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

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

[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 = 0.37, 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 = {6689, 43} \[ \frac {24 a^2 (a+b x)^{5/2}}{5 b^3 (a-c)^3}+\frac {2 a^2 (a+3 c) (a+b x)^{3/2}}{3 b^3 (a-c)^3}-\frac {8 a^3 (a+b x)^{3/2}}{3 b^3 (a-c)^3}-\frac {24 c^2 (b x+c)^{5/2}}{5 b^3 (a-c)^3}+\frac {8 c^3 (b x+c)^{3/2}}{3 b^3 (a-c)^3}-\frac {2 c^2 (3 a+c) (b x+c)^{3/2}}{3 b^3 (a-c)^3}+\frac {8 (a+b x)^{9/2}}{9 b^3 (a-c)^3}+\frac {2 (a+3 c) (a+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac {24 a (a+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac {4 a (a+3 c) (a+b x)^{5/2}}{5 b^3 (a-c)^3}-\frac {8 (b x+c)^{9/2}}{9 b^3 (a-c)^3}+\frac {24 c (b x+c)^{7/2}}{7 b^3 (a-c)^3}-\frac {2 (3 a+c) (b x+c)^{7/2}}{7 b^3 (a-c)^3}+\frac {4 c (3 a+c) (b x+c)^{5/2}}{5 b^3 (a-c)^3} \]

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 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])

Rule 6689

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 = 0.40, size = 282, normalized size = 0.75 \[ \frac {2 \left (-40 a^4 \sqrt {a+b x}+4 a^3 \sqrt {a+b x} (5 b x+18 c)-3 a^2 b x \sqrt {a+b x} (5 b x+12 c)+a \left (5 b^3 x^3 \left (13 \sqrt {a+b x}-27 \sqrt {b x+c}\right )+27 b^2 c x^2 \left (\sqrt {a+b x}-\sqrt {b x+c}\right )-72 c^3 \sqrt {b x+c}+36 b c^2 x \sqrt {b x+c}\right )+5 \left (28 b^4 x^4 \left (\sqrt {a+b x}-\sqrt {b x+c}\right )+b^3 c x^3 \left (27 \sqrt {a+b x}-13 \sqrt {b x+c}\right )+3 b^2 c^2 x^2 \sqrt {b x+c}+8 c^4 \sqrt {b x+c}-4 b c^3 x \sqrt {b x+c}\right )\right )}{315 b^3 (a-c)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-40*a^4*Sqrt[a + b*x] - 3*a^2*b*x*Sqrt[a + b*x]*(12*c + 5*b*x) + 4*a^3*Sqrt[a + b*x]*(18*c + 5*b*x) + a*(-
72*c^3*Sqrt[c + b*x] + 36*b*c^2*x*Sqrt[c + b*x] + 5*b^3*x^3*(13*Sqrt[a + b*x] - 27*Sqrt[c + b*x]) + 27*b^2*c*x
^2*(Sqrt[a + b*x] - Sqrt[c + b*x])) + 5*(8*c^4*Sqrt[c + b*x] - 4*b*c^3*x*Sqrt[c + b*x] + 3*b^2*c^2*x^2*Sqrt[c
+ b*x] + b^3*c*x^3*(27*Sqrt[a + b*x] - 13*Sqrt[c + b*x]) + 28*b^4*x^4*(Sqrt[a + b*x] - Sqrt[c + b*x]))))/(315*
b^3*(a - c)^3)

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 208, normalized size = 0.55 \[ \frac {2 \, {\left ({\left (140 \, b^{4} x^{4} - 40 \, a^{4} + 72 \, a^{3} c + 5 \, {\left (13 \, a b^{3} + 27 \, b^{3} c\right )} x^{3} - 3 \, {\left (5 \, a^{2} b^{2} - 9 \, a b^{2} c\right )} x^{2} + 4 \, {\left (5 \, a^{3} b - 9 \, a^{2} b c\right )} x\right )} \sqrt {b x + a} - {\left (140 \, b^{4} x^{4} + 72 \, a c^{3} - 40 \, c^{4} + 5 \, {\left (27 \, a b^{3} + 13 \, b^{3} c\right )} x^{3} + 3 \, {\left (9 \, a b^{2} c - 5 \, b^{2} c^{2}\right )} x^{2} - 4 \, {\left (9 \, a b c^{2} - 5 \, b c^{3}\right )} x\right )} \sqrt {b x + c}\right )}}{315 \, {\left (a^{3} b^{3} - 3 \, a^{2} b^{3} c + 3 \, a b^{3} c^{2} - b^{3} c^{3}\right )}} \]

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)

________________________________________________________________________________________

giac [B]  time = 0.81, size = 1447, normalized size = 3.86 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/((b*x+a)^(1/2)+(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)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 294, normalized size = 0.78 \[ \frac {2 \left (\frac {\left (b x +a \right )^{\frac {3}{2}} a^{2}}{3}-\frac {2 \left (b x +a \right )^{\frac {5}{2}} a}{5}+\frac {\left (b x +a \right )^{\frac {7}{2}}}{7}\right ) a}{\left (a -c \right )^{3} b^{3}}-\frac {6 \left (\frac {\left (b x +c \right )^{\frac {3}{2}} c^{2}}{3}-\frac {2 \left (b x +c \right )^{\frac {5}{2}} c}{5}+\frac {\left (b x +c \right )^{\frac {7}{2}}}{7}\right ) a}{\left (a -c \right )^{3} b^{3}}+\frac {6 \left (\frac {\left (b x +a \right )^{\frac {3}{2}} a^{2}}{3}-\frac {2 \left (b x +a \right )^{\frac {5}{2}} a}{5}+\frac {\left (b x +a \right )^{\frac {7}{2}}}{7}\right ) c}{\left (a -c \right )^{3} b^{3}}-\frac {2 \left (\frac {\left (b x +c \right )^{\frac {3}{2}} c^{2}}{3}-\frac {2 \left (b x +c \right )^{\frac {5}{2}} c}{5}+\frac {\left (b x +c \right )^{\frac {7}{2}}}{7}\right ) c}{\left (a -c \right )^{3} b^{3}}+\frac {-\frac {8 \left (b x +a \right )^{\frac {3}{2}} a^{3}}{3}+\frac {24 \left (b x +a \right )^{\frac {5}{2}} a^{2}}{5}-\frac {24 \left (b x +a \right )^{\frac {7}{2}} a}{7}+\frac {8 \left (b x +a \right )^{\frac {9}{2}}}{9}}{\left (a -c \right )^{3} b^{3}}-\frac {8 \left (-\frac {\left (b x +c \right )^{\frac {3}{2}} c^{3}}{3}+\frac {3 \left (b x +c \right )^{\frac {5}{2}} c^{2}}{5}-\frac {3 \left (b x +c \right )^{\frac {7}{2}} c}{7}+\frac {\left (b x +c \right )^{\frac {9}{2}}}{9}\right )}{\left (a -c \right )^{3} b^{3}} \]

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)

[Out]

2/(a-c)^3*a/b^3*(1/3*(b*x+a)^(3/2)*a^2-2/5*(b*x+a)^(5/2)*a+1/7*(b*x+a)^(7/2))+6/(a-c)^3*c/b^3*(1/3*(b*x+a)^(3/
2)*a^2-2/5*(b*x+a)^(5/2)*a+1/7*(b*x+a)^(7/2))-6/(a-c)^3*a/b^3*(1/3*(b*x+c)^(3/2)*c^2-2/5*(b*x+c)^(5/2)*c+1/7*(
b*x+c)^(7/2))-2/(a-c)^3*c/b^3*(1/3*(b*x+c)^(3/2)*c^2-2/5*(b*x+c)^(5/2)*c+1/7*(b*x+c)^(7/2))+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 \[ \int \frac {x^{2}}{{\left (\sqrt {b x + a} + \sqrt {b x + c}\right )}^{3}}\,{d x} \]

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________