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

Optimal. Leaf size=393 \[ \frac{(b g-a h)^2 (a+b x)^{m+1} (c+d x)^n (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+1;-n,-p;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^3 (m+1)}+\frac{2 h (b g-a h) (a+b x)^{m+2} (c+d x)^n (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+2;-n,-p;m+3;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^3 (m+2)}+\frac{h^2 (a+b x)^{m+3} (c+d x)^n (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+3;-n,-p;m+4;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^3 (m+3)} \]

[Out]

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

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Rubi [A]  time = 1.47267, antiderivative size = 393, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{(b g-a h)^2 (a+b x)^{m+1} (c+d x)^n (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+1;-n,-p;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^3 (m+1)}+\frac{2 h (b g-a h) (a+b x)^{m+2} (c+d x)^n (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+2;-n,-p;m+3;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^3 (m+2)}+\frac{h^2 (a+b x)^{m+3} (c+d x)^n (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+3;-n,-p;m+4;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^3 (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**m*(d*x+c)**n*(f*x+e)**p*(h*x+g)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 4.2179, size = 0, normalized size = 0. \[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^2 \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.251, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{n} \left ( fx+e \right ) ^{p} \left ( hx+g \right ) ^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^n*(f*x+e)^p*(h*x+g)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (h x + g\right )}^{2}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x + g)^2*(b*x + a)^m*(d*x + c)^n*(f*x + e)^p,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x + g)^2*(b*x + a)^m*(d*x + c)^n*(f*x + e)^p,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**m*(d*x+c)**n*(f*x+e)**p*(h*x+g)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (h x + g\right )}^{2}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x + g)^2*(b*x + a)^m*(d*x + c)^n*(f*x + e)^p,x, algorithm="giac")

[Out]

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