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

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

[Out]

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

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Rubi [A]  time = 0.03, 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 = {} \[ \int \frac {(d+e x)^m \sqrt {a+b x+c x^2}}{f+g x} \, dx \]

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*}

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Mathematica [A]  time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {(d+e x)^m \sqrt {a+b x+c x^2}}{f+g x} \, dx \]

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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fricas [A]  time = 1.38, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{m}}{g x + f}, x\right ) \]

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

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)

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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