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

Optimal. Leaf size=261 \[ \frac {8 a^2 (a+b x)^{3/2}}{3 b^2 (a-c)^3}-\frac {8 c^2 (b x+c)^{3/2}}{3 b^2 (a-c)^3}+\frac {8 (a+b x)^{7/2}}{7 b^2 (a-c)^3}+\frac {2 (a+3 c) (a+b x)^{5/2}}{5 b^2 (a-c)^3}-\frac {16 a (a+b x)^{5/2}}{5 b^2 (a-c)^3}-\frac {2 a (a+3 c) (a+b x)^{3/2}}{3 b^2 (a-c)^3}-\frac {8 (b x+c)^{7/2}}{7 b^2 (a-c)^3}+\frac {16 c (b x+c)^{5/2}}{5 b^2 (a-c)^3}-\frac {2 (3 a+c) (b x+c)^{5/2}}{5 b^2 (a-c)^3}+\frac {2 c (3 a+c) (b x+c)^{3/2}}{3 b^2 (a-c)^3} \]

[Out]

8/3*a^2*(b*x+a)^(3/2)/b^2/(a-c)^3-2/3*a*(a+3*c)*(b*x+a)^(3/2)/b^2/(a-c)^3-16/5*a*(b*x+a)^(5/2)/b^2/(a-c)^3+2/5
*(a+3*c)*(b*x+a)^(5/2)/b^2/(a-c)^3+8/7*(b*x+a)^(7/2)/b^2/(a-c)^3-8/3*c^2*(b*x+c)^(3/2)/b^2/(a-c)^3+2/3*c*(3*a+
c)*(b*x+c)^(3/2)/b^2/(a-c)^3+16/5*c*(b*x+c)^(5/2)/b^2/(a-c)^3-2/5*(3*a+c)*(b*x+c)^(5/2)/b^2/(a-c)^3-8/7*(b*x+c
)^(7/2)/b^2/(a-c)^3

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Rubi [A]  time = 0.24, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6689, 43} \[ \frac {8 a^2 (a+b x)^{3/2}}{3 b^2 (a-c)^3}-\frac {8 c^2 (b x+c)^{3/2}}{3 b^2 (a-c)^3}+\frac {8 (a+b x)^{7/2}}{7 b^2 (a-c)^3}+\frac {2 (a+3 c) (a+b x)^{5/2}}{5 b^2 (a-c)^3}-\frac {16 a (a+b x)^{5/2}}{5 b^2 (a-c)^3}-\frac {2 a (a+3 c) (a+b x)^{3/2}}{3 b^2 (a-c)^3}-\frac {8 (b x+c)^{7/2}}{7 b^2 (a-c)^3}+\frac {16 c (b x+c)^{5/2}}{5 b^2 (a-c)^3}-\frac {2 (3 a+c) (b x+c)^{5/2}}{5 b^2 (a-c)^3}+\frac {2 c (3 a+c) (b x+c)^{3/2}}{3 b^2 (a-c)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*a^2*(a + b*x)^(3/2))/(3*b^2*(a - c)^3) - (2*a*(a + 3*c)*(a + b*x)^(3/2))/(3*b^2*(a - c)^3) - (16*a*(a + b*x
)^(5/2))/(5*b^2*(a - c)^3) + (2*(a + 3*c)*(a + b*x)^(5/2))/(5*b^2*(a - c)^3) + (8*(a + b*x)^(7/2))/(7*b^2*(a -
c)^3) - (8*c^2*(c + b*x)^(3/2))/(3*b^2*(a - c)^3) + (2*c*(3*a + c)*(c + b*x)^(3/2))/(3*b^2*(a - c)^3) + (16*c
*(c + b*x)^(5/2))/(5*b^2*(a - c)^3) - (2*(3*a + c)*(c + b*x)^(5/2))/(5*b^2*(a - c)^3) - (8*(c + b*x)^(7/2))/(7
*b^2*(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}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx &=\frac {\int \left (a \left (1+\frac {3 c}{a}\right ) x \sqrt {a+b x}+4 b x^2 \sqrt {a+b x}-3 a \left (1+\frac {c}{3 a}\right ) x \sqrt {c+b x}-4 b x^2 \sqrt {c+b x}\right ) \, dx}{(a-c)^3}\\ &=\frac {(4 b) \int x^2 \sqrt {a+b x} \, dx}{(a-c)^3}-\frac {(4 b) \int x^2 \sqrt {c+b x} \, dx}{(a-c)^3}-\frac {(3 a+c) \int x \sqrt {c+b x} \, dx}{(a-c)^3}+\frac {(a+3 c) \int x \sqrt {a+b x} \, dx}{(a-c)^3}\\ &=\frac {(4 b) \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 {(4 b) \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 {(3 a+c) \int \left (-\frac {c \sqrt {c+b x}}{b}+\frac {(c+b x)^{3/2}}{b}\right ) \, dx}{(a-c)^3}+\frac {(a+3 c) \int \left (-\frac {a \sqrt {a+b x}}{b}+\frac {(a+b x)^{3/2}}{b}\right ) \, dx}{(a-c)^3}\\ &=\frac {8 a^2 (a+b x)^{3/2}}{3 b^2 (a-c)^3}-\frac {2 a (a+3 c) (a+b x)^{3/2}}{3 b^2 (a-c)^3}-\frac {16 a (a+b x)^{5/2}}{5 b^2 (a-c)^3}+\frac {2 (a+3 c) (a+b x)^{5/2}}{5 b^2 (a-c)^3}+\frac {8 (a+b x)^{7/2}}{7 b^2 (a-c)^3}-\frac {8 c^2 (c+b x)^{3/2}}{3 b^2 (a-c)^3}+\frac {2 c (3 a+c) (c+b x)^{3/2}}{3 b^2 (a-c)^3}+\frac {16 c (c+b x)^{5/2}}{5 b^2 (a-c)^3}-\frac {2 (3 a+c) (c+b x)^{5/2}}{5 b^2 (a-c)^3}-\frac {8 (c+b x)^{7/2}}{7 b^2 (a-c)^3}\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 214, normalized size = 0.82 \[ \frac {2 \left (6 a^3 \sqrt {a+b x}-a^2 \sqrt {a+b x} (3 b x+14 c)+20 b^3 x^3 \left (\sqrt {a+b x}-\sqrt {b x+c}\right )+a \left (b^2 x^2 \left (11 \sqrt {a+b x}-21 \sqrt {b x+c}\right )+7 b c x \left (\sqrt {a+b x}-\sqrt {b x+c}\right )+14 c^2 \sqrt {b x+c}\right )+b^2 c x^2 \left (21 \sqrt {a+b x}-11 \sqrt {b x+c}\right )-6 c^3 \sqrt {b x+c}+3 b c^2 x \sqrt {b x+c}\right )}{35 b^2 (a-c)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(6*a^3*Sqrt[a + b*x] - 6*c^3*Sqrt[c + b*x] + 3*b*c^2*x*Sqrt[c + b*x] - a^2*Sqrt[a + b*x]*(14*c + 3*b*x) + b
^2*c*x^2*(21*Sqrt[a + b*x] - 11*Sqrt[c + b*x]) + 20*b^3*x^3*(Sqrt[a + b*x] - Sqrt[c + b*x]) + a*(14*c^2*Sqrt[c
+ b*x] + b^2*x^2*(11*Sqrt[a + b*x] - 21*Sqrt[c + b*x]) + 7*b*c*x*(Sqrt[a + b*x] - Sqrt[c + b*x]))))/(35*b^2*(
a - c)^3)

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fricas [A]  time = 0.45, size = 159, normalized size = 0.61 \[ \frac {2 \, {\left ({\left (20 \, b^{3} x^{3} + 6 \, a^{3} - 14 \, a^{2} c + {\left (11 \, a b^{2} + 21 \, b^{2} c\right )} x^{2} - {\left (3 \, a^{2} b - 7 \, a b c\right )} x\right )} \sqrt {b x + a} - {\left (20 \, b^{3} x^{3} - 14 \, a c^{2} + 6 \, c^{3} + {\left (21 \, a b^{2} + 11 \, b^{2} c\right )} x^{2} + {\left (7 \, a b c - 3 \, b c^{2}\right )} x\right )} \sqrt {b x + c}\right )}}{35 \, {\left (a^{3} b^{2} - 3 \, a^{2} b^{2} c + 3 \, a b^{2} c^{2} - b^{2} c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*((20*b^3*x^3 + 6*a^3 - 14*a^2*c + (11*a*b^2 + 21*b^2*c)*x^2 - (3*a^2*b - 7*a*b*c)*x)*sqrt(b*x + a) - (20*
b^3*x^3 - 14*a*c^2 + 6*c^3 + (21*a*b^2 + 11*b^2*c)*x^2 + (7*a*b*c - 3*b*c^2)*x)*sqrt(b*x + c))/(a^3*b^2 - 3*a^
2*b^2*c + 3*a*b^2*c^2 - b^2*c^3)

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giac [B]  time = 0.38, size = 866, normalized size = 3.32 \[ -\frac {2 \, {\left ({\left ({\left ({\left (b x + a\right )} {\left (\frac {20 \, {\left (a^{6} b^{3} - 6 \, a^{5} b^{3} c + 15 \, a^{4} b^{3} c^{2} - 20 \, a^{3} b^{3} c^{3} + 15 \, a^{2} b^{3} c^{4} - 6 \, a b^{3} c^{5} + b^{3} c^{6}\right )} {\left (b x + a\right )}}{a^{9} b^{4} - 9 \, a^{8} b^{4} c + 36 \, a^{7} b^{4} c^{2} - 84 \, a^{6} b^{4} c^{3} + 126 \, a^{5} b^{4} c^{4} - 126 \, a^{4} b^{4} c^{5} + 84 \, a^{3} b^{4} c^{6} - 36 \, a^{2} b^{4} c^{7} + 9 \, a b^{4} c^{8} - b^{4} c^{9}} - \frac {39 \, a^{7} b^{3} - 245 \, a^{6} b^{3} c + 651 \, a^{5} b^{3} c^{2} - 945 \, a^{4} b^{3} c^{3} + 805 \, a^{3} b^{3} c^{4} - 399 \, a^{2} b^{3} c^{5} + 105 \, a b^{3} c^{6} - 11 \, b^{3} c^{7}}{a^{9} b^{4} - 9 \, a^{8} b^{4} c + 36 \, a^{7} b^{4} c^{2} - 84 \, a^{6} b^{4} c^{3} + 126 \, a^{5} b^{4} c^{4} - 126 \, a^{4} b^{4} c^{5} + 84 \, a^{3} b^{4} c^{6} - 36 \, a^{2} b^{4} c^{7} + 9 \, a b^{4} c^{8} - b^{4} c^{9}}\right )} + \frac {3 \, {\left (6 \, a^{8} b^{3} - 41 \, a^{7} b^{3} c + 119 \, a^{6} b^{3} c^{2} - 189 \, a^{5} b^{3} c^{3} + 175 \, a^{4} b^{3} c^{4} - 91 \, a^{3} b^{3} c^{5} + 21 \, a^{2} b^{3} c^{6} + a b^{3} c^{7} - b^{3} c^{8}\right )}}{a^{9} b^{4} - 9 \, a^{8} b^{4} c + 36 \, a^{7} b^{4} c^{2} - 84 \, a^{6} b^{4} c^{3} + 126 \, a^{5} b^{4} c^{4} - 126 \, a^{4} b^{4} c^{5} + 84 \, a^{3} b^{4} c^{6} - 36 \, a^{2} b^{4} c^{7} + 9 \, a b^{4} c^{8} - b^{4} c^{9}}\right )} {\left (b x + a\right )} + \frac {a^{9} b^{3} - 2 \, a^{8} b^{3} c - 20 \, a^{7} b^{3} c^{2} + 112 \, a^{6} b^{3} c^{3} - 266 \, a^{5} b^{3} c^{4} + 364 \, a^{4} b^{3} c^{5} - 308 \, a^{3} b^{3} c^{6} + 160 \, a^{2} b^{3} c^{7} - 47 \, a b^{3} c^{8} + 6 \, b^{3} c^{9}}{a^{9} b^{4} - 9 \, a^{8} b^{4} c + 36 \, a^{7} b^{4} c^{2} - 84 \, a^{6} b^{4} c^{3} + 126 \, a^{5} b^{4} c^{4} - 126 \, a^{4} b^{4} c^{5} + 84 \, a^{3} b^{4} c^{6} - 36 \, a^{2} b^{4} c^{7} + 9 \, a b^{4} c^{8} - b^{4} c^{9}}\right )} \sqrt {b x + c} - \frac {20 \, {\left (b x + a\right )}^{\frac {7}{2}} - 49 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} + 21 \, {\left (b x + a\right )}^{\frac {5}{2}} c - 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a c}{a^{3} b - 3 \, a^{2} b c + 3 \, a b c^{2} - b c^{3}}\right )}}{35 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*((((b*x + a)*(20*(a^6*b^3 - 6*a^5*b^3*c + 15*a^4*b^3*c^2 - 20*a^3*b^3*c^3 + 15*a^2*b^3*c^4 - 6*a*b^3*c^5
+ b^3*c^6)*(b*x + a)/(a^9*b^4 - 9*a^8*b^4*c + 36*a^7*b^4*c^2 - 84*a^6*b^4*c^3 + 126*a^5*b^4*c^4 - 126*a^4*b^4
*c^5 + 84*a^3*b^4*c^6 - 36*a^2*b^4*c^7 + 9*a*b^4*c^8 - b^4*c^9) - (39*a^7*b^3 - 245*a^6*b^3*c + 651*a^5*b^3*c^
2 - 945*a^4*b^3*c^3 + 805*a^3*b^3*c^4 - 399*a^2*b^3*c^5 + 105*a*b^3*c^6 - 11*b^3*c^7)/(a^9*b^4 - 9*a^8*b^4*c +
36*a^7*b^4*c^2 - 84*a^6*b^4*c^3 + 126*a^5*b^4*c^4 - 126*a^4*b^4*c^5 + 84*a^3*b^4*c^6 - 36*a^2*b^4*c^7 + 9*a*b
^4*c^8 - b^4*c^9)) + 3*(6*a^8*b^3 - 41*a^7*b^3*c + 119*a^6*b^3*c^2 - 189*a^5*b^3*c^3 + 175*a^4*b^3*c^4 - 91*a^
3*b^3*c^5 + 21*a^2*b^3*c^6 + a*b^3*c^7 - b^3*c^8)/(a^9*b^4 - 9*a^8*b^4*c + 36*a^7*b^4*c^2 - 84*a^6*b^4*c^3 + 1
26*a^5*b^4*c^4 - 126*a^4*b^4*c^5 + 84*a^3*b^4*c^6 - 36*a^2*b^4*c^7 + 9*a*b^4*c^8 - b^4*c^9))*(b*x + a) + (a^9*
b^3 - 2*a^8*b^3*c - 20*a^7*b^3*c^2 + 112*a^6*b^3*c^3 - 266*a^5*b^3*c^4 + 364*a^4*b^3*c^5 - 308*a^3*b^3*c^6 + 1
60*a^2*b^3*c^7 - 47*a*b^3*c^8 + 6*b^3*c^9)/(a^9*b^4 - 9*a^8*b^4*c + 36*a^7*b^4*c^2 - 84*a^6*b^4*c^3 + 126*a^5*
b^4*c^4 - 126*a^4*b^4*c^5 + 84*a^3*b^4*c^6 - 36*a^2*b^4*c^7 + 9*a*b^4*c^8 - b^4*c^9))*sqrt(b*x + c) - (20*(b*x
+ a)^(7/2) - 49*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 + 21*(b*x + a)^(5/2)*c - 35*(b*x + a)^(3/2)*a*c)/(
a^3*b - 3*a^2*b*c + 3*a*b*c^2 - b*c^3))/b

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maple [A]  time = 0.00, size = 222, normalized size = 0.85 \[ \frac {2 \left (-\frac {\left (b x +a \right )^{\frac {3}{2}} a}{3}+\frac {\left (b x +a \right )^{\frac {5}{2}}}{5}\right ) a}{\left (a -c \right )^{3} b^{2}}-\frac {6 \left (-\frac {\left (b x +c \right )^{\frac {3}{2}} c}{3}+\frac {\left (b x +c \right )^{\frac {5}{2}}}{5}\right ) a}{\left (a -c \right )^{3} b^{2}}+\frac {6 \left (-\frac {\left (b x +a \right )^{\frac {3}{2}} a}{3}+\frac {\left (b x +a \right )^{\frac {5}{2}}}{5}\right ) c}{\left (a -c \right )^{3} b^{2}}-\frac {2 \left (-\frac {\left (b x +c \right )^{\frac {3}{2}} c}{3}+\frac {\left (b x +c \right )^{\frac {5}{2}}}{5}\right ) c}{\left (a -c \right )^{3} b^{2}}+\frac {\frac {8 \left (b x +a \right )^{\frac {3}{2}} a^{2}}{3}-\frac {16 \left (b x +a \right )^{\frac {5}{2}} a}{5}+\frac {8 \left (b x +a \right )^{\frac {7}{2}}}{7}}{\left (a -c \right )^{3} b^{2}}-\frac {8 \left (\frac {\left (b x +c \right )^{\frac {3}{2}} c^{2}}{3}-\frac {2 \left (b x +c \right )^{\frac {5}{2}} c}{5}+\frac {\left (b x +c \right )^{\frac {7}{2}}}{7}\right )}{\left (a -c \right )^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.21, size = 385, normalized size = 1.48 \[ \frac {x^2\,\left (\frac {48\,b\,c}{7\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (3\,a+5\,c\right )}{{\left (a-c\right )}^3}\right )\,\sqrt {c+b\,x}}{5\,b}-\frac {x^2\,\left (\frac {48\,a\,b}{7\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (5\,a+3\,c\right )}{{\left (a-c\right )}^3}\right )\,\sqrt {a+b\,x}}{5\,b}-\frac {2\,a\,\left (\frac {2\,a\,\left (a+3\,c\right )}{{\left (a-c\right )}^3}+\frac {4\,a\,\left (\frac {48\,a\,b}{7\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (5\,a+3\,c\right )}{{\left (a-c\right )}^3}\right )}{5\,b}\right )\,\sqrt {a+b\,x}}{3\,b^2}+\frac {8\,b\,x^3\,\sqrt {a+b\,x}}{7\,{\left (a-c\right )}^3}+\frac {2\,c\,\left (\frac {2\,c\,\left (3\,a+c\right )}{{\left (a-c\right )}^3}+\frac {4\,c\,\left (\frac {48\,b\,c}{7\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (3\,a+5\,c\right )}{{\left (a-c\right )}^3}\right )}{5\,b}\right )\,\sqrt {c+b\,x}}{3\,b^2}-\frac {8\,b\,x^3\,\sqrt {c+b\,x}}{7\,{\left (a-c\right )}^3}+\frac {x\,\left (\frac {2\,a\,\left (a+3\,c\right )}{{\left (a-c\right )}^3}+\frac {4\,a\,\left (\frac {48\,a\,b}{7\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (5\,a+3\,c\right )}{{\left (a-c\right )}^3}\right )}{5\,b}\right )\,\sqrt {a+b\,x}}{3\,b}-\frac {x\,\left (\frac {2\,c\,\left (3\,a+c\right )}{{\left (a-c\right )}^3}+\frac {4\,c\,\left (\frac {48\,b\,c}{7\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (3\,a+5\,c\right )}{{\left (a-c\right )}^3}\right )}{5\,b}\right )\,\sqrt {c+b\,x}}{3\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2*((48*b*c)/(7*(a - c)^3) - (2*b*(3*a + 5*c))/(a - c)^3)*(c + b*x)^(1/2))/(5*b) - (x^2*((48*a*b)/(7*(a - c)
^3) - (2*b*(5*a + 3*c))/(a - c)^3)*(a + b*x)^(1/2))/(5*b) - (2*a*((2*a*(a + 3*c))/(a - c)^3 + (4*a*((48*a*b)/(
7*(a - c)^3) - (2*b*(5*a + 3*c))/(a - c)^3))/(5*b))*(a + b*x)^(1/2))/(3*b^2) + (8*b*x^3*(a + b*x)^(1/2))/(7*(a
- c)^3) + (2*c*((2*c*(3*a + c))/(a - c)^3 + (4*c*((48*b*c)/(7*(a - c)^3) - (2*b*(3*a + 5*c))/(a - c)^3))/(5*b
))*(c + b*x)^(1/2))/(3*b^2) - (8*b*x^3*(c + b*x)^(1/2))/(7*(a - c)^3) + (x*((2*a*(a + 3*c))/(a - c)^3 + (4*a*(
(48*a*b)/(7*(a - c)^3) - (2*b*(5*a + 3*c))/(a - c)^3))/(5*b))*(a + b*x)^(1/2))/(3*b) - (x*((2*c*(3*a + c))/(a
- c)^3 + (4*c*((48*b*c)/(7*(a - c)^3) - (2*b*(3*a + 5*c))/(a - c)^3))/(5*b))*(c + b*x)^(1/2))/(3*b)

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sympy [A]  time = 2.70, size = 942, normalized size = 3.61 \[ \begin {cases} \frac {12 a^{2}}{35 a b^{2} \sqrt {a + b x} + 105 a b^{2} \sqrt {b x + c} + 140 b^{3} x \sqrt {a + b x} + 140 b^{3} x \sqrt {b x + c} + 105 b^{2} c \sqrt {a + b x} + 35 b^{2} c \sqrt {b x + c}} + \frac {54 a b x}{35 a b^{2} \sqrt {a + b x} + 105 a b^{2} \sqrt {b x + c} + 140 b^{3} x \sqrt {a + b x} + 140 b^{3} x \sqrt {b x + c} + 105 b^{2} c \sqrt {a + b x} + 35 b^{2} c \sqrt {b x + c}} + \frac {44 a c}{35 a b^{2} \sqrt {a + b x} + 105 a b^{2} \sqrt {b x + c} + 140 b^{3} x \sqrt {a + b x} + 140 b^{3} x \sqrt {b x + c} + 105 b^{2} c \sqrt {a + b x} + 35 b^{2} c \sqrt {b x + c}} + \frac {36 a \sqrt {a + b x} \sqrt {b x + c}}{35 a b^{2} \sqrt {a + b x} + 105 a b^{2} \sqrt {b x + c} + 140 b^{3} x \sqrt {a + b x} + 140 b^{3} x \sqrt {b x + c} + 105 b^{2} c \sqrt {a + b x} + 35 b^{2} c \sqrt {b x + c}} + \frac {40 b^{2} x^{2}}{35 a b^{2} \sqrt {a + b x} + 105 a b^{2} \sqrt {b x + c} + 140 b^{3} x \sqrt {a + b x} + 140 b^{3} x \sqrt {b x + c} + 105 b^{2} c \sqrt {a + b x} + 35 b^{2} c \sqrt {b x + c}} + \frac {54 b c x}{35 a b^{2} \sqrt {a + b x} + 105 a b^{2} \sqrt {b x + c} + 140 b^{3} x \sqrt {a + b x} + 140 b^{3} x \sqrt {b x + c} + 105 b^{2} c \sqrt {a + b x} + 35 b^{2} c \sqrt {b x + c}} + \frac {30 b x \sqrt {a + b x} \sqrt {b x + c}}{35 a b^{2} \sqrt {a + b x} + 105 a b^{2} \sqrt {b x + c} + 140 b^{3} x \sqrt {a + b x} + 140 b^{3} x \sqrt {b x + c} + 105 b^{2} c \sqrt {a + b x} + 35 b^{2} c \sqrt {b x + c}} + \frac {12 c^{2}}{35 a b^{2} \sqrt {a + b x} + 105 a b^{2} \sqrt {b x + c} + 140 b^{3} x \sqrt {a + b x} + 140 b^{3} x \sqrt {b x + c} + 105 b^{2} c \sqrt {a + b x} + 35 b^{2} c \sqrt {b x + c}} + \frac {36 c \sqrt {a + b x} \sqrt {b x + c}}{35 a b^{2} \sqrt {a + b x} + 105 a b^{2} \sqrt {b x + c} + 140 b^{3} x \sqrt {a + b x} + 140 b^{3} x \sqrt {b x + c} + 105 b^{2} c \sqrt {a + b x} + 35 b^{2} c \sqrt {b x + c}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 \left (\sqrt {a} + \sqrt {c}\right )^{3}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((12*a**2/(35*a*b**2*sqrt(a + b*x) + 105*a*b**2*sqrt(b*x + c) + 140*b**3*x*sqrt(a + b*x) + 140*b**3*x
*sqrt(b*x + c) + 105*b**2*c*sqrt(a + b*x) + 35*b**2*c*sqrt(b*x + c)) + 54*a*b*x/(35*a*b**2*sqrt(a + b*x) + 105
*a*b**2*sqrt(b*x + c) + 140*b**3*x*sqrt(a + b*x) + 140*b**3*x*sqrt(b*x + c) + 105*b**2*c*sqrt(a + b*x) + 35*b*
*2*c*sqrt(b*x + c)) + 44*a*c/(35*a*b**2*sqrt(a + b*x) + 105*a*b**2*sqrt(b*x + c) + 140*b**3*x*sqrt(a + b*x) +
140*b**3*x*sqrt(b*x + c) + 105*b**2*c*sqrt(a + b*x) + 35*b**2*c*sqrt(b*x + c)) + 36*a*sqrt(a + b*x)*sqrt(b*x +
c)/(35*a*b**2*sqrt(a + b*x) + 105*a*b**2*sqrt(b*x + c) + 140*b**3*x*sqrt(a + b*x) + 140*b**3*x*sqrt(b*x + c)
+ 105*b**2*c*sqrt(a + b*x) + 35*b**2*c*sqrt(b*x + c)) + 40*b**2*x**2/(35*a*b**2*sqrt(a + b*x) + 105*a*b**2*sqr
t(b*x + c) + 140*b**3*x*sqrt(a + b*x) + 140*b**3*x*sqrt(b*x + c) + 105*b**2*c*sqrt(a + b*x) + 35*b**2*c*sqrt(b
*x + c)) + 54*b*c*x/(35*a*b**2*sqrt(a + b*x) + 105*a*b**2*sqrt(b*x + c) + 140*b**3*x*sqrt(a + b*x) + 140*b**3*
x*sqrt(b*x + c) + 105*b**2*c*sqrt(a + b*x) + 35*b**2*c*sqrt(b*x + c)) + 30*b*x*sqrt(a + b*x)*sqrt(b*x + c)/(35
*a*b**2*sqrt(a + b*x) + 105*a*b**2*sqrt(b*x + c) + 140*b**3*x*sqrt(a + b*x) + 140*b**3*x*sqrt(b*x + c) + 105*b
**2*c*sqrt(a + b*x) + 35*b**2*c*sqrt(b*x + c)) + 12*c**2/(35*a*b**2*sqrt(a + b*x) + 105*a*b**2*sqrt(b*x + c) +
140*b**3*x*sqrt(a + b*x) + 140*b**3*x*sqrt(b*x + c) + 105*b**2*c*sqrt(a + b*x) + 35*b**2*c*sqrt(b*x + c)) + 3
6*c*sqrt(a + b*x)*sqrt(b*x + c)/(35*a*b**2*sqrt(a + b*x) + 105*a*b**2*sqrt(b*x + c) + 140*b**3*x*sqrt(a + b*x)
+ 140*b**3*x*sqrt(b*x + c) + 105*b**2*c*sqrt(a + b*x) + 35*b**2*c*sqrt(b*x + c)), Ne(b, 0)), (x**2/(2*(sqrt(a
) + sqrt(c))**3), True))

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