3.948 \(\int \frac{(d+e x)^m \sqrt{a+b x+c x^2}}{f+g x} \, dx\)

Optimal. Leaf size=31 \[ \text{Unintegrable}\left (\frac{\sqrt{a+b x+c x^2} (d+e x)^m}{f+g x},x\right ) \]

[Out]

Defer[Int][((d + e*x)^m*Sqrt[a + b*x + c*x^2])/(f + g*x), x]

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Rubi [A]  time = 0.0344326, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{(d+e x)^m \sqrt{a+b x+c x^2}}{f+g x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((d + e*x)^m*Sqrt[a + b*x + c*x^2])/(f + g*x),x]

[Out]

Defer[Int][((d + e*x)^m*Sqrt[a + b*x + c*x^2])/(f + g*x), x]

Rubi steps

\begin{align*} \int \frac{(d+e x)^m \sqrt{a+b x+c x^2}}{f+g x} \, dx &=\int \frac{(d+e x)^m \sqrt{a+b x+c x^2}}{f+g x} \, dx\\ \end{align*}

Mathematica [A]  time = 0.109119, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^m \sqrt{a+b x+c x^2}}{f+g x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[((d + e*x)^m*Sqrt[a + b*x + c*x^2])/(f + g*x),x]

[Out]

Integrate[((d + e*x)^m*Sqrt[a + b*x + c*x^2])/(f + g*x), x]

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Maple [A]  time = 1.922, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{m}}{gx+f}\sqrt{c{x}^{2}+bx+a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{2} + b x + a}{\left (e x + d\right )}^{m}}{g x + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*x^2+b*x+a)^(1/2)/(g*x+f),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{2} + b x + a}{\left (e x + d\right )}^{m}}{g x + f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*x^2+b*x+a)^(1/2)/(g*x+f),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{m} \sqrt{a + b x + c x^{2}}}{f + g x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**m*(c*x**2+b*x+a)**(1/2)/(g*x+f),x)

[Out]

Integral((d + e*x)**m*sqrt(a + b*x + c*x**2)/(f + g*x), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{2} + b x + a}{\left (e x + d\right )}^{m}}{g x + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*x^2+b*x+a)^(1/2)/(g*x+f),x, algorithm="giac")

[Out]

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