### 3.17 $$\int (\frac{x^2}{\sqrt{\tan (a+b x^2)}}+\frac{\sqrt{\tan (a+b x^2)}}{b}+x^2 \tan ^{\frac{3}{2}}(a+b x^2)) \, dx$$

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

[Out]

(x*Sqrt[Tan[a + b*x^2]])/b

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Rubi [F]  time = 0.0346331, 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 = {} $\int \left (\frac{x^2}{\sqrt{\tan \left (a+b x^2\right )}}+\frac{\sqrt{\tan \left (a+b x^2\right )}}{b}+x^2 \tan ^{\frac{3}{2}}\left (a+b x^2\right )\right ) \, dx$

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*}

Mathematica [A]  time = 0.559597, size = 17, normalized size = 1. $\frac{x \sqrt{\tan \left (a+b x^2\right )}}{b}$

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 = 0.253, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2}{\frac{1}{\sqrt{\tan \left ( b{x}^{2}+a \right ) }}}}+{\frac{1}{b}\sqrt{\tan \left ( b{x}^{2}+a \right ) }}+{x}^{2} \left ( \tan \left ( b{x}^{2}+a \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int x^{2} \tan \left (b x^{2} + a\right )^{\frac{3}{2}} + \frac{x^{2}}{\sqrt{\tan \left (b x^{2} + a\right )}} + \frac{\sqrt{\tan \left (b x^{2} + a\right )}}{b}\,{d x} \end{align*}

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 = 1.55773, size = 35, normalized size = 2.06 \begin{align*} \frac{x \sqrt{\tan \left (b x^{2} + a\right )}}{b} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int x^{2} \tan \left (b x^{2} + a\right )^{\frac{3}{2}} + \frac{x^{2}}{\sqrt{\tan \left (b x^{2} + a\right )}} + \frac{\sqrt{\tan \left (b x^{2} + a\right )}}{b}\,{d x} \end{align*}

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)