3.402 \(\int \frac{x}{\sqrt{a+b x}+\sqrt{c+b x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0812099, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2104, 43} \[ \frac{2 (a+b x)^{5/2}}{5 b^2 (a-c)}-\frac{2 a (a+b x)^{3/2}}{3 b^2 (a-c)}-\frac{2 (b x+c)^{5/2}}{5 b^2 (a-c)}+\frac{2 c (b x+c)^{3/2}}{3 b^2 (a-c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0979928, size = 95, normalized size = 1. \[ \frac{2 (a+b x)^{5/2}}{5 b^2 (a-c)}-\frac{2 a (a+b x)^{3/2}}{3 b^2 (a-c)}-\frac{2 (b x+c)^{5/2}}{5 b^2 (a-c)}+\frac{2 c (b x+c)^{3/2}}{3 b^2 (a-c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\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/((b*x+a)^(1/2)+(b*x+c)^(1/2)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.940223, size = 149, normalized size = 1.57 \begin{align*} \frac{2 \,{\left ({\left (3 \, b^{2} x^{2} + a b x - 2 \, a^{2}\right )} \sqrt{b x + a} -{\left (3 \, b^{2} x^{2} + b c x - 2 \, c^{2}\right )} \sqrt{b x + c}\right )}}{15 \,{\left (a b^{2} - b^{2} c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: TypeError