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

Optimal. Leaf size=95 \[ \frac{2 (a+b x)^{5/2}}{5 b^2 (b-c)}-\frac{2 a (a+b x)^{3/2}}{3 b^2 (b-c)}-\frac{2 (a+c x)^{5/2}}{5 c^2 (b-c)}+\frac{2 a (a+c x)^{3/2}}{3 c^2 (b-c)} \]

[Out]

(-2*a*(a + b*x)^(3/2))/(3*b^2*(b - c)) + (2*(a + b*x)^(5/2))/(5*b^2*(b - c)) + (2*a*(a + c*x)^(3/2))/(3*(b - c
)*c^2) - (2*(a + c*x)^(5/2))/(5*(b - c)*c^2)

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Rubi [A]  time = 0.100534, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2103, 43} \[ \frac{2 (a+b x)^{5/2}}{5 b^2 (b-c)}-\frac{2 a (a+b x)^{3/2}}{3 b^2 (b-c)}-\frac{2 (a+c x)^{5/2}}{5 c^2 (b-c)}+\frac{2 a (a+c x)^{3/2}}{3 c^2 (b-c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*(a + b*x)^(3/2))/(3*b^2*(b - c)) + (2*(a + b*x)^(5/2))/(5*b^2*(b - c)) + (2*a*(a + c*x)^(3/2))/(3*(b - c
)*c^2) - (2*(a + c*x)^(5/2))/(5*(b - c)*c^2)

Rule 2103

Int[(u_)/((e_.)*Sqrt[(a_.) + (b_.)*(x_)] + (f_.)*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[c/(e*(b*c - a*d)
), Int[(u*Sqrt[a + b*x])/x, x], x] - Dist[a/(f*(b*c - a*d)), Int[(u*Sqrt[c + d*x])/x, x], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a*e^2 - c*f^2, 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.180861, size = 70, normalized size = 0.74 \[ \frac{2 \left (\frac{3 (a+b x)^{5/2}}{b^2}-\frac{5 a (a+b x)^{3/2}}{b^2}-\frac{3 (a+c x)^{5/2}}{c^2}+\frac{5 a (a+c x)^{3/2}}{c^2}\right )}{15 (b-c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((-5*a*(a + b*x)^(3/2))/b^2 + (3*(a + b*x)^(5/2))/b^2 + (5*a*(a + c*x)^(3/2))/c^2 - (3*(a + c*x)^(5/2))/c^2
))/(15*(b - c))

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Maple [A]  time = 0.003, size = 66, normalized size = 0.7 \begin{align*} 2\,{\frac{1/5\, \left ( bx+a \right ) ^{5/2}-1/3\,a \left ( bx+a \right ) ^{3/2}}{ \left ( b-c \right ){b}^{2}}}-2\,{\frac{1/5\, \left ( cx+a \right ) ^{5/2}-1/3\,a \left ( cx+a \right ) ^{3/2}}{ \left ( b-c \right ){c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(b-c)/b^2*(1/5*(b*x+a)^(5/2)-1/3*a*(b*x+a)^(3/2))-2/(b-c)/c^2*(1/5*(c*x+a)^(5/2)-1/3*a*(c*x+a)^(3/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{b x + a} + \sqrt{c x + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.10644, size = 186, normalized size = 1.96 \begin{align*} \frac{2 \,{\left ({\left (3 \, b^{2} c^{2} x^{2} + a b c^{2} x - 2 \, a^{2} c^{2}\right )} \sqrt{b x + a} -{\left (3 \, b^{2} c^{2} x^{2} + a b^{2} c x - 2 \, a^{2} b^{2}\right )} \sqrt{c x + a}\right )}}{15 \,{\left (b^{3} c^{2} - b^{2} c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*((3*b^2*c^2*x^2 + a*b*c^2*x - 2*a^2*c^2)*sqrt(b*x + a) - (3*b^2*c^2*x^2 + a*b^2*c*x - 2*a^2*b^2)*sqrt(c*x
+ a))/(b^3*c^2 - b^2*c^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError