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

Optimal. Leaf size=68 \[ \frac{\left (c x^2\right )^p (d x)^{m+1} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-n,m+2 p+1;m+2 p+2;-\frac{b x}{a}\right )}{d (m+2 p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0552771, antiderivative size = 64, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{x \left (c x^2\right )^p (d x)^m (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-n,m+2 p+1;m+2 p+2;-\frac{b x}{a}\right )}{m+2 p+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.9326, size = 66, normalized size = 0.97 \[ \frac{\left (c x^{2}\right )^{p} \left (d x\right )^{- 2 p} \left (d x\right )^{m + 2 p + 1} \left (1 + \frac{b x}{a}\right )^{- n} \left (a + b x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, m + 2 p + 1 \\ m + 2 p + 2 \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{d \left (m + 2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x**2)**p*(d*x)**(-2*p)*(d*x)**(m + 2*p + 1)*(1 + b*x/a)**(-n)*(a + b*x)**n*hy
per((-n, m + 2*p + 1), (m + 2*p + 2,), -b*x/a)/(d*(m + 2*p + 1))

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Mathematica [A]  time = 0.0391653, size = 64, normalized size = 0.94 \[ \frac{x \left (c x^2\right )^p (d x)^m (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-n,m+2 p+1;m+2 p+2;-\frac{b x}{a}\right )}{m+2 p+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.236, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m} \left ( c{x}^{2} \right ) ^{p} \left ( bx+a \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((c*x**2)**p*(d*x)**m*(a + b*x)**n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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