3.2.3 \(\int \frac {(a+b x^2)^{3/2} \sqrt {c+d x^2}}{\sqrt {e+f x^2}} \, dx\) [103]

Optimal. Leaf size=37 \[ \text {Int}\left (\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{\sqrt {e+f x^2}},x\right ) \]

[Out]

Unintegrable((b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(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 = {} \begin {gather*} \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{\sqrt {e+f x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/Sqrt[e + f*x^2],x]

[Out]

Defer[Int][((a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/Sqrt[e + f*x^2], x]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{\sqrt {e+f x^2}} \, dx &=\int \frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{\sqrt {e+f x^2}} \, dx\\ \end {align*}

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Mathematica [A]
time = 15.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{\sqrt {e+f x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/Sqrt[e + f*x^2],x]

[Out]

Integrate[((a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/Sqrt[e + f*x^2], x]

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Maple [A]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {d \,x^{2}+c}}{\sqrt {f \,x^{2}+e}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)

[Out]

int((b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}}}{\sqrt {e + f x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(3/2)*(d*x**2+c)**(1/2)/(f*x**2+e)**(1/2),x)

[Out]

Integral((a + b*x**2)**(3/2)*sqrt(c + d*x**2)/sqrt(e + f*x**2), x)

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c}}{\sqrt {f\,x^2+e}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2))/(e + f*x^2)^(1/2),x)

[Out]

int(((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2))/(e + f*x^2)^(1/2), x)

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