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

Optimal. Leaf size=147 \[ \frac{2 a^2 (a+b x)^{3/2}}{3 b^3 (a-c)}-\frac{2 c^2 (b x+c)^{3/2}}{3 b^3 (a-c)}+\frac{2 (a+b x)^{7/2}}{7 b^3 (a-c)}-\frac{4 a (a+b x)^{5/2}}{5 b^3 (a-c)}-\frac{2 (b x+c)^{7/2}}{7 b^3 (a-c)}+\frac{4 c (b x+c)^{5/2}}{5 b^3 (a-c)} \]

[Out]

(2*a^2*(a + b*x)^(3/2))/(3*b^3*(a - c)) - (4*a*(a + b*x)^(5/2))/(5*b^3*(a - c)) + (2*(a + b*x)^(7/2))/(7*b^3*(
a - c)) - (2*c^2*(c + b*x)^(3/2))/(3*b^3*(a - c)) + (4*c*(c + b*x)^(5/2))/(5*b^3*(a - c)) - (2*(c + b*x)^(7/2)
)/(7*b^3*(a - c))

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Rubi [A]  time = 0.132343, antiderivative size = 147, 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 = {2104, 43} \[ \frac{2 a^2 (a+b x)^{3/2}}{3 b^3 (a-c)}-\frac{2 c^2 (b x+c)^{3/2}}{3 b^3 (a-c)}+\frac{2 (a+b x)^{7/2}}{7 b^3 (a-c)}-\frac{4 a (a+b x)^{5/2}}{5 b^3 (a-c)}-\frac{2 (b x+c)^{7/2}}{7 b^3 (a-c)}+\frac{4 c (b x+c)^{5/2}}{5 b^3 (a-c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*(a + b*x)^(3/2))/(3*b^3*(a - c)) - (4*a*(a + b*x)^(5/2))/(5*b^3*(a - c)) + (2*(a + b*x)^(7/2))/(7*b^3*(
a - c)) - (2*c^2*(c + b*x)^(3/2))/(3*b^3*(a - c)) + (4*c*(c + b*x)^(5/2))/(5*b^3*(a - c)) - (2*(c + b*x)^(7/2)
)/(7*b^3*(a - c))

Rule 2104

Int[(u_)/((e_.)*Sqrt[(a_.) + (b_.)*(x_)] + (f_.)*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> -Dist[d/(e*(b*c - a*d
)), Int[u*Sqrt[a + b*x], x], x] + Dist[b/(f*(b*c - a*d)), Int[u*Sqrt[c + d*x], x], x] /; FreeQ[{a, b, c, d, e,
f}, x] && NeQ[b*c - a*d, 0] && EqQ[b*e^2 - d*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{c+b x}} \, dx &=-\frac{b \int x^2 \sqrt{a+b x} \, dx}{-a b+b c}+\frac{b \int x^2 \sqrt{c+b x} \, dx}{-a b+b c}\\ &=-\frac{b \int \left (\frac{a^2 \sqrt{a+b x}}{b^2}-\frac{2 a (a+b x)^{3/2}}{b^2}+\frac{(a+b x)^{5/2}}{b^2}\right ) \, dx}{-a b+b c}+\frac{b \int \left (\frac{c^2 \sqrt{c+b x}}{b^2}-\frac{2 c (c+b x)^{3/2}}{b^2}+\frac{(c+b x)^{5/2}}{b^2}\right ) \, dx}{-a b+b c}\\ &=\frac{2 a^2 (a+b x)^{3/2}}{3 b^3 (a-c)}-\frac{4 a (a+b x)^{5/2}}{5 b^3 (a-c)}+\frac{2 (a+b x)^{7/2}}{7 b^3 (a-c)}-\frac{2 c^2 (c+b x)^{3/2}}{3 b^3 (a-c)}+\frac{4 c (c+b x)^{5/2}}{5 b^3 (a-c)}-\frac{2 (c+b x)^{7/2}}{7 b^3 (a-c)}\\ \end{align*}

Mathematica [A]  time = 0.170526, size = 140, normalized size = 0.95 \[ \frac{2 \left (8 a^3 \sqrt{a+b x}-4 a^2 b x \sqrt{a+b x}+15 b^3 x^3 \left (\sqrt{a+b x}-\sqrt{b x+c}\right )+3 a b^2 x^2 \sqrt{a+b x}-3 b^2 c x^2 \sqrt{b x+c}-8 c^3 \sqrt{b x+c}+4 b c^2 x \sqrt{b x+c}\right )}{105 b^3 (a-c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(8*a^3*Sqrt[a + b*x] - 4*a^2*b*x*Sqrt[a + b*x] + 3*a*b^2*x^2*Sqrt[a + b*x] - 8*c^3*Sqrt[c + b*x] + 4*b*c^2*
x*Sqrt[c + b*x] - 3*b^2*c*x^2*Sqrt[c + b*x] + 15*b^3*x^3*(Sqrt[a + b*x] - Sqrt[c + b*x])))/(105*b^3*(a - c))

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Maple [A]  time = 0.004, size = 90, normalized size = 0.6 \begin{align*} 2\,{\frac{1/7\, \left ( bx+a \right ) ^{7/2}-2/5\,a \left ( bx+a \right ) ^{5/2}+1/3\,{a}^{2} \left ( bx+a \right ) ^{3/2}}{ \left ( a-c \right ){b}^{3}}}-2\,{\frac{1/7\, \left ( bx+c \right ) ^{7/2}-2/5\,c \left ( bx+c \right ) ^{5/2}+1/3\,{c}^{2} \left ( bx+c \right ) ^{3/2}}{ \left ( a-c \right ){b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(a-c)/b^3*(1/7*(b*x+a)^(7/2)-2/5*a*(b*x+a)^(5/2)+1/3*a^2*(b*x+a)^(3/2))-2/(a-c)/b^3*(1/7*(b*x+c)^(7/2)-2/5*c
*(b*x+c)^(5/2)+1/3*c^2*(b*x+c)^(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{b x + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.927076, size = 201, normalized size = 1.37 \begin{align*} \frac{2 \,{\left ({\left (15 \, b^{3} x^{3} + 3 \, a b^{2} x^{2} - 4 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x + a} -{\left (15 \, b^{3} x^{3} + 3 \, b^{2} c x^{2} - 4 \, b c^{2} x + 8 \, c^{3}\right )} \sqrt{b x + c}\right )}}{105 \,{\left (a b^{3} - b^{3} c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*((15*b^3*x^3 + 3*a*b^2*x^2 - 4*a^2*b*x + 8*a^3)*sqrt(b*x + a) - (15*b^3*x^3 + 3*b^2*c*x^2 - 4*b*c^2*x +
8*c^3)*sqrt(b*x + c))/(a*b^3 - b^3*c)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: TypeError