3.440 \(\int \frac {x^2}{(\sqrt {a+b x}+\sqrt {a+c x})^3} \, dx\)

Optimal. Leaf size=155 \[ -\frac {8 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{(b-c)^3}+\frac {8 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )}{(b-c)^3}+\frac {8 a \sqrt {a+b x}}{(b-c)^3}-\frac {8 a \sqrt {a+c x}}{(b-c)^3}+\frac {2 (b+3 c) (a+b x)^{3/2}}{3 b (b-c)^3}-\frac {2 (3 b+c) (a+c x)^{3/2}}{3 c (b-c)^3} \]

[Out]

2/3*(b+3*c)*(b*x+a)^(3/2)/b/(b-c)^3-2/3*(3*b+c)*(c*x+a)^(3/2)/(b-c)^3/c-8*a^(3/2)*arctanh((b*x+a)^(1/2)/a^(1/2
))/(b-c)^3+8*a^(3/2)*arctanh((c*x+a)^(1/2)/a^(1/2))/(b-c)^3+8*a*(b*x+a)^(1/2)/(b-c)^3-8*a*(c*x+a)^(1/2)/(b-c)^
3

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Rubi [A]  time = 0.20, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6690, 50, 63, 208} \[ -\frac {8 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{(b-c)^3}+\frac {8 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )}{(b-c)^3}+\frac {8 a \sqrt {a+b x}}{(b-c)^3}-\frac {8 a \sqrt {a+c x}}{(b-c)^3}+\frac {2 (b+3 c) (a+b x)^{3/2}}{3 b (b-c)^3}-\frac {2 (3 b+c) (a+c x)^{3/2}}{3 c (b-c)^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(Sqrt[a + b*x] + Sqrt[a + c*x])^3,x]

[Out]

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

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]

Rubi steps

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

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Mathematica [A]  time = 0.27, size = 119, normalized size = 0.77 \[ \frac {2 \left (-12 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+12 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )+\frac {(b+3 c) (a+b x)^{3/2}}{b}-\frac {(3 b+c) (a+c x)^{3/2}}{c}+12 a \sqrt {a+b x}-12 a \sqrt {a+c x}\right )}{3 (b-c)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(Sqrt[a + b*x] + Sqrt[a + c*x])^3,x]

[Out]

(2*(12*a*Sqrt[a + b*x] + ((b + 3*c)*(a + b*x)^(3/2))/b - 12*a*Sqrt[a + c*x] - ((3*b + c)*(a + c*x)^(3/2))/c -
12*a^(3/2)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]] + 12*a^(3/2)*ArcTanh[Sqrt[a + c*x]/Sqrt[a]]))/(3*(b - c)^3)

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fricas [A]  time = 0.46, size = 321, normalized size = 2.07 \[ \left [-\frac {2 \, {\left (6 \, a^{\frac {3}{2}} b c \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 6 \, a^{\frac {3}{2}} b c \log \left (\frac {c x - 2 \, \sqrt {c x + a} \sqrt {a} + 2 \, a}{x}\right ) - {\left (13 \, a b c + 3 \, a c^{2} + {\left (b^{2} c + 3 \, b c^{2}\right )} x\right )} \sqrt {b x + a} + {\left (3 \, a b^{2} + 13 \, a b c + {\left (3 \, b^{2} c + b c^{2}\right )} x\right )} \sqrt {c x + a}\right )}}{3 \, {\left (b^{4} c - 3 \, b^{3} c^{2} + 3 \, b^{2} c^{3} - b c^{4}\right )}}, \frac {2 \, {\left (12 \, \sqrt {-a} a b c \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - 12 \, \sqrt {-a} a b c \arctan \left (\frac {\sqrt {c x + a} \sqrt {-a}}{a}\right ) + {\left (13 \, a b c + 3 \, a c^{2} + {\left (b^{2} c + 3 \, b c^{2}\right )} x\right )} \sqrt {b x + a} - {\left (3 \, a b^{2} + 13 \, a b c + {\left (3 \, b^{2} c + b c^{2}\right )} x\right )} \sqrt {c x + a}\right )}}{3 \, {\left (b^{4} c - 3 \, b^{3} c^{2} + 3 \, b^{2} c^{3} - b c^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-2/3*(6*a^(3/2)*b*c*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 6*a^(3/2)*b*c*log((c*x - 2*sqrt(c*x + a)*s
qrt(a) + 2*a)/x) - (13*a*b*c + 3*a*c^2 + (b^2*c + 3*b*c^2)*x)*sqrt(b*x + a) + (3*a*b^2 + 13*a*b*c + (3*b^2*c +
b*c^2)*x)*sqrt(c*x + a))/(b^4*c - 3*b^3*c^2 + 3*b^2*c^3 - b*c^4), 2/3*(12*sqrt(-a)*a*b*c*arctan(sqrt(b*x + a)
*sqrt(-a)/a) - 12*sqrt(-a)*a*b*c*arctan(sqrt(c*x + a)*sqrt(-a)/a) + (13*a*b*c + 3*a*c^2 + (b^2*c + 3*b*c^2)*x)
*sqrt(b*x + a) - (3*a*b^2 + 13*a*b*c + (3*b^2*c + b*c^2)*x)*sqrt(c*x + a))/(b^4*c - 3*b^3*c^2 + 3*b^2*c^3 - b*
c^4)]

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giac [B]  time = 10.12, size = 2374, normalized size = 15.32 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(a*b^2 + (b*x + a)*b*c - a*b*c)*((3*b^7*c*abs(b) - 8*b^6*c^2*abs(b) + 6*b^5*c^3*abs(b) - b^3*c^5*abs(
b))*(b*x + a)/(b^12*c - 6*b^11*c^2 + 15*b^10*c^3 - 20*b^9*c^4 + 15*b^8*c^5 - 6*b^7*c^6 + b^6*c^7) + (3*a*b^8*a
bs(b) + a*b^7*c*abs(b) - 22*a*b^6*c^2*abs(b) + 30*a*b^5*c^3*abs(b) - 13*a*b^4*c^4*abs(b) + a*b^3*c^5*abs(b))/(
b^12*c - 6*b^11*c^2 + 15*b^10*c^3 - 20*b^9*c^4 + 15*b^8*c^5 - 6*b^7*c^6 + b^6*c^7)) + 8*a^2*arctan(sqrt(b*x +
a)/sqrt(-a))/((b^3 - 3*b^2*c + 3*b*c^2 - c^3)*sqrt(-a)) + 2/3*((b*x + a)^(3/2)*b^9 + 12*sqrt(b*x + a)*a*b^9 -
3*(b*x + a)^(3/2)*b^8*c - 72*sqrt(b*x + a)*a*b^8*c - 3*(b*x + a)^(3/2)*b^7*c^2 + 180*sqrt(b*x + a)*a*b^7*c^2 +
25*(b*x + a)^(3/2)*b^6*c^3 - 240*sqrt(b*x + a)*a*b^6*c^3 - 45*(b*x + a)^(3/2)*b^5*c^4 + 180*sqrt(b*x + a)*a*b
^5*c^4 + 39*(b*x + a)^(3/2)*b^4*c^5 - 72*sqrt(b*x + a)*a*b^4*c^5 - 17*(b*x + a)^(3/2)*b^3*c^6 + 12*sqrt(b*x +
a)*a*b^3*c^6 + 3*(b*x + a)^(3/2)*b^2*c^7)/(b^12 - 9*b^11*c + 36*b^10*c^2 - 84*b^9*c^3 + 126*b^8*c^4 - 126*b^7*
c^5 + 84*b^6*c^6 - 36*b^5*c^7 + 9*b^4*c^8 - b^3*c^9) - 8*(2*(a*b^4 - 3*a*b^3*c + 3*a*b^2*c^2 - a*b*c^3)^2*(a*b
^3*c - a*b^2*c^2)*sqrt(-a)*abs(b)*sgn(b^3 - 3*b^2*c + 3*b*c^2 - c^3) + 2*(a*b^4 - 3*a*b^3*c + 3*a*b^2*c^2 - a*
b*c^3)^2*(a*b^3 - a*b^2*c)*sqrt(-a*b*c)*abs(b) + (a^2*b^7 - 5*a^2*b^6*c + 10*a^2*b^5*c^2 - 10*a^2*b^4*c^3 + 5*
a^2*b^3*c^4 - a^2*b^2*c^5)*sqrt(-a*b*c)*abs(-a*b^4 + 3*a*b^3*c - 3*a*b^2*c^2 + a*b*c^3)*abs(b)*sgn(b^3 - 3*b^2
*c + 3*b*c^2 - c^3) + (a^2*b^8 - 5*a^2*b^7*c + 10*a^2*b^6*c^2 - 10*a^2*b^5*c^3 + 5*a^2*b^4*c^4 - a^2*b^3*c^5)*
sqrt(-a)*abs(-a*b^4 + 3*a*b^3*c - 3*a*b^2*c^2 + a*b*c^3)*abs(b) + (a^3*b^11*c - 6*a^3*b^10*c^2 + 14*a^3*b^9*c^
3 - 14*a^3*b^8*c^4 + 14*a^3*b^6*c^6 - 14*a^3*b^5*c^7 + 6*a^3*b^4*c^8 - a^3*b^3*c^9)*sqrt(-a)*abs(b)*sgn(b^3 -
3*b^2*c + 3*b*c^2 - c^3) + (a^3*b^11 - 6*a^3*b^10*c + 14*a^3*b^9*c^2 - 14*a^3*b^8*c^3 + 14*a^3*b^6*c^5 - 14*a^
3*b^5*c^6 + 6*a^3*b^4*c^7 - a^3*b^3*c^8)*sqrt(-a*b*c)*abs(b))*arctan(-(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 +
(b*x + a)*b*c - a*b*c))/sqrt(-(a*b^5 - 2*a*b^4*c + 2*a*b^2*c^3 - a*b*c^4 + sqrt((a*b^5 - 2*a*b^4*c + 2*a*b^2*c
^3 - a*b*c^4)^2 - (a^2*b^7 - 5*a^2*b^6*c + 10*a^2*b^5*c^2 - 10*a^2*b^4*c^3 + 5*a^2*b^3*c^4 - a^2*b^2*c^5)*(b^3
- 3*b^2*c + 3*b*c^2 - c^3)))/(b^3 - 3*b^2*c + 3*b*c^2 - c^3)))/((b^12 - 9*b^11*c + 36*b^10*c^2 - 84*b^9*c^3 +
126*b^8*c^4 - 126*b^7*c^5 + 84*b^6*c^6 - 36*b^5*c^7 + 9*b^4*c^8 - b^3*c^9)*a*abs(-a*b^4 + 3*a*b^3*c - 3*a*b^2
*c^2 + a*b*c^3)) + 8*(2*(a*b^4 - 3*a*b^3*c + 3*a*b^2*c^2 - a*b*c^3)^2*(a*b^3*c - a*b^2*c^2)*sqrt(-a)*abs(b)*sg
n(b^3 - 3*b^2*c + 3*b*c^2 - c^3) + 2*(a*b^4 - 3*a*b^3*c + 3*a*b^2*c^2 - a*b*c^3)^2*(a*b^3 - a*b^2*c)*sqrt(-a*b
*c)*abs(b) + (a^2*b^7 - 5*a^2*b^6*c + 10*a^2*b^5*c^2 - 10*a^2*b^4*c^3 + 5*a^2*b^3*c^4 - a^2*b^2*c^5)*sqrt(-a*b
*c)*abs(-a*b^4 + 3*a*b^3*c - 3*a*b^2*c^2 + a*b*c^3)*abs(b)*sgn(b^3 - 3*b^2*c + 3*b*c^2 - c^3) + (a^2*b^8 - 5*a
^2*b^7*c + 10*a^2*b^6*c^2 - 10*a^2*b^5*c^3 + 5*a^2*b^4*c^4 - a^2*b^3*c^5)*sqrt(-a)*abs(-a*b^4 + 3*a*b^3*c - 3*
a*b^2*c^2 + a*b*c^3)*abs(b) + (a^3*b^11*c - 6*a^3*b^10*c^2 + 14*a^3*b^9*c^3 - 14*a^3*b^8*c^4 + 14*a^3*b^6*c^6
- 14*a^3*b^5*c^7 + 6*a^3*b^4*c^8 - a^3*b^3*c^9)*sqrt(-a)*abs(b)*sgn(b^3 - 3*b^2*c + 3*b*c^2 - c^3) + (a^3*b^11
- 6*a^3*b^10*c + 14*a^3*b^9*c^2 - 14*a^3*b^8*c^3 + 14*a^3*b^6*c^5 - 14*a^3*b^5*c^6 + 6*a^3*b^4*c^7 - a^3*b^3*
c^8)*sqrt(-a*b*c)*abs(b))*arctan(-(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))/sqrt(-(a*b^5
- 2*a*b^4*c + 2*a*b^2*c^3 - a*b*c^4 - sqrt((a*b^5 - 2*a*b^4*c + 2*a*b^2*c^3 - a*b*c^4)^2 - (a^2*b^7 - 5*a^2*b
^6*c + 10*a^2*b^5*c^2 - 10*a^2*b^4*c^3 + 5*a^2*b^3*c^4 - a^2*b^2*c^5)*(b^3 - 3*b^2*c + 3*b*c^2 - c^3)))/(b^3 -
3*b^2*c + 3*b*c^2 - c^3)))/((b^12 - 9*b^11*c + 36*b^10*c^2 - 84*b^9*c^3 + 126*b^8*c^4 - 126*b^7*c^5 + 84*b^6*
c^6 - 36*b^5*c^7 + 9*b^4*c^8 - b^3*c^9)*a*abs(-a*b^4 + 3*a*b^3*c - 3*a*b^2*c^2 + a*b*c^3))

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maple [A]  time = 0.00, size = 148, normalized size = 0.95 \[ \frac {4 \left (-2 \sqrt {a}\, \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+2 \sqrt {b x +a}\right ) a}{\left (b -c \right )^{3}}-\frac {4 \left (-2 \sqrt {a}\, \arctanh \left (\frac {\sqrt {c x +a}}{\sqrt {a}}\right )+2 \sqrt {c x +a}\right ) a}{\left (b -c \right )^{3}}-\frac {2 \left (c x +a \right )^{\frac {3}{2}} b}{\left (b -c \right )^{3} c}+\frac {2 \left (b x +a \right )^{\frac {3}{2}} c}{\left (b -c \right )^{3} b}+\frac {2 \left (b x +a \right )^{\frac {3}{2}}}{3 \left (b -c \right )^{3}}-\frac {2 \left (c x +a \right )^{\frac {3}{2}}}{3 \left (b -c \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3/(b-c)^3*(b*x+a)^(3/2)+2/(b-c)^3*c*(b*x+a)^(3/2)/b-2/(b-c)^3*b*(c*x+a)^(3/2)/c-2/3/(b-c)^3*(c*x+a)^(3/2)+4/
(b-c)^3*a*(2*(b*x+a)^(1/2)-2*a^(1/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))-4/(b-c)^3*a*(2*(c*x+a)^(1/2)-2*a^(1/2)*ar
ctanh((c*x+a)^(1/2)/a^(1/2)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (\sqrt {b x + a} + \sqrt {c x + a}\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 7.02, size = 762, normalized size = 4.92 \[ \frac {4\,a^{3/2}\,b^4-\frac {4\,a^{3/2}\,c^4\,\left (\frac {4\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^3}-\frac {15\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^4}+\frac {24\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^5}+\frac {6\,\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {a+c\,x}-\sqrt {a}}\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^6}\right )}{3}-\frac {4\,a^{3/2}\,b^2\,c^2\,\left (\frac {24\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {a+c\,x}-\sqrt {a}}+\frac {12\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}+\frac {12\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^3}-\frac {15\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^4}+\frac {18\,\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {a+c\,x}-\sqrt {a}}\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}-3\right )}{3}+\frac {4\,a^{3/2}\,b\,c^3\,\left (\frac {6\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}-\frac {12\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^3}+\frac {66\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^4}-\frac {24\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^5}+\frac {18\,\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {a+c\,x}-\sqrt {a}}\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^4}\right )}{3}+\frac {4\,a^{3/2}\,b^3\,c\,\left (6\,\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {a+c\,x}-\sqrt {a}}\right )-\frac {24\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {a+c\,x}-\sqrt {a}}+\frac {6\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}-\frac {4\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^3}+26\right )}{3}}{c\,{\left (b-c\right )}^3\,{\left (b-\frac {c\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*a^(3/2)*b^4 - (4*a^(3/2)*c^4*((4*((a + b*x)^(1/2) - a^(1/2))^3)/((a + c*x)^(1/2) - a^(1/2))^3 - (15*((a + b
*x)^(1/2) - a^(1/2))^4)/((a + c*x)^(1/2) - a^(1/2))^4 + (24*((a + b*x)^(1/2) - a^(1/2))^5)/((a + c*x)^(1/2) -
a^(1/2))^5 + (6*log(((a + b*x)^(1/2) - a^(1/2))/((a + c*x)^(1/2) - a^(1/2)))*((a + b*x)^(1/2) - a^(1/2))^6)/((
a + c*x)^(1/2) - a^(1/2))^6))/3 - (4*a^(3/2)*b^2*c^2*((24*((a + b*x)^(1/2) - a^(1/2)))/((a + c*x)^(1/2) - a^(1
/2)) + (12*((a + b*x)^(1/2) - a^(1/2))^2)/((a + c*x)^(1/2) - a^(1/2))^2 + (12*((a + b*x)^(1/2) - a^(1/2))^3)/(
(a + c*x)^(1/2) - a^(1/2))^3 - (15*((a + b*x)^(1/2) - a^(1/2))^4)/((a + c*x)^(1/2) - a^(1/2))^4 + (18*log(((a
+ b*x)^(1/2) - a^(1/2))/((a + c*x)^(1/2) - a^(1/2)))*((a + b*x)^(1/2) - a^(1/2))^2)/((a + c*x)^(1/2) - a^(1/2)
)^2 - 3))/3 + (4*a^(3/2)*b*c^3*((6*((a + b*x)^(1/2) - a^(1/2))^2)/((a + c*x)^(1/2) - a^(1/2))^2 - (12*((a + b*
x)^(1/2) - a^(1/2))^3)/((a + c*x)^(1/2) - a^(1/2))^3 + (66*((a + b*x)^(1/2) - a^(1/2))^4)/((a + c*x)^(1/2) - a
^(1/2))^4 - (24*((a + b*x)^(1/2) - a^(1/2))^5)/((a + c*x)^(1/2) - a^(1/2))^5 + (18*log(((a + b*x)^(1/2) - a^(1
/2))/((a + c*x)^(1/2) - a^(1/2)))*((a + b*x)^(1/2) - a^(1/2))^4)/((a + c*x)^(1/2) - a^(1/2))^4))/3 + (4*a^(3/2
)*b^3*c*(6*log(((a + b*x)^(1/2) - a^(1/2))/((a + c*x)^(1/2) - a^(1/2))) - (24*((a + b*x)^(1/2) - a^(1/2)))/((a
+ c*x)^(1/2) - a^(1/2)) + (6*((a + b*x)^(1/2) - a^(1/2))^2)/((a + c*x)^(1/2) - a^(1/2))^2 - (4*((a + b*x)^(1/
2) - a^(1/2))^3)/((a + c*x)^(1/2) - a^(1/2))^3 + 26))/3)/(c*(b - c)^3*(b - (c*((a + b*x)^(1/2) - a^(1/2))^2)/(
(a + c*x)^(1/2) - a^(1/2))^2)^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (\sqrt {a + b x} + \sqrt {a + c x}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/(sqrt(a + b*x) + sqrt(a + c*x))**3, x)

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