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

Optimal. Leaf size=108 \[ \frac{f (b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+2)}-\frac{(b x)^{m+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} (c f (m+1)-d e (m+n+2)) \, _2F_1\left (m+1,-n;m+2;-\frac{d x}{c}\right )}{b d (m+1) (m+n+2)} \]

[Out]

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

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Rubi [A]  time = 0.137626, antiderivative size = 99, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{(b x)^{m+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} \left (\frac{e}{m+1}-\frac{c f}{d (m+n+2)}\right ) \, _2F_1\left (m+1,-n;m+2;-\frac{d x}{c}\right )}{b}+\frac{f (b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.1035, size = 87, normalized size = 0.81 \[ \frac{f \left (b x\right )^{m + 1} \left (c + d x\right )^{n + 1}}{b d \left (m + n + 2\right )} - \frac{\left (b x\right )^{m + 1} \left (1 + \frac{d x}{c}\right )^{- n} \left (c + d x\right )^{n} \left (c f \left (m + 1\right ) - d e \left (m + n + 2\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{d x}{c}} \right )}}{b d \left (m + 1\right ) \left (m + n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0868754, size = 82, normalized size = 0.76 \[ \frac{x (b x)^m (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} \left (e (m+2) \, _2F_1\left (m+1,-n;m+2;-\frac{d x}{c}\right )+f (m+1) x \, _2F_1\left (m+2,-n;m+3;-\frac{d x}{c}\right )\right )}{(m+1) (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.071, size = 0, normalized size = 0. \[ \int \left ( bx \right ) ^{m} \left ( dx+c \right ) ^{n} \left ( fx+e \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 40.5567, size = 82, normalized size = 0.76 \[ \frac{b^{m} c^{n} e x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 2\right )} + \frac{b^{m} c^{n} f x^{2} x^{m} \Gamma \left (m + 2\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 2 \\ m + 3 \end{matrix}\middle |{\frac{d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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