3.1.17 \(\int (\frac {x^2}{\sqrt {\tan (a+b x^2)}}+\frac {\sqrt {\tan (a+b x^2)}}{b}+x^2 \tan ^{\frac {3}{2}}(a+b x^2)) \, dx\) [17]

Optimal. Leaf size=17 \[ \frac {x \sqrt {\tan \left (a+b x^2\right )}}{b} \]

[Out]

x*tan(b*x^2+a)^(1/2)/b

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Rubi [F]
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 = {} \begin {gather*} \int \left (\frac {x^2}{\sqrt {\tan \left (a+b x^2\right )}}+\frac {\sqrt {\tan \left (a+b x^2\right )}}{b}+x^2 \tan ^{\frac {3}{2}}\left (a+b x^2\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[x^2/Sqrt[Tan[a + b*x^2]] + Sqrt[Tan[a + b*x^2]]/b + x^2*Tan[a + b*x^2]^(3/2),x]

[Out]

Defer[Int][x^2/Sqrt[Tan[a + b*x^2]], x] + Defer[Int][Sqrt[Tan[a + b*x^2]], x]/b + Defer[Int][x^2*Tan[a + b*x^2
]^(3/2), x]

Rubi steps

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

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Mathematica [A]
time = 0.51, size = 17, normalized size = 1.00 \begin {gather*} \frac {x \sqrt {\tan \left (a+b x^2\right )}}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[Tan[a + b*x^2]] + Sqrt[Tan[a + b*x^2]]/b + x^2*Tan[a + b*x^2]^(3/2),x]

[Out]

(x*Sqrt[Tan[a + b*x^2]])/b

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Maple [F]
time = 0.43, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\sqrt {\tan \left (b \,x^{2}+a \right )}}+\frac {\sqrt {\tan }\left (b \,x^{2}+a \right )}{b}+x^{2} \left (\tan ^{\frac {3}{2}}\left (b \,x^{2}+a \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/tan(b*x^2+a)^(1/2)+tan(b*x^2+a)^(1/2)/b+x^2*tan(b*x^2+a)^(3/2),x)

[Out]

int(x^2/tan(b*x^2+a)^(1/2)+tan(b*x^2+a)^(1/2)/b+x^2*tan(b*x^2+a)^(3/2),x)

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Maxima [F]
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(x^2/tan(b*x^2+a)^(1/2)+tan(b*x^2+a)^(1/2)/b+x^2*tan(b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^2*tan(b*x^2 + a)^(3/2) + x^2/sqrt(tan(b*x^2 + a)) + sqrt(tan(b*x^2 + a))/b, x)

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Fricas [A]
time = 0.35, size = 15, normalized size = 0.88 \begin {gather*} \frac {x \sqrt {\tan \left (b x^{2} + a\right )}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/tan(b*x^2+a)^(1/2)+tan(b*x^2+a)^(1/2)/b+x^2*tan(b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

x*sqrt(tan(b*x^2 + a))/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/tan(b*x**2+a)**(1/2)+tan(b*x**2+a)**(1/2)/b+x**2*tan(b*x**2+a)**(3/2),x)

[Out]

(Integral(b*x**2/sqrt(tan(a + b*x**2)), x) + Integral(b*x**2*tan(a + b*x**2)**(3/2), x) + Integral(sqrt(tan(a
+ b*x**2)), x))/b

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Giac [F]
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(x^2/tan(b*x^2+a)^(1/2)+tan(b*x^2+a)^(1/2)/b+x^2*tan(b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

integrate(x^2*tan(b*x^2 + a)^(3/2) + x^2/sqrt(tan(b*x^2 + a)) + sqrt(tan(b*x^2 + a))/b, x)

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Mupad [B]
time = 4.17, size = 45, normalized size = 2.65 \begin {gather*} \frac {x\,\sqrt {-\frac {{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}}{{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}+1}}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(a + b*x^2)^(1/2)/b + x^2/tan(a + b*x^2)^(1/2) + x^2*tan(a + b*x^2)^(3/2),x)

[Out]

(x*(-(exp(a*2i + b*x^2*2i)*1i - 1i)/(exp(a*2i + b*x^2*2i) + 1))^(1/2))/b

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