3.438 \(\int \frac{x^4}{(\sqrt{a+b x}+\sqrt{a+c x})^3} \, dx\)

Optimal. Leaf size=277 \[ \frac{2 a^2 (b+3 c) (a+b x)^{3/2}}{3 b^3 (b-c)^3}-\frac{8 a^2 (a+b x)^{3/2}}{3 b^2 (b-c)^3}-\frac{2 a^2 (3 b+c) (a+c x)^{3/2}}{3 c^3 (b-c)^3}+\frac{8 a^2 (a+c x)^{3/2}}{3 c^2 (b-c)^3}+\frac{2 (b+3 c) (a+b x)^{7/2}}{7 b^3 (b-c)^3}-\frac{4 a (b+3 c) (a+b x)^{5/2}}{5 b^3 (b-c)^3}+\frac{8 a (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac{2 (3 b+c) (a+c x)^{7/2}}{7 c^3 (b-c)^3}+\frac{4 a (3 b+c) (a+c x)^{5/2}}{5 c^3 (b-c)^3}-\frac{8 a (a+c x)^{5/2}}{5 c^2 (b-c)^3} \]

[Out]

(-8*a^2*(a + b*x)^(3/2))/(3*b^2*(b - c)^3) + (2*a^2*(b + 3*c)*(a + b*x)^(3/2))/(3*b^3*(b - c)^3) + (8*a*(a + b
*x)^(5/2))/(5*b^2*(b - c)^3) - (4*a*(b + 3*c)*(a + b*x)^(5/2))/(5*b^3*(b - c)^3) + (2*(b + 3*c)*(a + b*x)^(7/2
))/(7*b^3*(b - c)^3) + (8*a^2*(a + c*x)^(3/2))/(3*(b - c)^3*c^2) - (2*a^2*(3*b + c)*(a + c*x)^(3/2))/(3*(b - c
)^3*c^3) - (8*a*(a + c*x)^(5/2))/(5*(b - c)^3*c^2) + (4*a*(3*b + c)*(a + c*x)^(5/2))/(5*(b - c)^3*c^3) - (2*(3
*b + c)*(a + c*x)^(7/2))/(7*(b - c)^3*c^3)

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Rubi [A]  time = 0.319216, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6690, 43} \[ \frac{2 a^2 (b+3 c) (a+b x)^{3/2}}{3 b^3 (b-c)^3}-\frac{8 a^2 (a+b x)^{3/2}}{3 b^2 (b-c)^3}-\frac{2 a^2 (3 b+c) (a+c x)^{3/2}}{3 c^3 (b-c)^3}+\frac{8 a^2 (a+c x)^{3/2}}{3 c^2 (b-c)^3}+\frac{2 (b+3 c) (a+b x)^{7/2}}{7 b^3 (b-c)^3}-\frac{4 a (b+3 c) (a+b x)^{5/2}}{5 b^3 (b-c)^3}+\frac{8 a (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac{2 (3 b+c) (a+c x)^{7/2}}{7 c^3 (b-c)^3}+\frac{4 a (3 b+c) (a+c x)^{5/2}}{5 c^3 (b-c)^3}-\frac{8 a (a+c x)^{5/2}}{5 c^2 (b-c)^3} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(Sqrt[a + b*x] + Sqrt[a + c*x])^3,x]

[Out]

(-8*a^2*(a + b*x)^(3/2))/(3*b^2*(b - c)^3) + (2*a^2*(b + 3*c)*(a + b*x)^(3/2))/(3*b^3*(b - c)^3) + (8*a*(a + b
*x)^(5/2))/(5*b^2*(b - c)^3) - (4*a*(b + 3*c)*(a + b*x)^(5/2))/(5*b^3*(b - c)^3) + (2*(b + 3*c)*(a + b*x)^(7/2
))/(7*b^3*(b - c)^3) + (8*a^2*(a + c*x)^(3/2))/(3*(b - c)^3*c^2) - (2*a^2*(3*b + c)*(a + c*x)^(3/2))/(3*(b - c
)^3*c^3) - (8*a*(a + c*x)^(5/2))/(5*(b - c)^3*c^2) + (4*a*(3*b + c)*(a + c*x)^(5/2))/(5*(b - c)^3*c^3) - (2*(3
*b + c)*(a + c*x)^(7/2))/(7*(b - c)^3*c^3)

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^4}{\left (\sqrt{a+b x}+\sqrt{a+c x}\right )^3} \, dx &=\frac{\int \left (4 a x \sqrt{a+b x}+b \left (1+\frac{3 c}{b}\right ) x^2 \sqrt{a+b x}-4 a x \sqrt{a+c x}-3 b \left (1+\frac{c}{3 b}\right ) x^2 \sqrt{a+c x}\right ) \, dx}{(b-c)^3}\\ &=\frac{(4 a) \int x \sqrt{a+b x} \, dx}{(b-c)^3}-\frac{(4 a) \int x \sqrt{a+c x} \, dx}{(b-c)^3}-\frac{(3 b+c) \int x^2 \sqrt{a+c x} \, dx}{(b-c)^3}+\frac{(b+3 c) \int x^2 \sqrt{a+b x} \, dx}{(b-c)^3}\\ &=\frac{(4 a) \int \left (-\frac{a \sqrt{a+b x}}{b}+\frac{(a+b x)^{3/2}}{b}\right ) \, dx}{(b-c)^3}-\frac{(4 a) \int \left (-\frac{a \sqrt{a+c x}}{c}+\frac{(a+c x)^{3/2}}{c}\right ) \, dx}{(b-c)^3}-\frac{(3 b+c) \int \left (\frac{a^2 \sqrt{a+c x}}{c^2}-\frac{2 a (a+c x)^{3/2}}{c^2}+\frac{(a+c x)^{5/2}}{c^2}\right ) \, dx}{(b-c)^3}+\frac{(b+3 c) \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}{(b-c)^3}\\ &=-\frac{8 a^2 (a+b x)^{3/2}}{3 b^2 (b-c)^3}+\frac{2 a^2 (b+3 c) (a+b x)^{3/2}}{3 b^3 (b-c)^3}+\frac{8 a (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac{4 a (b+3 c) (a+b x)^{5/2}}{5 b^3 (b-c)^3}+\frac{2 (b+3 c) (a+b x)^{7/2}}{7 b^3 (b-c)^3}+\frac{8 a^2 (a+c x)^{3/2}}{3 (b-c)^3 c^2}-\frac{2 a^2 (3 b+c) (a+c x)^{3/2}}{3 (b-c)^3 c^3}-\frac{8 a (a+c x)^{5/2}}{5 (b-c)^3 c^2}+\frac{4 a (3 b+c) (a+c x)^{5/2}}{5 (b-c)^3 c^3}-\frac{2 (3 b+c) (a+c x)^{7/2}}{7 (b-c)^3 c^3}\\ \end{align*}

Mathematica [A]  time = 0.389686, size = 271, normalized size = 0.98 \[ \frac{2 \left (8 a^3 \left (b^4 \left (-\sqrt{a+c x}\right )+2 b^3 c \sqrt{a+c x}-2 b c^3 \sqrt{a+b x}+c^4 \sqrt{a+b x}\right )+4 a^2 b c x \left (b^3 \sqrt{a+c x}-2 b^2 c \sqrt{a+c x}+2 b c^2 \sqrt{a+b x}-c^3 \sqrt{a+b x}\right )+a b^2 c^2 x^2 \left (-3 b^2 \sqrt{a+c x}+3 c^2 \sqrt{a+b x}+29 b c \left (\sqrt{a+b x}-\sqrt{a+c x}\right )\right )+5 b^3 c^3 x^3 \left (-3 b \sqrt{a+c x}+3 c \sqrt{a+b x}+b \sqrt{a+b x}-c \sqrt{a+c x}\right )\right )}{35 b^3 c^3 (b-c)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(Sqrt[a + b*x] + Sqrt[a + c*x])^3,x]

[Out]

(2*(5*b^3*c^3*x^3*(b*Sqrt[a + b*x] + 3*c*Sqrt[a + b*x] - 3*b*Sqrt[a + c*x] - c*Sqrt[a + c*x]) + 4*a^2*b*c*x*(2
*b*c^2*Sqrt[a + b*x] - c^3*Sqrt[a + b*x] + b^3*Sqrt[a + c*x] - 2*b^2*c*Sqrt[a + c*x]) + 8*a^3*(-2*b*c^3*Sqrt[a
+ b*x] + c^4*Sqrt[a + b*x] - b^4*Sqrt[a + c*x] + 2*b^3*c*Sqrt[a + c*x]) + a*b^2*c^2*x^2*(3*c^2*Sqrt[a + b*x]
- 3*b^2*Sqrt[a + c*x] + 29*b*c*(Sqrt[a + b*x] - Sqrt[a + c*x]))))/(35*b^3*(b - c)^3*c^3)

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Maple [A]  time = 0.003, size = 246, normalized size = 0.9 \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 ( b-c \right ) ^{3}{b}^{2}}}+8\,{\frac{a \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}}}-8\,{\frac{a \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}}}+6\,{\frac{c \left ( 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} \right ) }{ \left ( b-c \right ) ^{3}{b}^{3}}}-6\,{\frac{b \left ( 1/7\, \left ( cx+a \right ) ^{7/2}-2/5\,a \left ( cx+a \right ) ^{5/2}+1/3\,{a}^{2} \left ( cx+a \right ) ^{3/2} \right ) }{ \left ( b-c \right ) ^{3}{c}^{3}}}-2\,{\frac{1/7\, \left ( cx+a \right ) ^{7/2}-2/5\,a \left ( cx+a \right ) ^{5/2}+1/3\,{a}^{2} \left ( cx+a \right ) ^{3/2}}{ \left ( b-c \right ) ^{3}{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x)

[Out]

2/(b-c)^3/b^2*(1/7*(b*x+a)^(7/2)-2/5*a*(b*x+a)^(5/2)+1/3*a^2*(b*x+a)^(3/2))+8/(b-c)^3*a/b^2*(1/5*(b*x+a)^(5/2)
-1/3*a*(b*x+a)^(3/2))-8/(b-c)^3*a/c^2*(1/5*(c*x+a)^(5/2)-1/3*a*(c*x+a)^(3/2))+6/(b-c)^3*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))-6/(b-c)^3*b/c^3*(1/7*(c*x+a)^(7/2)-2/5*a*(c*x+a)^(5/2)+1/3*a^2*(
c*x+a)^(3/2))-2/(b-c)^3/c^2*(1/7*(c*x+a)^(7/2)-2/5*a*(c*x+a)^(5/2)+1/3*a^2*(c*x+a)^(3/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\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^4/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x, algorithm="maxima")

[Out]

integrate(x^4/(sqrt(b*x + a) + sqrt(c*x + a))^3, x)

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Fricas [A]  time = 1.26517, size = 452, normalized size = 1.63 \begin{align*} -\frac{2 \,{\left ({\left (16 \, a^{3} b c^{3} - 8 \, a^{3} c^{4} - 5 \,{\left (b^{4} c^{3} + 3 \, b^{3} c^{4}\right )} x^{3} -{\left (29 \, a b^{3} c^{3} + 3 \, a b^{2} c^{4}\right )} x^{2} - 4 \,{\left (2 \, a^{2} b^{2} c^{3} - a^{2} b c^{4}\right )} x\right )} \sqrt{b x + a} +{\left (8 \, a^{3} b^{4} - 16 \, a^{3} b^{3} c + 5 \,{\left (3 \, b^{4} c^{3} + b^{3} c^{4}\right )} x^{3} +{\left (3 \, a b^{4} c^{2} + 29 \, a b^{3} c^{3}\right )} x^{2} - 4 \,{\left (a^{2} b^{4} c - 2 \, a^{2} b^{3} c^{2}\right )} x\right )} \sqrt{c x + a}\right )}}{35 \,{\left (b^{6} c^{3} - 3 \, b^{5} c^{4} + 3 \, b^{4} c^{5} - b^{3} c^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x, algorithm="fricas")

[Out]

-2/35*((16*a^3*b*c^3 - 8*a^3*c^4 - 5*(b^4*c^3 + 3*b^3*c^4)*x^3 - (29*a*b^3*c^3 + 3*a*b^2*c^4)*x^2 - 4*(2*a^2*b
^2*c^3 - a^2*b*c^4)*x)*sqrt(b*x + a) + (8*a^3*b^4 - 16*a^3*b^3*c + 5*(3*b^4*c^3 + b^3*c^4)*x^3 + (3*a*b^4*c^2
+ 29*a*b^3*c^3)*x^2 - 4*(a^2*b^4*c - 2*a^2*b^3*c^2)*x)*sqrt(c*x + a))/(b^6*c^3 - 3*b^5*c^4 + 3*b^4*c^5 - b^3*c
^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\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**4/((b*x+a)**(1/2)+(c*x+a)**(1/2))**3,x)

[Out]

Integral(x**4/(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^4/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x, algorithm="giac")

[Out]

Timed out