^{2} + e g +{\left (f g + e h\right )} x\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}{\left (h x + g\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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