3.399 \(\int (a+b x^2)^{5/2} \, dx\)

Optimal. Leaf size=84 \[ \frac{5}{16} a^2 x \sqrt{a+b x^2}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 \sqrt{b}}+\frac{5}{24} a x \left (a+b x^2\right )^{3/2}+\frac{1}{6} x \left (a+b x^2\right )^{5/2} \]

[Out]

(5*a^2*x*Sqrt[a + b*x^2])/16 + (5*a*x*(a + b*x^2)^(3/2))/24 + (x*(a + b*x^2)^(5/2))/6 + (5*a^3*ArcTanh[(Sqrt[b
]*x)/Sqrt[a + b*x^2]])/(16*Sqrt[b])

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Rubi [A]  time = 0.0216615, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {195, 217, 206} \[ \frac{5}{16} a^2 x \sqrt{a+b x^2}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 \sqrt{b}}+\frac{5}{24} a x \left (a+b x^2\right )^{3/2}+\frac{1}{6} x \left (a+b x^2\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(5/2),x]

[Out]

(5*a^2*x*Sqrt[a + b*x^2])/16 + (5*a*x*(a + b*x^2)^(3/2))/24 + (x*(a + b*x^2)^(5/2))/6 + (5*a^3*ArcTanh[(Sqrt[b
]*x)/Sqrt[a + b*x^2]])/(16*Sqrt[b])

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \left (a+b x^2\right )^{5/2} \, dx &=\frac{1}{6} x \left (a+b x^2\right )^{5/2}+\frac{1}{6} (5 a) \int \left (a+b x^2\right )^{3/2} \, dx\\ &=\frac{5}{24} a x \left (a+b x^2\right )^{3/2}+\frac{1}{6} x \left (a+b x^2\right )^{5/2}+\frac{1}{8} \left (5 a^2\right ) \int \sqrt{a+b x^2} \, dx\\ &=\frac{5}{16} a^2 x \sqrt{a+b x^2}+\frac{5}{24} a x \left (a+b x^2\right )^{3/2}+\frac{1}{6} x \left (a+b x^2\right )^{5/2}+\frac{1}{16} \left (5 a^3\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx\\ &=\frac{5}{16} a^2 x \sqrt{a+b x^2}+\frac{5}{24} a x \left (a+b x^2\right )^{3/2}+\frac{1}{6} x \left (a+b x^2\right )^{5/2}+\frac{1}{16} \left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )\\ &=\frac{5}{16} a^2 x \sqrt{a+b x^2}+\frac{5}{24} a x \left (a+b x^2\right )^{3/2}+\frac{1}{6} x \left (a+b x^2\right )^{5/2}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.106692, size = 76, normalized size = 0.9 \[ \frac{1}{48} \sqrt{a+b x^2} \left (\frac{15 a^{5/2} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} \sqrt{\frac{b x^2}{a}+1}}+33 a^2 x+26 a b x^3+8 b^2 x^5\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(5/2),x]

[Out]

(Sqrt[a + b*x^2]*(33*a^2*x + 26*a*b*x^3 + 8*b^2*x^5 + (15*a^(5/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*Sqrt[
1 + (b*x^2)/a])))/48

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Maple [A]  time = 0.003, size = 66, normalized size = 0.8 \begin{align*}{\frac{x}{6} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,ax}{24} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}x}{16}\sqrt{b{x}^{2}+a}}+{\frac{5\,{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x*(b*x^2+a)^(5/2)+5/24*a*x*(b*x^2+a)^(3/2)+5/16*a^2*x*(b*x^2+a)^(1/2)+5/16*a^3/b^(1/2)*ln(x*b^(1/2)+(b*x^2
+a)^(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((b*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.594, size = 347, normalized size = 4.13 \begin{align*} \left [\frac{15 \, a^{3} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (8 \, b^{3} x^{5} + 26 \, a b^{2} x^{3} + 33 \, a^{2} b x\right )} \sqrt{b x^{2} + a}}{96 \, b}, -\frac{15 \, a^{3} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (8 \, b^{3} x^{5} + 26 \, a b^{2} x^{3} + 33 \, a^{2} b x\right )} \sqrt{b x^{2} + a}}{48 \, b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/96*(15*a^3*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(8*b^3*x^5 + 26*a*b^2*x^3 + 33*a^2*b
*x)*sqrt(b*x^2 + a))/b, -1/48*(15*a^3*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (8*b^3*x^5 + 26*a*b^2*x^3
+ 33*a^2*b*x)*sqrt(b*x^2 + a))/b]

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Sympy [A]  time = 4.23791, size = 97, normalized size = 1.15 \begin{align*} \frac{11 a^{\frac{5}{2}} x \sqrt{1 + \frac{b x^{2}}{a}}}{16} + \frac{13 a^{\frac{3}{2}} b x^{3} \sqrt{1 + \frac{b x^{2}}{a}}}{24} + \frac{\sqrt{a} b^{2} x^{5} \sqrt{1 + \frac{b x^{2}}{a}}}{6} + \frac{5 a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 \sqrt{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11*a**(5/2)*x*sqrt(1 + b*x**2/a)/16 + 13*a**(3/2)*b*x**3*sqrt(1 + b*x**2/a)/24 + sqrt(a)*b**2*x**5*sqrt(1 + b*
x**2/a)/6 + 5*a**3*asinh(sqrt(b)*x/sqrt(a))/(16*sqrt(b))

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Giac [A]  time = 1.71163, size = 85, normalized size = 1.01 \begin{align*} -\frac{5 \, a^{3} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, \sqrt{b}} + \frac{1}{48} \,{\left (2 \,{\left (4 \, b^{2} x^{2} + 13 \, a b\right )} x^{2} + 33 \, a^{2}\right )} \sqrt{b x^{2} + a} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/16*a^3*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/sqrt(b) + 1/48*(2*(4*b^2*x^2 + 13*a*b)*x^2 + 33*a^2)*sqrt(b*x
^2 + a)*x