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

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

[Out]

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

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Rubi [A]  time = 0.0355541, 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 = {} \[ \int \frac{(d+e x)^m}{(f+g x) \sqrt{a+b x+c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.122449, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^m}{(f+g x) \sqrt{a+b x+c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.486, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{m}}{gx+f}{\frac{1}{\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/(g*x+f)/(c*x^2+b*x+a)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x + d)^m/(sqrt(c*x^2 + b*x + a)*(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}}{c g x^{3} +{\left (c f + b g\right )} x^{2} + a f +{\left (b f + a g\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*x^2 + b*x + a)*(e*x + d)^m/(c*g*x^3 + (c*f + b*g)*x^2 + a*f + (b*f + a*g)*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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