3.940 \(\int (b x)^m (c+d x)^n (e+f x)^2 \, dx\)

Optimal. Leaf size=209 \[ \frac{(b x)^{m+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} \left (c^2 f^2 \left (m^2+3 m+2\right )-2 c d e f (m+1) (m+n+3)+d^2 e^2 \left (m^2+m (2 n+5)+n^2+5 n+6\right )\right ) \, _2F_1\left (m+1,-n;m+2;-\frac{d x}{c}\right )}{b d^2 (m+1) (m+n+2) (m+n+3)}-\frac{f (b x)^{m+1} (c+d x)^{n+1} (c f (m+2)-d e (m+n+4))}{b d^2 (m+n+2) (m+n+3)}+\frac{f (b x)^{m+1} (e+f x) (c+d x)^{n+1}}{b d (m+n+3)} \]

[Out]

-((f*(c*f*(2 + m) - d*e*(4 + m + n))*(b*x)^(1 + m)*(c + d*x)^(1 + n))/(b*d^2*(2
+ m + n)*(3 + m + n))) + (f*(b*x)^(1 + m)*(c + d*x)^(1 + n)*(e + f*x))/(b*d*(3 +
m + n)) + ((c^2*f^2*(2 + 3*m + m^2) - 2*c*d*e*f*(1 + m)*(3 + m + n) + d^2*e^2*(
6 + m^2 + 5*n + n^2 + m*(5 + 2*n)))*(b*x)^(1 + m)*(c + d*x)^n*Hypergeometric2F1[
1 + m, -n, 2 + m, -((d*x)/c)])/(b*d^2*(1 + m)*(2 + m + n)*(3 + m + n)*(1 + (d*x)
/c)^n)

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Rubi [A]  time = 0.459052, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{(b x)^{m+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} \left (c^2 f^2 \left (m^2+3 m+2\right )-2 c d e f (m+1) (m+n+3)+d^2 e^2 \left (m^2+m (2 n+5)+n^2+5 n+6\right )\right ) \, _2F_1\left (m+1,-n;m+2;-\frac{d x}{c}\right )}{b d^2 (m+1) (m+n+2) (m+n+3)}-\frac{f (b x)^{m+1} (c+d x)^{n+1} (c f (m+2)-d e (m+n+4))}{b d^2 (m+n+2) (m+n+3)}+\frac{f (b x)^{m+1} (e+f x) (c+d x)^{n+1}}{b d (m+n+3)} \]

Antiderivative was successfully verified.

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

[Out]

-((f*(c*f*(2 + m) - d*e*(4 + m + n))*(b*x)^(1 + m)*(c + d*x)^(1 + n))/(b*d^2*(2
+ m + n)*(3 + m + n))) + (f*(b*x)^(1 + m)*(c + d*x)^(1 + n)*(e + f*x))/(b*d*(3 +
m + n)) + ((c^2*f^2*(2 + 3*m + m^2) - 2*c*d*e*f*(1 + m)*(3 + m + n) + d^2*e^2*(
6 + m^2 + 5*n + n^2 + m*(5 + 2*n)))*(b*x)^(1 + m)*(c + d*x)^n*Hypergeometric2F1[
1 + m, -n, 2 + m, -((d*x)/c)])/(b*d^2*(1 + m)*(2 + m + n)*(3 + m + n)*(1 + (d*x)
/c)^n)

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Rubi in Sympy [A]  time = 56.6678, size = 177, normalized size = 0.85 \[ \frac{f \left (b x\right )^{m + 1} \left (c + d x\right )^{n + 1} \left (e + f x\right )}{b d \left (m + n + 3\right )} - \frac{f \left (b x\right )^{m + 1} \left (c + d x\right )^{n + 1} \left (c f \left (m + 2\right ) - d e \left (m + n + 4\right )\right )}{b d^{2} \left (m + n + 2\right ) \left (m + n + 3\right )} + \frac{\left (b x\right )^{m + 1} \left (1 + \frac{d x}{c}\right )^{- n} \left (c + d x\right )^{n} \left (c f \left (m + 1\right ) \left (c f \left (m + 2\right ) - d e \left (m + n + 4\right )\right ) - d e \left (c f \left (m + 1\right ) - d e \left (m + n + 3\right )\right ) \left (m + n + 2\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{d x}{c}} \right )}}{b d^{2} \left (m + 1\right ) \left (m + n + 2\right ) \left (m + n + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*(b*x)**(m + 1)*(c + d*x)**(n + 1)*(e + f*x)/(b*d*(m + n + 3)) - f*(b*x)**(m +
1)*(c + d*x)**(n + 1)*(c*f*(m + 2) - d*e*(m + n + 4))/(b*d**2*(m + n + 2)*(m + n
+ 3)) + (b*x)**(m + 1)*(1 + d*x/c)**(-n)*(c + d*x)**n*(c*f*(m + 1)*(c*f*(m + 2)
- d*e*(m + n + 4)) - d*e*(c*f*(m + 1) - d*e*(m + n + 3))*(m + n + 2))*hyper((-n
, m + 1), (m + 2,), -d*x/c)/(b*d**2*(m + 1)*(m + n + 2)*(m + n + 3))

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Mathematica [A]  time = 0.200948, size = 124, normalized size = 0.59 \[ \frac{x (b x)^m (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} \left (e^2 \left (m^2+5 m+6\right ) \, _2F_1\left (m+1,-n;m+2;-\frac{d x}{c}\right )+f (m+1) x \left (2 e (m+3) \, _2F_1\left (m+2,-n;m+3;-\frac{d x}{c}\right )+f (m+2) x \, _2F_1\left (m+3,-n;m+4;-\frac{d x}{c}\right )\right )\right )}{(m+1) (m+2) (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

(x*(b*x)^m*(c + d*x)^n*(e^2*(6 + 5*m + m^2)*Hypergeometric2F1[1 + m, -n, 2 + m,
-((d*x)/c)] + f*(1 + m)*x*(2*e*(3 + m)*Hypergeometric2F1[2 + m, -n, 3 + m, -((d*
x)/c)] + f*(2 + m)*x*Hypergeometric2F1[3 + m, -n, 4 + m, -((d*x)/c)])))/((1 + m)
*(2 + m)*(3 + m)*(1 + (d*x)/c)^n)

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Maple [F]  time = 0.09, size = 0, normalized size = 0. \[ \int \left ( bx \right ) ^{m} \left ( dx+c \right ) ^{n} \left ( fx+e \right ) ^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 83.2779, size = 131, normalized size = 0.63 \[ \frac{b^{m} c^{n} e^{2} x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 2\right )} + \frac{2 b^{m} c^{n} e f x^{2} x^{m} \Gamma \left (m + 2\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 2 \\ m + 3 \end{matrix}\middle |{\frac{d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 3\right )} + \frac{b^{m} c^{n} f^{2} x^{3} x^{m} \Gamma \left (m + 3\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 3 \\ m + 4 \end{matrix}\middle |{\frac{d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**m*c**n*e**2*x*x**m*gamma(m + 1)*hyper((-n, m + 1), (m + 2,), d*x*exp_polar(I*
pi)/c)/gamma(m + 2) + 2*b**m*c**n*e*f*x**2*x**m*gamma(m + 2)*hyper((-n, m + 2),
(m + 3,), d*x*exp_polar(I*pi)/c)/gamma(m + 3) + b**m*c**n*f**2*x**3*x**m*gamma(m
+ 3)*hyper((-n, m + 3), (m + 4,), d*x*exp_polar(I*pi)/c)/gamma(m + 4)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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