3.128 \(\int (a+b x)^m (c+d x)^{-2-m} (e+f x) (g+h x) \, dx\)

Optimal. Leaf size=203 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1} \left (-c d (a f h (m+1)+b (e h+f g))+d f h (m+1) x (b c-a d)+b c^2 f h (m+2)+b d^2 e g\right )}{b d^2 (m+1) (b c-a d)}-\frac{(a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right ) (a d f h m+b (d (e h+f g)-c f h (m+2)))}{b d^3 m} \]

[Out]

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

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Rubi [A]  time = 0.106725, antiderivative size = 205, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {143, 70, 69} \[ -\frac{(a+b x)^{m+1} (c+d x)^{-m-1} \left (-d f h (m+1) x (b c-a d)+a c d f h (m+1)-b \left (c^2 f h (m+2)-c d (e h+f g)+d^2 e g\right )\right )}{b d^2 (m+1) (b c-a d)}-\frac{(a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right ) (a d f h m-b c f h (m+2)+b d (e h+f g))}{b d^3 m} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 143

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

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

Mathematica [A]  time = 0.249263, size = 198, normalized size = 0.98 \[ \frac{(a+b x)^m (c+d x)^{-m} \left (\frac{(m+1) (b c-a d) \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right ) (-a d f h m+b c f h (m+2)-b d (e h+f g))}{m}-\frac{d (a+b x) \left (a d f h (m+1) (c+d x)-b \left (c^2 f h (m+2)+c d (-e h-f g+f h (m+1) x)+d^2 e g\right )\right )}{c+d x}\right )}{b d^3 (m+1) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^m*(-((d*(a + b*x)*(a*d*f*h*(1 + m)*(c + d*x) - b*(d^2*e*g + c^2*f*h*(2 + m) + c*d*(-(f*g) - e*h + f
*h*(1 + m)*x))))/(c + d*x)) + ((b*c - a*d)*(1 + m)*(-(b*d*(f*g + e*h)) - a*d*f*h*m + b*c*f*h*(2 + m))*Hypergeo
metric2F1[-m, -m, 1 - m, (b*(c + d*x))/(b*c - a*d)])/(m*((d*(a + b*x))/(-(b*c) + a*d))^m)))/(b*d^3*(b*c - a*d)
*(1 + m)*(c + d*x)^m)

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-2-m} \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)^(-2-m)*(f*x+e)*(h*x+g),x)

[Out]

int((b*x+a)^m*(d*x+c)^(-2-m)*(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 )}^{-m - 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((f*x + e)*(h*x + g)*(b*x + a)^m*(d*x + c)^(-m - 2), 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 )}^{-m - 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-2-m)*(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)^(-m - 2), 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)**(-2-m)*(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 )}^{-m - 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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