3.47 \(\int \sqrt{a+b x^2} (c+d x^2) \, dx\)

Optimal. Leaf size=87 \[ \frac{a (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{3/2}}+\frac{x \sqrt{a+b x^2} (4 b c-a d)}{8 b}+\frac{d x \left (a+b x^2\right )^{3/2}}{4 b} \]

[Out]

((4*b*c - a*d)*x*Sqrt[a + b*x^2])/(8*b) + (d*x*(a + b*x^2)^(3/2))/(4*b) + (a*(4*b*c - a*d)*ArcTanh[(Sqrt[b]*x)
/Sqrt[a + b*x^2]])/(8*b^(3/2))

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Rubi [A]  time = 0.0276762, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {388, 195, 217, 206} \[ \frac{a (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{3/2}}+\frac{x \sqrt{a+b x^2} (4 b c-a d)}{8 b}+\frac{d x \left (a+b x^2\right )^{3/2}}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*x^2]*(c + d*x^2),x]

[Out]

((4*b*c - a*d)*x*Sqrt[a + b*x^2])/(8*b) + (d*x*(a + b*x^2)^(3/2))/(4*b) + (a*(4*b*c - a*d)*ArcTanh[(Sqrt[b]*x)
/Sqrt[a + b*x^2]])/(8*b^(3/2))

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

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 \sqrt{a+b x^2} \left (c+d x^2\right ) \, dx &=\frac{d x \left (a+b x^2\right )^{3/2}}{4 b}-\frac{(-4 b c+a d) \int \sqrt{a+b x^2} \, dx}{4 b}\\ &=\frac{(4 b c-a d) x \sqrt{a+b x^2}}{8 b}+\frac{d x \left (a+b x^2\right )^{3/2}}{4 b}+\frac{(a (4 b c-a d)) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{8 b}\\ &=\frac{(4 b c-a d) x \sqrt{a+b x^2}}{8 b}+\frac{d x \left (a+b x^2\right )^{3/2}}{4 b}+\frac{(a (4 b c-a d)) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{8 b}\\ &=\frac{(4 b c-a d) x \sqrt{a+b x^2}}{8 b}+\frac{d x \left (a+b x^2\right )^{3/2}}{4 b}+\frac{a (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.154802, size = 85, normalized size = 0.98 \[ \frac{\sqrt{a+b x^2} \left (\sqrt{b} x \left (a d+4 b c+2 b d x^2\right )-\frac{\sqrt{a} (a d-4 b c) \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{\frac{b x^2}{a}+1}}\right )}{8 b^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x^2]*(Sqrt[b]*x*(4*b*c + a*d + 2*b*d*x^2) - (Sqrt[a]*(-4*b*c + a*d)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/
Sqrt[1 + (b*x^2)/a]))/(8*b^(3/2))

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Maple [A]  time = 0.005, size = 96, normalized size = 1.1 \begin{align*}{\frac{dx}{4\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{adx}{8\,b}\sqrt{b{x}^{2}+a}}-{\frac{d{a}^{2}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{cx}{2}\sqrt{b{x}^{2}+a}}+{\frac{ac}{2}\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)^(1/2)*(d*x^2+c),x)

[Out]

1/4*d*x*(b*x^2+a)^(3/2)/b-1/8*d/b*a*x*(b*x^2+a)^(1/2)-1/8*d/b^(3/2)*a^2*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/2*c*x*
(b*x^2+a)^(1/2)+1/2*c*a/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)^(1/2)*(d*x^2+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.6509, size = 371, normalized size = 4.26 \begin{align*} \left [-\frac{{\left (4 \, a b c - a^{2} d\right )} \sqrt{b} \log \left (-2 \, b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (2 \, b^{2} d x^{3} +{\left (4 \, b^{2} c + a b d\right )} x\right )} \sqrt{b x^{2} + a}}{16 \, b^{2}}, -\frac{{\left (4 \, a b c - a^{2} d\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (2 \, b^{2} d x^{3} +{\left (4 \, b^{2} c + a b d\right )} x\right )} \sqrt{b x^{2} + a}}{8 \, b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/16*((4*a*b*c - a^2*d)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(2*b^2*d*x^3 + (4*b^2*c
+ a*b*d)*x)*sqrt(b*x^2 + a))/b^2, -1/8*((4*a*b*c - a^2*d)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (2*b^2
*d*x^3 + (4*b^2*c + a*b*d)*x)*sqrt(b*x^2 + a))/b^2]

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Sympy [A]  time = 5.13809, size = 144, normalized size = 1.66 \begin{align*} \frac{a^{\frac{3}{2}} d x}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{\sqrt{a} c x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{3 \sqrt{a} d x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{a^{2} d \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{3}{2}}} + \frac{a c \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{b}} + \frac{b d x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**(3/2)*d*x/(8*b*sqrt(1 + b*x**2/a)) + sqrt(a)*c*x*sqrt(1 + b*x**2/a)/2 + 3*sqrt(a)*d*x**3/(8*sqrt(1 + b*x**2
/a)) - a**2*d*asinh(sqrt(b)*x/sqrt(a))/(8*b**(3/2)) + a*c*asinh(sqrt(b)*x/sqrt(a))/(2*sqrt(b)) + b*d*x**5/(4*s
qrt(a)*sqrt(1 + b*x**2/a))

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Giac [A]  time = 1.14035, size = 95, normalized size = 1.09 \begin{align*} \frac{1}{8} \, \sqrt{b x^{2} + a}{\left (2 \, d x^{2} + \frac{4 \, b^{2} c + a b d}{b^{2}}\right )} x - \frac{{\left (4 \, a b c - a^{2} d\right )} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, b^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(b*x^2 + a)*(2*d*x^2 + (4*b^2*c + a*b*d)/b^2)*x - 1/8*(4*a*b*c - a^2*d)*log(abs(-sqrt(b)*x + sqrt(b*x^
2 + a)))/b^(3/2)