3.5.11 \(\int \frac {x^2}{(\sqrt {a+b x}+\sqrt {c+b x})^3} \, dx\) [411]

Optimal. Leaf size=375 \[ -\frac {8 a^3 (a+b x)^{3/2}}{3 b^3 (a-c)^3}+\frac {2 a^2 (a+3 c) (a+b x)^{3/2}}{3 b^3 (a-c)^3}+\frac {24 a^2 (a+b x)^{5/2}}{5 b^3 (a-c)^3}-\frac {4 a (a+3 c) (a+b x)^{5/2}}{5 b^3 (a-c)^3}-\frac {24 a (a+b x)^{7/2}}{7 b^3 (a-c)^3}+\frac {2 (a+3 c) (a+b x)^{7/2}}{7 b^3 (a-c)^3}+\frac {8 (a+b x)^{9/2}}{9 b^3 (a-c)^3}+\frac {8 c^3 (c+b x)^{3/2}}{3 b^3 (a-c)^3}-\frac {2 c^2 (3 a+c) (c+b x)^{3/2}}{3 b^3 (a-c)^3}-\frac {24 c^2 (c+b x)^{5/2}}{5 b^3 (a-c)^3}+\frac {4 c (3 a+c) (c+b x)^{5/2}}{5 b^3 (a-c)^3}+\frac {24 c (c+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac {2 (3 a+c) (c+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac {8 (c+b x)^{9/2}}{9 b^3 (a-c)^3} \]

[Out]

-8/3*a^3*(b*x+a)^(3/2)/b^3/(a-c)^3+2/3*a^2*(a+3*c)*(b*x+a)^(3/2)/b^3/(a-c)^3+24/5*a^2*(b*x+a)^(5/2)/b^3/(a-c)^
3-4/5*a*(a+3*c)*(b*x+a)^(5/2)/b^3/(a-c)^3-24/7*a*(b*x+a)^(7/2)/b^3/(a-c)^3+2/7*(a+3*c)*(b*x+a)^(7/2)/b^3/(a-c)
^3+8/9*(b*x+a)^(9/2)/b^3/(a-c)^3+8/3*c^3*(b*x+c)^(3/2)/b^3/(a-c)^3-2/3*c^2*(3*a+c)*(b*x+c)^(3/2)/b^3/(a-c)^3-2
4/5*c^2*(b*x+c)^(5/2)/b^3/(a-c)^3+4/5*c*(3*a+c)*(b*x+c)^(5/2)/b^3/(a-c)^3+24/7*c*(b*x+c)^(7/2)/b^3/(a-c)^3-2/7
*(3*a+c)*(b*x+c)^(7/2)/b^3/(a-c)^3-8/9*(b*x+c)^(9/2)/b^3/(a-c)^3

________________________________________________________________________________________

Rubi [A]
time = 0.28, antiderivative size = 375, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6821, 45} \begin {gather*} -\frac {8 a^3 (a+b x)^{3/2}}{3 b^3 (a-c)^3}+\frac {24 a^2 (a+b x)^{5/2}}{5 b^3 (a-c)^3}+\frac {2 a^2 (a+3 c) (a+b x)^{3/2}}{3 b^3 (a-c)^3}+\frac {8 c^3 (b x+c)^{3/2}}{3 b^3 (a-c)^3}-\frac {24 c^2 (b x+c)^{5/2}}{5 b^3 (a-c)^3}-\frac {2 c^2 (3 a+c) (b x+c)^{3/2}}{3 b^3 (a-c)^3}+\frac {8 (a+b x)^{9/2}}{9 b^3 (a-c)^3}+\frac {2 (a+3 c) (a+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac {24 a (a+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac {4 a (a+3 c) (a+b x)^{5/2}}{5 b^3 (a-c)^3}-\frac {8 (b x+c)^{9/2}}{9 b^3 (a-c)^3}+\frac {24 c (b x+c)^{7/2}}{7 b^3 (a-c)^3}-\frac {2 (3 a+c) (b x+c)^{7/2}}{7 b^3 (a-c)^3}+\frac {4 c (3 a+c) (b x+c)^{5/2}}{5 b^3 (a-c)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*a^3*(a + b*x)^(3/2))/(3*b^3*(a - c)^3) + (2*a^2*(a + 3*c)*(a + b*x)^(3/2))/(3*b^3*(a - c)^3) + (24*a^2*(a
+ b*x)^(5/2))/(5*b^3*(a - c)^3) - (4*a*(a + 3*c)*(a + b*x)^(5/2))/(5*b^3*(a - c)^3) - (24*a*(a + b*x)^(7/2))/(
7*b^3*(a - c)^3) + (2*(a + 3*c)*(a + b*x)^(7/2))/(7*b^3*(a - c)^3) + (8*(a + b*x)^(9/2))/(9*b^3*(a - c)^3) + (
8*c^3*(c + b*x)^(3/2))/(3*b^3*(a - c)^3) - (2*c^2*(3*a + c)*(c + b*x)^(3/2))/(3*b^3*(a - c)^3) - (24*c^2*(c +
b*x)^(5/2))/(5*b^3*(a - c)^3) + (4*c*(3*a + c)*(c + b*x)^(5/2))/(5*b^3*(a - c)^3) + (24*c*(c + b*x)^(7/2))/(7*
b^3*(a - c)^3) - (2*(3*a + c)*(c + b*x)^(7/2))/(7*b^3*(a - c)^3) - (8*(c + b*x)^(9/2))/(9*b^3*(a - c)^3)

Rule 45

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])

Rule 6821

Int[(u_.)*((e_.)*Sqrt[(a_.) + (b_.)*(x_)^(n_.)] + (f_.)*Sqrt[(c_.) + (d_.)*(x_)^(n_.)])^(m_), x_Symbol] :> Dis
t[(a*e^2 - c*f^2)^m, Int[ExpandIntegrand[u/(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[b*e^2 - d*f^2, 0]

Rubi steps

\begin {align*} \int \frac {x^2}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx &=\frac {\int \left (a \left (1+\frac {3 c}{a}\right ) x^2 \sqrt {a+b x}+4 b x^3 \sqrt {a+b x}-3 a \left (1+\frac {c}{3 a}\right ) x^2 \sqrt {c+b x}-4 b x^3 \sqrt {c+b x}\right ) \, dx}{(a-c)^3}\\ &=\frac {(4 b) \int x^3 \sqrt {a+b x} \, dx}{(a-c)^3}-\frac {(4 b) \int x^3 \sqrt {c+b x} \, dx}{(a-c)^3}-\frac {(3 a+c) \int x^2 \sqrt {c+b x} \, dx}{(a-c)^3}+\frac {(a+3 c) \int x^2 \sqrt {a+b x} \, dx}{(a-c)^3}\\ &=\frac {(4 b) \int \left (-\frac {a^3 \sqrt {a+b x}}{b^3}+\frac {3 a^2 (a+b x)^{3/2}}{b^3}-\frac {3 a (a+b x)^{5/2}}{b^3}+\frac {(a+b x)^{7/2}}{b^3}\right ) \, dx}{(a-c)^3}-\frac {(4 b) \int \left (-\frac {c^3 \sqrt {c+b x}}{b^3}+\frac {3 c^2 (c+b x)^{3/2}}{b^3}-\frac {3 c (c+b x)^{5/2}}{b^3}+\frac {(c+b x)^{7/2}}{b^3}\right ) \, dx}{(a-c)^3}-\frac {(3 a+c) \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-c)^3}+\frac {(a+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}{(a-c)^3}\\ &=-\frac {8 a^3 (a+b x)^{3/2}}{3 b^3 (a-c)^3}+\frac {2 a^2 (a+3 c) (a+b x)^{3/2}}{3 b^3 (a-c)^3}+\frac {24 a^2 (a+b x)^{5/2}}{5 b^3 (a-c)^3}-\frac {4 a (a+3 c) (a+b x)^{5/2}}{5 b^3 (a-c)^3}-\frac {24 a (a+b x)^{7/2}}{7 b^3 (a-c)^3}+\frac {2 (a+3 c) (a+b x)^{7/2}}{7 b^3 (a-c)^3}+\frac {8 (a+b x)^{9/2}}{9 b^3 (a-c)^3}+\frac {8 c^3 (c+b x)^{3/2}}{3 b^3 (a-c)^3}-\frac {2 c^2 (3 a+c) (c+b x)^{3/2}}{3 b^3 (a-c)^3}-\frac {24 c^2 (c+b x)^{5/2}}{5 b^3 (a-c)^3}+\frac {4 c (3 a+c) (c+b x)^{5/2}}{5 b^3 (a-c)^3}+\frac {24 c (c+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac {2 (3 a+c) (c+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac {8 (c+b x)^{9/2}}{9 b^3 (a-c)^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.78, size = 138, normalized size = 0.37 \begin {gather*} \frac {2 \left ((a+b x)^{3/2} \left (-40 a^3+12 a^2 (6 c+5 b x)-3 a b x (36 c+25 b x)+5 b^2 x^2 (27 c+28 b x)\right )+(c+b x)^{3/2} \left (-9 a \left (8 c^2-12 b c x+15 b^2 x^2\right )+5 \left (8 c^3-12 b c^2 x+15 b^2 c x^2-28 b^3 x^3\right )\right )\right )}{315 b^3 (a-c)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((a + b*x)^(3/2)*(-40*a^3 + 12*a^2*(6*c + 5*b*x) - 3*a*b*x*(36*c + 25*b*x) + 5*b^2*x^2*(27*c + 28*b*x)) + (
c + b*x)^(3/2)*(-9*a*(8*c^2 - 12*b*c*x + 15*b^2*x^2) + 5*(8*c^3 - 12*b*c^2*x + 15*b^2*c*x^2 - 28*b^3*x^3))))/(
315*b^3*(a - c)^3)

________________________________________________________________________________________

Maple [A]
time = 0.05, size = 294, normalized size = 0.78

method	result	size
default	\(\frac {2 a \left (\frac {\left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {2 a \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {a^{2} \left (b x +a \right )^{\frac {3}{2}}}{3}\right )}{\left (a -c \right )^{3} b^{3}}+\frac {6 c \left (\frac {\left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {2 a \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {a^{2} \left (b x +a \right )^{\frac {3}{2}}}{3}\right )}{\left (a -c \right )^{3} b^{3}}-\frac {6 a \left (\frac {\left (b x +c \right )^{\frac {7}{2}}}{7}-\frac {2 \left (b x +c \right )^{\frac {5}{2}} c}{5}+\frac {c^{2} \left (b x +c \right )^{\frac {3}{2}}}{3}\right )}{\left (a -c \right )^{3} b^{3}}-\frac {2 c \left (\frac {\left (b x +c \right )^{\frac {7}{2}}}{7}-\frac {2 \left (b x +c \right )^{\frac {5}{2}} c}{5}+\frac {c^{2} \left (b x +c \right )^{\frac {3}{2}}}{3}\right )}{\left (a -c \right )^{3} b^{3}}+\frac {\frac {8 \left (b x +a \right )^{\frac {9}{2}}}{9}-\frac {24 a \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {24 a^{2} \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {8 a^{3} \left (b x +a \right )^{\frac {3}{2}}}{3}}{\left (a -c \right )^{3} b^{3}}-\frac {8 \left (\frac {\left (b x +c \right )^{\frac {9}{2}}}{9}-\frac {3 c \left (b x +c \right )^{\frac {7}{2}}}{7}+\frac {3 c^{2} \left (b x +c \right )^{\frac {5}{2}}}{5}-\frac {c^{3} \left (b x +c \right )^{\frac {3}{2}}}{3}\right )}{\left (a -c \right )^{3} b^{3}}\)	\(294\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(a-c)^3*a/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/(a-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/(a-c)^3*a/b^3*(1/7*(b*x+c)^(7/2)-2/5*(b*x+c)^(5/2)*c+1/3*c^2*(
b*x+c)^(3/2))-2/(a-c)^3*c/b^3*(1/7*(b*x+c)^(7/2)-2/5*(b*x+c)^(5/2)*c+1/3*c^2*(b*x+c)^(3/2))+8/(a-c)^3/b^3*(1/9
*(b*x+a)^(9/2)-3/7*a*(b*x+a)^(7/2)+3/5*a^2*(b*x+a)^(5/2)-1/3*a^3*(b*x+a)^(3/2))-8/(a-c)^3/b^3*(1/9*(b*x+c)^(9/
2)-3/7*c*(b*x+c)^(7/2)+3/5*c^2*(b*x+c)^(5/2)-1/3*c^3*(b*x+c)^(3/2))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.33, size = 208, normalized size = 0.55 \begin {gather*} \frac {2 \, {\left ({\left (140 \, b^{4} x^{4} - 40 \, a^{4} + 72 \, a^{3} c + 5 \, {\left (13 \, a b^{3} + 27 \, b^{3} c\right )} x^{3} - 3 \, {\left (5 \, a^{2} b^{2} - 9 \, a b^{2} c\right )} x^{2} + 4 \, {\left (5 \, a^{3} b - 9 \, a^{2} b c\right )} x\right )} \sqrt {b x + a} - {\left (140 \, b^{4} x^{4} + 72 \, a c^{3} - 40 \, c^{4} + 5 \, {\left (27 \, a b^{3} + 13 \, b^{3} c\right )} x^{3} + 3 \, {\left (9 \, a b^{2} c - 5 \, b^{2} c^{2}\right )} x^{2} - 4 \, {\left (9 \, a b c^{2} - 5 \, b c^{3}\right )} x\right )} \sqrt {b x + c}\right )}}{315 \, {\left (a^{3} b^{3} - 3 \, a^{2} b^{3} c + 3 \, a b^{3} c^{2} - b^{3} c^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*((140*b^4*x^4 - 40*a^4 + 72*a^3*c + 5*(13*a*b^3 + 27*b^3*c)*x^3 - 3*(5*a^2*b^2 - 9*a*b^2*c)*x^2 + 4*(5*a
^3*b - 9*a^2*b*c)*x)*sqrt(b*x + a) - (140*b^4*x^4 + 72*a*c^3 - 40*c^4 + 5*(27*a*b^3 + 13*b^3*c)*x^3 + 3*(9*a*b
^2*c - 5*b^2*c^2)*x^2 - 4*(9*a*b*c^2 - 5*b*c^3)*x)*sqrt(b*x + c))/(a^3*b^3 - 3*a^2*b^3*c + 3*a*b^3*c^2 - b^3*c
^3)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1447 vs. \(2 (319) = 638\).
time = 5.58, size = 1447, normalized size = 3.86 \begin {gather*} -\frac {2}{315} \, {\left ({\left ({\left (5 \, {\left (b x + a\right )} {\left (\frac {28 \, {\left (a^{9} b^{12} - 9 \, a^{8} b^{12} c + 36 \, a^{7} b^{12} c^{2} - 84 \, a^{6} b^{12} c^{3} + 126 \, a^{5} b^{12} c^{4} - 126 \, a^{4} b^{12} c^{5} + 84 \, a^{3} b^{12} c^{6} - 36 \, a^{2} b^{12} c^{7} + 9 \, a b^{12} c^{8} - b^{12} c^{9}\right )} {\left (b x + a\right )}}{a^{12} b^{15} - 12 \, a^{11} b^{15} c + 66 \, a^{10} b^{15} c^{2} - 220 \, a^{9} b^{15} c^{3} + 495 \, a^{8} b^{15} c^{4} - 792 \, a^{7} b^{15} c^{5} + 924 \, a^{6} b^{15} c^{6} - 792 \, a^{5} b^{15} c^{7} + 495 \, a^{4} b^{15} c^{8} - 220 \, a^{3} b^{15} c^{9} + 66 \, a^{2} b^{15} c^{10} - 12 \, a b^{15} c^{11} + b^{15} c^{12}} - \frac {85 \, a^{10} b^{12} - 778 \, a^{9} b^{12} c + 3177 \, a^{8} b^{12} c^{2} - 7608 \, a^{7} b^{12} c^{3} + 11802 \, a^{6} b^{12} c^{4} - 12348 \, a^{5} b^{12} c^{5} + 8778 \, a^{4} b^{12} c^{6} - 4152 \, a^{3} b^{12} c^{7} + 1233 \, a^{2} b^{12} c^{8} - 202 \, a b^{12} c^{9} + 13 \, b^{12} c^{10}}{a^{12} b^{15} - 12 \, a^{11} b^{15} c + 66 \, a^{10} b^{15} c^{2} - 220 \, a^{9} b^{15} c^{3} + 495 \, a^{8} b^{15} c^{4} - 792 \, a^{7} b^{15} c^{5} + 924 \, a^{6} b^{15} c^{6} - 792 \, a^{5} b^{15} c^{7} + 495 \, a^{4} b^{15} c^{8} - 220 \, a^{3} b^{15} c^{9} + 66 \, a^{2} b^{15} c^{10} - 12 \, a b^{15} c^{11} + b^{15} c^{12}}\right )} + \frac {3 \, {\left (145 \, a^{11} b^{12} - 1361 \, a^{10} b^{12} c + 5719 \, a^{9} b^{12} c^{2} - 14151 \, a^{8} b^{12} c^{3} + 22794 \, a^{7} b^{12} c^{4} - 24906 \, a^{6} b^{12} c^{5} + 18606 \, a^{5} b^{12} c^{6} - 9294 \, a^{4} b^{12} c^{7} + 2901 \, a^{3} b^{12} c^{8} - 469 \, a^{2} b^{12} c^{9} + 11 \, a b^{12} c^{10} + 5 \, b^{12} c^{11}\right )}}{a^{12} b^{15} - 12 \, a^{11} b^{15} c + 66 \, a^{10} b^{15} c^{2} - 220 \, a^{9} b^{15} c^{3} + 495 \, a^{8} b^{15} c^{4} - 792 \, a^{7} b^{15} c^{5} + 924 \, a^{6} b^{15} c^{6} - 792 \, a^{5} b^{15} c^{7} + 495 \, a^{4} b^{15} c^{8} - 220 \, a^{3} b^{15} c^{9} + 66 \, a^{2} b^{15} c^{10} - 12 \, a b^{15} c^{11} + b^{15} c^{12}}\right )} {\left (b x + a\right )} - \frac {155 \, a^{12} b^{12} - 1536 \, a^{11} b^{12} c + 6855 \, a^{10} b^{12} c^{2} - 18170 \, a^{9} b^{12} c^{3} + 31770 \, a^{8} b^{12} c^{4} - 38520 \, a^{7} b^{12} c^{5} + 33222 \, a^{6} b^{12} c^{6} - 20700 \, a^{5} b^{12} c^{7} + 9495 \, a^{4} b^{12} c^{8} - 3320 \, a^{3} b^{12} c^{9} + 915 \, a^{2} b^{12} c^{10} - 186 \, a b^{12} c^{11} + 20 \, b^{12} c^{12}}{a^{12} b^{15} - 12 \, a^{11} b^{15} c + 66 \, a^{10} b^{15} c^{2} - 220 \, a^{9} b^{15} c^{3} + 495 \, a^{8} b^{15} c^{4} - 792 \, a^{7} b^{15} c^{5} + 924 \, a^{6} b^{15} c^{6} - 792 \, a^{5} b^{15} c^{7} + 495 \, a^{4} b^{15} c^{8} - 220 \, a^{3} b^{15} c^{9} + 66 \, a^{2} b^{15} c^{10} - 12 \, a b^{15} c^{11} + b^{15} c^{12}}\right )} {\left (b x + a\right )} + \frac {5 \, a^{13} b^{12} - 83 \, a^{12} b^{12} c + 543 \, a^{11} b^{12} c^{2} - 1925 \, a^{10} b^{12} c^{3} + 4070 \, a^{9} b^{12} c^{4} - 4950 \, a^{8} b^{12} c^{5} + 2046 \, a^{7} b^{12} c^{6} + 3894 \, a^{6} b^{12} c^{7} - 8415 \, a^{5} b^{12} c^{8} + 8305 \, a^{4} b^{12} c^{9} - 5005 \, a^{3} b^{12} c^{10} + 1887 \, a^{2} b^{12} c^{11} - 412 \, a b^{12} c^{12} + 40 \, b^{12} c^{13}}{a^{12} b^{15} - 12 \, a^{11} b^{15} c + 66 \, a^{10} b^{15} c^{2} - 220 \, a^{9} b^{15} c^{3} + 495 \, a^{8} b^{15} c^{4} - 792 \, a^{7} b^{15} c^{5} + 924 \, a^{6} b^{15} c^{6} - 792 \, a^{5} b^{15} c^{7} + 495 \, a^{4} b^{15} c^{8} - 220 \, a^{3} b^{15} c^{9} + 66 \, a^{2} b^{15} c^{10} - 12 \, a b^{15} c^{11} + b^{15} c^{12}}\right )} \sqrt {b x + c} + \frac {2 \, {\left (140 \, {\left (b x + a\right )}^{\frac {9}{2}} - 495 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 630 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 315 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 135 \, {\left (b x + a\right )}^{\frac {7}{2}} c - 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a c + 315 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} c\right )}}{315 \, {\left (a^{3} b^{3} - 3 \, a^{2} b^{3} c + 3 \, a b^{3} c^{2} - b^{3} c^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(((5*(b*x + a)*(28*(a^9*b^12 - 9*a^8*b^12*c + 36*a^7*b^12*c^2 - 84*a^6*b^12*c^3 + 126*a^5*b^12*c^4 - 12
6*a^4*b^12*c^5 + 84*a^3*b^12*c^6 - 36*a^2*b^12*c^7 + 9*a*b^12*c^8 - b^12*c^9)*(b*x + a)/(a^12*b^15 - 12*a^11*b
^15*c + 66*a^10*b^15*c^2 - 220*a^9*b^15*c^3 + 495*a^8*b^15*c^4 - 792*a^7*b^15*c^5 + 924*a^6*b^15*c^6 - 792*a^5
*b^15*c^7 + 495*a^4*b^15*c^8 - 220*a^3*b^15*c^9 + 66*a^2*b^15*c^10 - 12*a*b^15*c^11 + b^15*c^12) - (85*a^10*b^
12 - 778*a^9*b^12*c + 3177*a^8*b^12*c^2 - 7608*a^7*b^12*c^3 + 11802*a^6*b^12*c^4 - 12348*a^5*b^12*c^5 + 8778*a
^4*b^12*c^6 - 4152*a^3*b^12*c^7 + 1233*a^2*b^12*c^8 - 202*a*b^12*c^9 + 13*b^12*c^10)/(a^12*b^15 - 12*a^11*b^15
*c + 66*a^10*b^15*c^2 - 220*a^9*b^15*c^3 + 495*a^8*b^15*c^4 - 792*a^7*b^15*c^5 + 924*a^6*b^15*c^6 - 792*a^5*b^
15*c^7 + 495*a^4*b^15*c^8 - 220*a^3*b^15*c^9 + 66*a^2*b^15*c^10 - 12*a*b^15*c^11 + b^15*c^12)) + 3*(145*a^11*b
^12 - 1361*a^10*b^12*c + 5719*a^9*b^12*c^2 - 14151*a^8*b^12*c^3 + 22794*a^7*b^12*c^4 - 24906*a^6*b^12*c^5 + 18
606*a^5*b^12*c^6 - 9294*a^4*b^12*c^7 + 2901*a^3*b^12*c^8 - 469*a^2*b^12*c^9 + 11*a*b^12*c^10 + 5*b^12*c^11)/(a
^12*b^15 - 12*a^11*b^15*c + 66*a^10*b^15*c^2 - 220*a^9*b^15*c^3 + 495*a^8*b^15*c^4 - 792*a^7*b^15*c^5 + 924*a^
6*b^15*c^6 - 792*a^5*b^15*c^7 + 495*a^4*b^15*c^8 - 220*a^3*b^15*c^9 + 66*a^2*b^15*c^10 - 12*a*b^15*c^11 + b^15
*c^12))*(b*x + a) - (155*a^12*b^12 - 1536*a^11*b^12*c + 6855*a^10*b^12*c^2 - 18170*a^9*b^12*c^3 + 31770*a^8*b^
12*c^4 - 38520*a^7*b^12*c^5 + 33222*a^6*b^12*c^6 - 20700*a^5*b^12*c^7 + 9495*a^4*b^12*c^8 - 3320*a^3*b^12*c^9
+ 915*a^2*b^12*c^10 - 186*a*b^12*c^11 + 20*b^12*c^12)/(a^12*b^15 - 12*a^11*b^15*c + 66*a^10*b^15*c^2 - 220*a^9
*b^15*c^3 + 495*a^8*b^15*c^4 - 792*a^7*b^15*c^5 + 924*a^6*b^15*c^6 - 792*a^5*b^15*c^7 + 495*a^4*b^15*c^8 - 220
*a^3*b^15*c^9 + 66*a^2*b^15*c^10 - 12*a*b^15*c^11 + b^15*c^12))*(b*x + a) + (5*a^13*b^12 - 83*a^12*b^12*c + 54
3*a^11*b^12*c^2 - 1925*a^10*b^12*c^3 + 4070*a^9*b^12*c^4 - 4950*a^8*b^12*c^5 + 2046*a^7*b^12*c^6 + 3894*a^6*b^
12*c^7 - 8415*a^5*b^12*c^8 + 8305*a^4*b^12*c^9 - 5005*a^3*b^12*c^10 + 1887*a^2*b^12*c^11 - 412*a*b^12*c^12 + 4
0*b^12*c^13)/(a^12*b^15 - 12*a^11*b^15*c + 66*a^10*b^15*c^2 - 220*a^9*b^15*c^3 + 495*a^8*b^15*c^4 - 792*a^7*b^
15*c^5 + 924*a^6*b^15*c^6 - 792*a^5*b^15*c^7 + 495*a^4*b^15*c^8 - 220*a^3*b^15*c^9 + 66*a^2*b^15*c^10 - 12*a*b
^15*c^11 + b^15*c^12))*sqrt(b*x + c) + 2/315*(140*(b*x + a)^(9/2) - 495*(b*x + a)^(7/2)*a + 630*(b*x + a)^(5/2
)*a^2 - 315*(b*x + a)^(3/2)*a^3 + 135*(b*x + a)^(7/2)*c - 378*(b*x + a)^(5/2)*a*c + 315*(b*x + a)^(3/2)*a^2*c)
/(a^3*b^3 - 3*a^2*b^3*c + 3*a*b^3*c^2 - b^3*c^3)

________________________________________________________________________________________

Mupad [B]
time = 3.30, size = 529, normalized size = 1.41 \begin {gather*} \frac {x^3\,\left (\frac {64\,b\,c}{9\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (3\,a+5\,c\right )}{{\left (a-c\right )}^3}\right )\,\sqrt {c+b\,x}}{7\,b}-\frac {x^3\,\left (\frac {64\,a\,b}{9\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (5\,a+3\,c\right )}{{\left (a-c\right )}^3}\right )\,\sqrt {a+b\,x}}{7\,b}-\frac {8\,c^2\,\left (\frac {2\,c\,\left (3\,a+c\right )}{{\left (a-c\right )}^3}+\frac {6\,c\,\left (\frac {64\,b\,c}{9\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (3\,a+5\,c\right )}{{\left (a-c\right )}^3}\right )}{7\,b}\right )\,\sqrt {c+b\,x}}{15\,b^3}-\frac {x^2\,\left (\frac {2\,c\,\left (3\,a+c\right )}{{\left (a-c\right )}^3}+\frac {6\,c\,\left (\frac {64\,b\,c}{9\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (3\,a+5\,c\right )}{{\left (a-c\right )}^3}\right )}{7\,b}\right )\,\sqrt {c+b\,x}}{5\,b}+\frac {8\,a^2\,\left (\frac {2\,\left (a^2+3\,c\,a\right )}{{\left (a-c\right )}^3}+\frac {6\,a\,\left (\frac {64\,a\,b}{9\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (5\,a+3\,c\right )}{{\left (a-c\right )}^3}\right )}{7\,b}\right )\,\sqrt {a+b\,x}}{15\,b^3}+\frac {x^2\,\left (\frac {2\,\left (a^2+3\,c\,a\right )}{{\left (a-c\right )}^3}+\frac {6\,a\,\left (\frac {64\,a\,b}{9\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (5\,a+3\,c\right )}{{\left (a-c\right )}^3}\right )}{7\,b}\right )\,\sqrt {a+b\,x}}{5\,b}+\frac {8\,b\,x^4\,\sqrt {a+b\,x}}{9\,{\left (a-c\right )}^3}-\frac {8\,b\,x^4\,\sqrt {c+b\,x}}{9\,{\left (a-c\right )}^3}+\frac {4\,c\,x\,\left (\frac {2\,c\,\left (3\,a+c\right )}{{\left (a-c\right )}^3}+\frac {6\,c\,\left (\frac {64\,b\,c}{9\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (3\,a+5\,c\right )}{{\left (a-c\right )}^3}\right )}{7\,b}\right )\,\sqrt {c+b\,x}}{15\,b^2}-\frac {4\,a\,x\,\left (\frac {2\,\left (a^2+3\,c\,a\right )}{{\left (a-c\right )}^3}+\frac {6\,a\,\left (\frac {64\,a\,b}{9\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (5\,a+3\,c\right )}{{\left (a-c\right )}^3}\right )}{7\,b}\right )\,\sqrt {a+b\,x}}{15\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^3*((64*b*c)/(9*(a - c)^3) - (2*b*(3*a + 5*c))/(a - c)^3)*(c + b*x)^(1/2))/(7*b) - (x^3*((64*a*b)/(9*(a - c)
^3) - (2*b*(5*a + 3*c))/(a - c)^3)*(a + b*x)^(1/2))/(7*b) - (8*c^2*((2*c*(3*a + c))/(a - c)^3 + (6*c*((64*b*c)
/(9*(a - c)^3) - (2*b*(3*a + 5*c))/(a - c)^3))/(7*b))*(c + b*x)^(1/2))/(15*b^3) - (x^2*((2*c*(3*a + c))/(a - c
)^3 + (6*c*((64*b*c)/(9*(a - c)^3) - (2*b*(3*a + 5*c))/(a - c)^3))/(7*b))*(c + b*x)^(1/2))/(5*b) + (8*a^2*((2*
(3*a*c + a^2))/(a - c)^3 + (6*a*((64*a*b)/(9*(a - c)^3) - (2*b*(5*a + 3*c))/(a - c)^3))/(7*b))*(a + b*x)^(1/2)
)/(15*b^3) + (x^2*((2*(3*a*c + a^2))/(a - c)^3 + (6*a*((64*a*b)/(9*(a - c)^3) - (2*b*(5*a + 3*c))/(a - c)^3))/
(7*b))*(a + b*x)^(1/2))/(5*b) + (8*b*x^4*(a + b*x)^(1/2))/(9*(a - c)^3) - (8*b*x^4*(c + b*x)^(1/2))/(9*(a - c)
^3) + (4*c*x*((2*c*(3*a + c))/(a - c)^3 + (6*c*((64*b*c)/(9*(a - c)^3) - (2*b*(3*a + 5*c))/(a - c)^3))/(7*b))*
(c + b*x)^(1/2))/(15*b^2) - (4*a*x*((2*(3*a*c + a^2))/(a - c)^3 + (6*a*((64*a*b)/(9*(a - c)^3) - (2*b*(5*a + 3
*c))/(a - c)^3))/(7*b))*(a + b*x)^(1/2))/(15*b^2)

________________________________________________________________________________________