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

Optimal. Leaf size=32 \[ \text {Int}\left (\frac {(d+e x)^m}{(f+g x) \sqrt {a+b x+c x^2}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {(d+e x)^m}{(f+g x) \sqrt {a+b x+c x^2}} \, dx \]

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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fricas [A]  time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{m}}{c g x^{3} + {\left (c f + b g\right )} x^{2} + a f + {\left (b f + a g\right )} x}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{m}}{\sqrt {c x^{2} + b x + a} {\left (g x + f\right )}}\,{d x} \]

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)

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maple [A]  time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d \right )^{m}}{\left (g x +f \right ) \sqrt {c \,x^{2}+b x +a}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{m}}{\sqrt {c x^{2} + b x + a} {\left (g x + f\right )}}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (d+e\,x\right )}^m}{\left (f+g\,x\right )\,\sqrt {c\,x^2+b\,x+a}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x\right )^{m}}{\left (f + g x\right ) \sqrt {a + b x + c x^{2}}}\, dx \]

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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