3.439 \(\int \frac{x^3}{(\sqrt{a+b x}+\sqrt{a+c x})^3} \, dx\)

Optimal. Leaf size=163 \[ \frac{2 (b+3 c) (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac{2 a (b+3 c) (a+b x)^{3/2}}{3 b^2 (b-c)^3}-\frac{2 (3 b+c) (a+c x)^{5/2}}{5 c^2 (b-c)^3}+\frac{2 a (3 b+c) (a+c x)^{3/2}}{3 c^2 (b-c)^3}+\frac{8 a (a+b x)^{3/2}}{3 b (b-c)^3}-\frac{8 a (a+c x)^{3/2}}{3 c (b-c)^3} \]

[Out]

(8*a*(a + b*x)^(3/2))/(3*b*(b - c)^3) - (2*a*(b + 3*c)*(a + b*x)^(3/2))/(3*b^2*(b - c)^3) + (2*(b + 3*c)*(a +
b*x)^(5/2))/(5*b^2*(b - c)^3) - (8*a*(a + c*x)^(3/2))/(3*(b - c)^3*c) + (2*a*(3*b + c)*(a + c*x)^(3/2))/(3*(b
- c)^3*c^2) - (2*(3*b + c)*(a + c*x)^(5/2))/(5*(b - c)^3*c^2)

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Rubi [A]  time = 0.217549, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6690, 43} \[ \frac{2 (b+3 c) (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac{2 a (b+3 c) (a+b x)^{3/2}}{3 b^2 (b-c)^3}-\frac{2 (3 b+c) (a+c x)^{5/2}}{5 c^2 (b-c)^3}+\frac{2 a (3 b+c) (a+c x)^{3/2}}{3 c^2 (b-c)^3}+\frac{8 a (a+b x)^{3/2}}{3 b (b-c)^3}-\frac{8 a (a+c x)^{3/2}}{3 c (b-c)^3} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(Sqrt[a + b*x] + Sqrt[a + c*x])^3,x]

[Out]

(8*a*(a + b*x)^(3/2))/(3*b*(b - c)^3) - (2*a*(b + 3*c)*(a + b*x)^(3/2))/(3*b^2*(b - c)^3) + (2*(b + 3*c)*(a +
b*x)^(5/2))/(5*b^2*(b - c)^3) - (8*a*(a + c*x)^(3/2))/(3*(b - c)^3*c) + (2*a*(3*b + c)*(a + c*x)^(3/2))/(3*(b
- c)^3*c^2) - (2*(3*b + c)*(a + c*x)^(5/2))/(5*(b - c)^3*c^2)

Rule 6690

Int[(u_.)*((e_.)*Sqrt[(a_.) + (b_.)*(x_)^(n_.)] + (f_.)*Sqrt[(c_.) + (d_.)*(x_)^(n_.)])^(m_), x_Symbol] :> Dis
t[(b*e^2 - d*f^2)^m, Int[ExpandIntegrand[(u*x^(m*n))/(e*Sqrt[a + b*x^n] - f*Sqrt[c + d*x^n])^m, x], x], x] /;
FreeQ[{a, b, c, d, e, f, n}, x] && ILtQ[m, 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^3}{\left (\sqrt{a+b x}+\sqrt{a+c x}\right )^3} \, dx &=\frac{\int \left (4 a \sqrt{a+b x}+b \left (1+\frac{3 c}{b}\right ) x \sqrt{a+b x}-4 a \sqrt{a+c x}-3 b \left (1+\frac{c}{3 b}\right ) x \sqrt{a+c x}\right ) \, dx}{(b-c)^3}\\ &=\frac{8 a (a+b x)^{3/2}}{3 b (b-c)^3}-\frac{8 a (a+c x)^{3/2}}{3 (b-c)^3 c}-\frac{(3 b+c) \int x \sqrt{a+c x} \, dx}{(b-c)^3}+\frac{(b+3 c) \int x \sqrt{a+b x} \, dx}{(b-c)^3}\\ &=\frac{8 a (a+b x)^{3/2}}{3 b (b-c)^3}-\frac{8 a (a+c x)^{3/2}}{3 (b-c)^3 c}-\frac{(3 b+c) \int \left (-\frac{a \sqrt{a+c x}}{c}+\frac{(a+c x)^{3/2}}{c}\right ) \, dx}{(b-c)^3}+\frac{(b+3 c) \int \left (-\frac{a \sqrt{a+b x}}{b}+\frac{(a+b x)^{3/2}}{b}\right ) \, dx}{(b-c)^3}\\ &=\frac{8 a (a+b x)^{3/2}}{3 b (b-c)^3}-\frac{2 a (b+3 c) (a+b x)^{3/2}}{3 b^2 (b-c)^3}+\frac{2 (b+3 c) (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac{8 a (a+c x)^{3/2}}{3 (b-c)^3 c}+\frac{2 a (3 b+c) (a+c x)^{3/2}}{3 (b-c)^3 c^2}-\frac{2 (3 b+c) (a+c x)^{5/2}}{5 (b-c)^3 c^2}\\ \end{align*}

Mathematica [A]  time = 0.435509, size = 120, normalized size = 0.74 \[ \frac{2 \left (\frac{3 (b+3 c) (a+b x)^{5/2}}{b^2}-\frac{5 a (b+3 c) (a+b x)^{3/2}}{b^2}-\frac{3 (3 b+c) (a+c x)^{5/2}}{c^2}+\frac{5 a (3 b+c) (a+c x)^{3/2}}{c^2}+\frac{20 a (a+b x)^{3/2}}{b}-\frac{20 a (a+c x)^{3/2}}{c}\right )}{15 (b-c)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(Sqrt[a + b*x] + Sqrt[a + c*x])^3,x]

[Out]

(2*((20*a*(a + b*x)^(3/2))/b - (5*a*(b + 3*c)*(a + b*x)^(3/2))/b^2 + (3*(b + 3*c)*(a + b*x)^(5/2))/b^2 - (20*a
*(a + c*x)^(3/2))/c + (5*a*(3*b + c)*(a + c*x)^(3/2))/c^2 - (3*(3*b + c)*(a + c*x)^(5/2))/c^2))/(15*(b - c)^3)

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Maple [A]  time = 0.003, size = 172, normalized size = 1.1 \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 ) ^{3}b}}+{\frac{8\,a}{3\, \left ( b-c \right ) ^{3}b} \left ( bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{8\,a}{3\, \left ( b-c \right ) ^{3}c} \left ( cx+a \right ) ^{{\frac{3}{2}}}}+6\,{\frac{c \left ( 1/5\, \left ( bx+a \right ) ^{5/2}-1/3\,a \left ( bx+a \right ) ^{3/2} \right ) }{ \left ( b-c \right ) ^{3}{b}^{2}}}-6\,{\frac{b \left ( 1/5\, \left ( cx+a \right ) ^{5/2}-1/3\,a \left ( cx+a \right ) ^{3/2} \right ) }{ \left ( b-c \right ) ^{3}{c}^{2}}}-2\,{\frac{1/5\, \left ( cx+a \right ) ^{5/2}-1/3\,a \left ( cx+a \right ) ^{3/2}}{ \left ( b-c \right ) ^{3}c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x)

[Out]

2/(b-c)^3/b*(1/5*(b*x+a)^(5/2)-1/3*a*(b*x+a)^(3/2))+8/3*a*(b*x+a)^(3/2)/b/(b-c)^3-8/3*a*(c*x+a)^(3/2)/(b-c)^3/
c+6/(b-c)^3*c/b^2*(1/5*(b*x+a)^(5/2)-1/3*a*(b*x+a)^(3/2))-6/(b-c)^3*b/c^2*(1/5*(c*x+a)^(5/2)-1/3*a*(c*x+a)^(3/
2))-2/(b-c)^3/c*(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^{3}}{{\left (\sqrt{b x + a} + \sqrt{c x + a}\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x, algorithm="maxima")

[Out]

integrate(x^3/(sqrt(b*x + a) + sqrt(c*x + a))^3, x)

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Fricas [A]  time = 1.30071, size = 331, normalized size = 2.03 \begin{align*} \frac{2 \,{\left ({\left (6 \, a^{2} b c^{2} - 2 \, a^{2} c^{3} +{\left (b^{3} c^{2} + 3 \, b^{2} c^{3}\right )} x^{2} +{\left (7 \, a b^{2} c^{2} + a b c^{3}\right )} x\right )} \sqrt{b x + a} +{\left (2 \, a^{2} b^{3} - 6 \, a^{2} b^{2} c -{\left (3 \, b^{3} c^{2} + b^{2} c^{3}\right )} x^{2} -{\left (a b^{3} c + 7 \, a b^{2} c^{2}\right )} x\right )} \sqrt{c x + a}\right )}}{5 \,{\left (b^{5} c^{2} - 3 \, b^{4} c^{3} + 3 \, b^{3} c^{4} - b^{2} c^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x, algorithm="fricas")

[Out]

2/5*((6*a^2*b*c^2 - 2*a^2*c^3 + (b^3*c^2 + 3*b^2*c^3)*x^2 + (7*a*b^2*c^2 + a*b*c^3)*x)*sqrt(b*x + a) + (2*a^2*
b^3 - 6*a^2*b^2*c - (3*b^3*c^2 + b^2*c^3)*x^2 - (a*b^3*c + 7*a*b^2*c^2)*x)*sqrt(c*x + a))/(b^5*c^2 - 3*b^4*c^3
+ 3*b^3*c^4 - b^2*c^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\left (\sqrt{a + b x} + \sqrt{a + c x}\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/((b*x+a)**(1/2)+(c*x+a)**(1/2))**3,x)

[Out]

Integral(x**3/(sqrt(a + b*x) + sqrt(a + c*x))**3, x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x, algorithm="giac")

[Out]

Timed out