3.984 \(\int \frac{(d x)^m (a+b x)^n}{\sqrt{c x^2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{x (d x)^m (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m,-n;m+1;-\frac{b x}{a}\right )}{m \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0550144, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x (d x)^m (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m,-n;m+1;-\frac{b x}{a}\right )}{m \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.1166, size = 44, normalized size = 0.83 \[ \frac{\sqrt{c x^{2}} \left (d x\right )^{m} \left (1 + \frac{b x}{a}\right )^{- n} \left (a + b x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, m \\ m + 1 \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{c m x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(b*x+a)**n/(c*x**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0349226, size = 53, normalized size = 1. \[ \frac{x (d x)^m (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m,-n;m+1;-\frac{b x}{a}\right )}{m \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \[ \int{ \left ( dx \right ) ^{m} \left ( bx+a \right ) ^{n}{\frac{1}{\sqrt{c{x}^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(b*x+a)^n/(c*x^2)^(1/2),x)

[Out]

int((d*x)^m*(b*x+a)^n/(c*x^2)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**m*(b*x+a)**n/(c*x**2)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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