3.413 \(\int \frac {1}{(\sqrt {a+b x}+\sqrt {c+b x})^3} \, dx\)

Optimal. Leaf size=64 \[ \frac {(a-c)^2}{10 b \left (\sqrt {a+b x}+\sqrt {b x+c}\right )^5}-\frac {1}{2 b \left (\sqrt {a+b x}+\sqrt {b x+c}\right )} \]

[Out]

1/10*(a-c)^2/b/((b*x+a)^(1/2)+(b*x+c)^(1/2))^5-1/2/b/((b*x+a)^(1/2)+(b*x+c)^(1/2))

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Rubi [B]  time = 0.09, antiderivative size = 151, normalized size of antiderivative = 2.36, number of steps used = 6, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6689, 43} \[ \frac {8 (a+b x)^{5/2}}{5 b (a-c)^3}+\frac {2 (a+3 c) (a+b x)^{3/2}}{3 b (a-c)^3}-\frac {8 a (a+b x)^{3/2}}{3 b (a-c)^3}-\frac {8 (b x+c)^{5/2}}{5 b (a-c)^3}+\frac {8 c (b x+c)^{3/2}}{3 b (a-c)^3}-\frac {2 (3 a+c) (b x+c)^{3/2}}{3 b (a-c)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [B]  time = 0.15, size = 151, normalized size = 2.36 \[ \frac {8 (a+b x)^{5/2}}{5 b (a-c)^3}+\frac {2 (a+3 c) (a+b x)^{3/2}}{3 b (a-c)^3}-\frac {8 a (a+b x)^{3/2}}{3 b (a-c)^3}-\frac {8 (b x+c)^{5/2}}{5 b (a-c)^3}+\frac {8 c (b x+c)^{3/2}}{3 b (a-c)^3}-\frac {2 (3 a+c) (b x+c)^{3/2}}{3 b (a-c)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.44, size = 106, normalized size = 1.66 \[ \frac {2 \, {\left ({\left (4 \, b^{2} x^{2} - a^{2} + 5 \, a c + {\left (3 \, a b + 5 \, b c\right )} x\right )} \sqrt {b x + a} - {\left (4 \, b^{2} x^{2} + 5 \, a c - c^{2} + {\left (5 \, a b + 3 \, b c\right )} x\right )} \sqrt {b x + c}\right )}}{5 \, {\left (a^{3} b - 3 \, a^{2} b c + 3 \, a b c^{2} - b c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.30, size = 427, normalized size = 6.67 \[ -\frac {2}{5} \, {\left ({\left (b x + a\right )} {\left (\frac {4 \, {\left (a^{3} b^{2} - 3 \, a^{2} b^{2} c + 3 \, a b^{2} c^{2} - b^{2} c^{3}\right )} {\left (b x + a\right )}}{a^{6} b^{3} - 6 \, a^{5} b^{3} c + 15 \, a^{4} b^{3} c^{2} - 20 \, a^{3} b^{3} c^{3} + 15 \, a^{2} b^{3} c^{4} - 6 \, a b^{3} c^{5} + b^{3} c^{6}} - \frac {3 \, {\left (a^{4} b^{2} - 4 \, a^{3} b^{2} c + 6 \, a^{2} b^{2} c^{2} - 4 \, a b^{2} c^{3} + b^{2} c^{4}\right )}}{a^{6} b^{3} - 6 \, a^{5} b^{3} c + 15 \, a^{4} b^{3} c^{2} - 20 \, a^{3} b^{3} c^{3} + 15 \, a^{2} b^{3} c^{4} - 6 \, a b^{3} c^{5} + b^{3} c^{6}}\right )} - \frac {a^{5} b^{2} - 5 \, a^{4} b^{2} c + 10 \, a^{3} b^{2} c^{2} - 10 \, a^{2} b^{2} c^{3} + 5 \, a b^{2} c^{4} - b^{2} c^{5}}{a^{6} b^{3} - 6 \, a^{5} b^{3} c + 15 \, a^{4} b^{3} c^{2} - 20 \, a^{3} b^{3} c^{3} + 15 \, a^{2} b^{3} c^{4} - 6 \, a b^{3} c^{5} + b^{3} c^{6}}\right )} \sqrt {b x + c} + \frac {2 \, {\left (4 \, {\left (b x + a\right )}^{\frac {5}{2}} - 5 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 5 \, {\left (b x + a\right )}^{\frac {3}{2}} c\right )}}{5 \, {\left (a^{3} b - 3 \, a^{2} b c + 3 \, a b c^{2} - b c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*((b*x + a)*(4*(a^3*b^2 - 3*a^2*b^2*c + 3*a*b^2*c^2 - b^2*c^3)*(b*x + a)/(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) - 3*(a^4*b^2 - 4*a^3*b^2*c + 6*a^2*b^2*c^2 -
4*a*b^2*c^3 + b^2*c^4)/(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)) - (a^5*b^2 - 5*a^4*b^2*c + 10*a^3*b^2*c^2 - 10*a^2*b^2*c^3 + 5*a*b^2*c^4 - b^2*c^5)/(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))*sqrt(b*x + c) + 2/5*(4
*(b*x + a)^(5/2) - 5*(b*x + a)^(3/2)*a + 5*(b*x + a)^(3/2)*c)/(a^3*b - 3*a^2*b*c + 3*a*b*c^2 - b*c^3)

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maple [B]  time = 0.01, size = 146, normalized size = 2.28 \[ \frac {2 \left (b x +a \right )^{\frac {3}{2}} a}{3 \left (a -c \right )^{3} b}-\frac {2 \left (b x +c \right )^{\frac {3}{2}} a}{\left (a -c \right )^{3} b}+\frac {2 \left (b x +a \right )^{\frac {3}{2}} c}{\left (a -c \right )^{3} b}-\frac {2 \left (b x +c \right )^{\frac {3}{2}} c}{3 \left (a -c \right )^{3} b}+\frac {-\frac {8 \left (b x +a \right )^{\frac {3}{2}} a}{3}+\frac {8 \left (b x +a \right )^{\frac {5}{2}}}{5}}{\left (a -c \right )^{3} b}-\frac {8 \left (-\frac {\left (b x +c \right )^{\frac {3}{2}} c}{3}+\frac {\left (b x +c \right )^{\frac {5}{2}}}{5}\right )}{\left (a -c \right )^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (\sqrt {b x + a} + \sqrt {b x + c}\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.99, size = 252, normalized size = 3.94 \[ \frac {\left (\frac {2\,\left (a^2+3\,c\,a\right )}{{\left (a-c\right )}^3}+\frac {2\,a\,\left (\frac {32\,a\,b}{5\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (5\,a+3\,c\right )}{{\left (a-c\right )}^3}\right )}{3\,b}\right )\,\sqrt {a+b\,x}}{b}-\frac {\left (\frac {2\,c\,\left (3\,a+c\right )}{{\left (a-c\right )}^3}+\frac {2\,c\,\left (\frac {32\,b\,c}{5\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (3\,a+5\,c\right )}{{\left (a-c\right )}^3}\right )}{3\,b}\right )\,\sqrt {c+b\,x}}{b}+\frac {8\,b\,x^2\,\sqrt {a+b\,x}}{5\,{\left (a-c\right )}^3}-\frac {8\,b\,x^2\,\sqrt {c+b\,x}}{5\,{\left (a-c\right )}^3}-\frac {x\,\left (\frac {32\,a\,b}{5\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (5\,a+3\,c\right )}{{\left (a-c\right )}^3}\right )\,\sqrt {a+b\,x}}{3\,b}+\frac {x\,\left (\frac {32\,b\,c}{5\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (3\,a+5\,c\right )}{{\left (a-c\right )}^3}\right )\,\sqrt {c+b\,x}}{3\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((2*(3*a*c + a^2))/(a - c)^3 + (2*a*((32*a*b)/(5*(a - c)^3) - (2*b*(5*a + 3*c))/(a - c)^3))/(3*b))*(a + b*x)^
(1/2))/b - (((2*c*(3*a + c))/(a - c)^3 + (2*c*((32*b*c)/(5*(a - c)^3) - (2*b*(3*a + 5*c))/(a - c)^3))/(3*b))*(
c + b*x)^(1/2))/b + (8*b*x^2*(a + b*x)^(1/2))/(5*(a - c)^3) - (8*b*x^2*(c + b*x)^(1/2))/(5*(a - c)^3) - (x*((3
2*a*b)/(5*(a - c)^3) - (2*b*(5*a + 3*c))/(a - c)^3)*(a + b*x)^(1/2))/(3*b) + (x*((32*b*c)/(5*(a - c)^3) - (2*b
*(3*a + 5*c))/(a - c)^3)*(c + b*x)^(1/2))/(3*b)

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sympy [A]  time = 1.82, size = 384, normalized size = 6.00 \[ \begin {cases} - \frac {2 a}{5 a b \sqrt {a + b x} + 15 a b \sqrt {b x + c} + 20 b^{2} x \sqrt {a + b x} + 20 b^{2} x \sqrt {b x + c} + 15 b c \sqrt {a + b x} + 5 b c \sqrt {b x + c}} - \frac {4 b x}{5 a b \sqrt {a + b x} + 15 a b \sqrt {b x + c} + 20 b^{2} x \sqrt {a + b x} + 20 b^{2} x \sqrt {b x + c} + 15 b c \sqrt {a + b x} + 5 b c \sqrt {b x + c}} - \frac {2 c}{5 a b \sqrt {a + b x} + 15 a b \sqrt {b x + c} + 20 b^{2} x \sqrt {a + b x} + 20 b^{2} x \sqrt {b x + c} + 15 b c \sqrt {a + b x} + 5 b c \sqrt {b x + c}} - \frac {6 \sqrt {a + b x} \sqrt {b x + c}}{5 a b \sqrt {a + b x} + 15 a b \sqrt {b x + c} + 20 b^{2} x \sqrt {a + b x} + 20 b^{2} x \sqrt {b x + c} + 15 b c \sqrt {a + b x} + 5 b c \sqrt {b x + c}} & \text {for}\: b \neq 0 \\\frac {x}{\left (\sqrt {a} + \sqrt {c}\right )^{3}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*a/(5*a*b*sqrt(a + b*x) + 15*a*b*sqrt(b*x + c) + 20*b**2*x*sqrt(a + b*x) + 20*b**2*x*sqrt(b*x + c
) + 15*b*c*sqrt(a + b*x) + 5*b*c*sqrt(b*x + c)) - 4*b*x/(5*a*b*sqrt(a + b*x) + 15*a*b*sqrt(b*x + c) + 20*b**2*
x*sqrt(a + b*x) + 20*b**2*x*sqrt(b*x + c) + 15*b*c*sqrt(a + b*x) + 5*b*c*sqrt(b*x + c)) - 2*c/(5*a*b*sqrt(a +
b*x) + 15*a*b*sqrt(b*x + c) + 20*b**2*x*sqrt(a + b*x) + 20*b**2*x*sqrt(b*x + c) + 15*b*c*sqrt(a + b*x) + 5*b*c
*sqrt(b*x + c)) - 6*sqrt(a + b*x)*sqrt(b*x + c)/(5*a*b*sqrt(a + b*x) + 15*a*b*sqrt(b*x + c) + 20*b**2*x*sqrt(a
+ b*x) + 20*b**2*x*sqrt(b*x + c) + 15*b*c*sqrt(a + b*x) + 5*b*c*sqrt(b*x + c)), Ne(b, 0)), (x/(sqrt(a) + sqrt
(c))**3, True))

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