3.671 \(\int \frac{x^5}{\sqrt{d x^2} (a+b x^2)} \, dx\)

Optimal. Leaf size=72 \[ \frac{a^{3/2} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2} \sqrt{d x^2}}-\frac{a x^2}{b^2 \sqrt{d x^2}}+\frac{x^4}{3 b \sqrt{d x^2}} \]

[Out]

-((a*x^2)/(b^2*Sqrt[d*x^2])) + x^4/(3*b*Sqrt[d*x^2]) + (a^(3/2)*x*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b^(5/2)*Sqrt[d
*x^2])

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Rubi [A]  time = 0.0256232, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {15, 302, 205} \[ \frac{a^{3/2} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2} \sqrt{d x^2}}-\frac{a x^2}{b^2 \sqrt{d x^2}}+\frac{x^4}{3 b \sqrt{d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(Sqrt[d*x^2]*(a + b*x^2)),x]

[Out]

-((a*x^2)/(b^2*Sqrt[d*x^2])) + x^4/(3*b*Sqrt[d*x^2]) + (a^(3/2)*x*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b^(5/2)*Sqrt[d
*x^2])

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^5}{\sqrt{d x^2} \left (a+b x^2\right )} \, dx &=\frac{x \int \frac{x^4}{a+b x^2} \, dx}{\sqrt{d x^2}}\\ &=\frac{x \int \left (-\frac{a}{b^2}+\frac{x^2}{b}+\frac{a^2}{b^2 \left (a+b x^2\right )}\right ) \, dx}{\sqrt{d x^2}}\\ &=-\frac{a x^2}{b^2 \sqrt{d x^2}}+\frac{x^4}{3 b \sqrt{d x^2}}+\frac{\left (a^2 x\right ) \int \frac{1}{a+b x^2} \, dx}{b^2 \sqrt{d x^2}}\\ &=-\frac{a x^2}{b^2 \sqrt{d x^2}}+\frac{x^4}{3 b \sqrt{d x^2}}+\frac{a^{3/2} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2} \sqrt{d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0240915, size = 56, normalized size = 0.78 \[ \frac{x \left (3 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+\sqrt{b} x \left (b x^2-3 a\right )\right )}{3 b^{5/2} \sqrt{d x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(Sqrt[d*x^2]*(a + b*x^2)),x]

[Out]

(x*(Sqrt[b]*x*(-3*a + b*x^2) + 3*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]]))/(3*b^(5/2)*Sqrt[d*x^2])

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Maple [A]  time = 0.007, size = 53, normalized size = 0.7 \begin{align*}{\frac{x}{3\,{b}^{2}} \left ( \sqrt{ab}{x}^{3}b-3\,\sqrt{ab}xa+3\,{a}^{2}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) \right ){\frac{1}{\sqrt{d{x}^{2}}}}{\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x^2+a)/(d*x^2)^(1/2),x)

[Out]

1/3*x*((a*b)^(1/2)*x^3*b-3*(a*b)^(1/2)*x*a+3*a^2*arctan(b*x/(a*b)^(1/2)))/(d*x^2)^(1/2)/b^2/(a*b)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)/(d*x^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.33494, size = 315, normalized size = 4.38 \begin{align*} \left [\frac{3 \, a d \sqrt{-\frac{a}{b d}} \log \left (\frac{b x^{2} + 2 \, \sqrt{d x^{2}} b \sqrt{-\frac{a}{b d}} - a}{b x^{2} + a}\right ) + 2 \,{\left (b x^{2} - 3 \, a\right )} \sqrt{d x^{2}}}{6 \, b^{2} d}, \frac{3 \, a d \sqrt{\frac{a}{b d}} \arctan \left (\frac{\sqrt{d x^{2}} b \sqrt{\frac{a}{b d}}}{a}\right ) +{\left (b x^{2} - 3 \, a\right )} \sqrt{d x^{2}}}{3 \, b^{2} d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)/(d*x^2)^(1/2),x, algorithm="fricas")

[Out]

[1/6*(3*a*d*sqrt(-a/(b*d))*log((b*x^2 + 2*sqrt(d*x^2)*b*sqrt(-a/(b*d)) - a)/(b*x^2 + a)) + 2*(b*x^2 - 3*a)*sqr
t(d*x^2))/(b^2*d), 1/3*(3*a*d*sqrt(a/(b*d))*arctan(sqrt(d*x^2)*b*sqrt(a/(b*d))/a) + (b*x^2 - 3*a)*sqrt(d*x^2))
/(b^2*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\sqrt{d x^{2}} \left (a + b x^{2}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x**2+a)/(d*x**2)**(1/2),x)

[Out]

Integral(x**5/(sqrt(d*x**2)*(a + b*x**2)), x)

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Giac [A]  time = 1.12183, size = 95, normalized size = 1.32 \begin{align*} \frac{a^{2} \arctan \left (\frac{\sqrt{d x^{2}} b}{\sqrt{a b d}}\right )}{\sqrt{a b d} b^{2}} + \frac{\sqrt{d x^{2}} b^{2} d^{5} x^{2} - 3 \, \sqrt{d x^{2}} a b d^{5}}{3 \, b^{3} d^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)/(d*x^2)^(1/2),x, algorithm="giac")

[Out]

a^2*arctan(sqrt(d*x^2)*b/sqrt(a*b*d))/(sqrt(a*b*d)*b^2) + 1/3*(sqrt(d*x^2)*b^2*d^5*x^2 - 3*sqrt(d*x^2)*a*b*d^5
)/(b^3*d^6)