3.124 \(\int (d x)^m \left (b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=55 \[ \frac{x (d x)^m \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,m+p+1;m+p+2;-\frac{c x}{b}\right )}{m+p+1} \]

[Out]

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

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Rubi [A]  time = 0.0696622, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{x (d x)^m \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,m+p+1;m+p+2;-\frac{c x}{b}\right )}{m+p+1} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m*(b*x + c*x^2)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.8825, size = 58, normalized size = 1.05 \[ \frac{x^{- m - p} x^{m + p + 1} \left (d x\right )^{m} \left (1 + \frac{c x}{b}\right )^{- p} \left (b x + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, m + p + 1 \\ m + p + 2 \end{matrix}\middle |{- \frac{c x}{b}} \right )}}{m + p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(c*x**2+b*x)**p,x)

[Out]

x**(-m - p)*x**(m + p + 1)*(d*x)**m*(1 + c*x/b)**(-p)*(b*x + c*x**2)**p*hyper((-
p, m + p + 1), (m + p + 2,), -c*x/b)/(m + p + 1)

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Mathematica [A]  time = 0.0550313, size = 53, normalized size = 0.96 \[ \frac{x (d x)^m (x (b+c x))^p \left (\frac{c x}{b}+1\right )^{-p} \, _2F_1\left (-p,m+p+1;m+p+2;-\frac{c x}{b}\right )}{m+p+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.137, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m} \left ( c{x}^{2}+bx \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(c*x^2+b*x)^p,x)

[Out]

int((d*x)^m*(c*x^2+b*x)^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x)^p*(d*x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x^2 + b*x)^p*(d*x)^m, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{m} \left (x \left (b + c x\right )\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**m*(c*x**2+b*x)**p,x)

[Out]

Integral((d*x)**m*(x*(b + c*x))**p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x)^p*(d*x)^m, x)