Optimal. Leaf size=163 \[ \frac {8 a (a+b x)^{3/2}}{3 b (b-c)^3}-\frac {2 a (b+3 c) (a+b x)^{3/2}}{3 b^2 (b-c)^3}+\frac {2 (b+3 c) (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac {8 a (a+c x)^{3/2}}{3 (b-c)^3 c}+\frac {2 a (3 b+c) (a+c x)^{3/2}}{3 (b-c)^3 c^2}-\frac {2 (3 b+c) (a+c x)^{5/2}}{5 (b-c)^3 c^2} \]
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Rubi [A]
time = 0.16, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6822, 45}
\begin {gather*} \frac {2 (b+3 c) (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac {2 a (b+3 c) (a+b x)^{3/2}}{3 b^2 (b-c)^3}-\frac {2 (3 b+c) (a+c x)^{5/2}}{5 c^2 (b-c)^3}+\frac {2 a (3 b+c) (a+c x)^{3/2}}{3 c^2 (b-c)^3}+\frac {8 a (a+b x)^{3/2}}{3 b (b-c)^3}-\frac {8 a (a+c x)^{3/2}}{3 c (b-c)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6822
Rubi steps
\begin {align*} \int \frac {x^3}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx &=\frac {\int \left (4 a \sqrt {a+b x}+b \left (1+\frac {3 c}{b}\right ) x \sqrt {a+b x}-4 a \sqrt {a+c x}-3 b \left (1+\frac {c}{3 b}\right ) x \sqrt {a+c x}\right ) \, dx}{(b-c)^3}\\ &=\frac {8 a (a+b x)^{3/2}}{3 b (b-c)^3}-\frac {8 a (a+c x)^{3/2}}{3 (b-c)^3 c}-\frac {(3 b+c) \int x \sqrt {a+c x} \, dx}{(b-c)^3}+\frac {(b+3 c) \int x \sqrt {a+b x} \, dx}{(b-c)^3}\\ &=\frac {8 a (a+b x)^{3/2}}{3 b (b-c)^3}-\frac {8 a (a+c x)^{3/2}}{3 (b-c)^3 c}-\frac {(3 b+c) \int \left (-\frac {a \sqrt {a+c x}}{c}+\frac {(a+c x)^{3/2}}{c}\right ) \, dx}{(b-c)^3}+\frac {(b+3 c) \int \left (-\frac {a \sqrt {a+b x}}{b}+\frac {(a+b x)^{3/2}}{b}\right ) \, dx}{(b-c)^3}\\ &=\frac {8 a (a+b x)^{3/2}}{3 b (b-c)^3}-\frac {2 a (b+3 c) (a+b x)^{3/2}}{3 b^2 (b-c)^3}+\frac {2 (b+3 c) (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac {8 a (a+c x)^{3/2}}{3 (b-c)^3 c}+\frac {2 a (3 b+c) (a+c x)^{3/2}}{3 (b-c)^3 c^2}-\frac {2 (3 b+c) (a+c x)^{5/2}}{5 (b-c)^3 c^2}\\ \end {align*}
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Mathematica [A]
time = 1.06, size = 197, normalized size = 1.21 \begin {gather*} \frac {2 \sqrt {a-\frac {a b}{c}+\frac {b (a+c x)}{c}} \left (a^2 b^3-4 a^2 b^2 c+5 a^2 b c^2-2 a^2 c^3-2 a b^3 (a+c x)+a b^2 c (a+c x)+a b c^2 (a+c x)+b^3 (a+c x)^2+3 b^2 c (a+c x)^2\right )}{5 b^2 (b-c)^3 c^2}+\frac {2 \left (5 a b (a+c x)^{3/2}-5 a c (a+c x)^{3/2}-3 b (a+c x)^{5/2}-c (a+c x)^{5/2}\right )}{5 (b-c)^3 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 172, normalized size = 1.06
method | result | size |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {2 \left (b x +a \right )^{\frac {3}{2}} a}{3}}{\left (b -c \right )^{3} b}+\frac {8 a \left (b x +a \right )^{\frac {3}{2}}}{3 b \left (b -c \right )^{3}}-\frac {8 a \left (c x +a \right )^{\frac {3}{2}}}{3 \left (b -c \right )^{3} c}+\frac {6 c \left (\frac {\left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {\left (b x +a \right )^{\frac {3}{2}} a}{3}\right )}{\left (b -c \right )^{3} b^{2}}-\frac {6 b \left (\frac {\left (c x +a \right )^{\frac {5}{2}}}{5}-\frac {\left (c x +a \right )^{\frac {3}{2}} a}{3}\right )}{\left (b -c \right )^{3} c^{2}}-\frac {2 \left (\frac {\left (c x +a \right )^{\frac {5}{2}}}{5}-\frac {\left (c x +a \right )^{\frac {3}{2}} a}{3}\right )}{\left (b -c \right )^{3} c}\) | \(172\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A]
time = 0.34, size = 167, normalized size = 1.02 \begin {gather*} \frac {2 \, {\left ({\left (6 \, a^{2} b c^{2} - 2 \, a^{2} c^{3} + {\left (b^{3} c^{2} + 3 \, b^{2} c^{3}\right )} x^{2} + {\left (7 \, a b^{2} c^{2} + a b c^{3}\right )} x\right )} \sqrt {b x + a} + {\left (2 \, a^{2} b^{3} - 6 \, a^{2} b^{2} c - {\left (3 \, b^{3} c^{2} + b^{2} c^{3}\right )} x^{2} - {\left (a b^{3} c + 7 \, a b^{2} c^{2}\right )} x\right )} \sqrt {c x + a}\right )}}{5 \, {\left (b^{5} c^{2} - 3 \, b^{4} c^{3} + 3 \, b^{3} c^{4} - b^{2} c^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\left (\sqrt {a + b x} + \sqrt {a + c x}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 480 vs.
\(2 (139) = 278\).
time = 6.79, size = 480, normalized size = 2.94 \begin {gather*} -\frac {2}{5} \, \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c} {\left ({\left (b x + a\right )} {\left (\frac {{\left (3 \, b^{12} c^{3} {\left | b \right |} - 8 \, b^{11} c^{4} {\left | b \right |} + 6 \, b^{10} c^{5} {\left | b \right |} - b^{8} c^{7} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{18} c^{3} - 6 \, b^{17} c^{4} + 15 \, b^{16} c^{5} - 20 \, b^{15} c^{6} + 15 \, b^{14} c^{7} - 6 \, b^{13} c^{8} + b^{12} c^{9}} + \frac {a b^{13} c^{2} {\left | b \right |} - 2 \, a b^{12} c^{3} {\left | b \right |} - 2 \, a b^{11} c^{4} {\left | b \right |} + 8 \, a b^{10} c^{5} {\left | b \right |} - 7 \, a b^{9} c^{6} {\left | b \right |} + 2 \, a b^{8} c^{7} {\left | b \right |}}{b^{18} c^{3} - 6 \, b^{17} c^{4} + 15 \, b^{16} c^{5} - 20 \, b^{15} c^{6} + 15 \, b^{14} c^{7} - 6 \, b^{13} c^{8} + b^{12} c^{9}}\right )} - \frac {2 \, a^{2} b^{14} c {\left | b \right |} - 11 \, a^{2} b^{13} c^{2} {\left | b \right |} + 25 \, a^{2} b^{12} c^{3} {\left | b \right |} - 30 \, a^{2} b^{11} c^{4} {\left | b \right |} + 20 \, a^{2} b^{10} c^{5} {\left | b \right |} - 7 \, a^{2} b^{9} c^{6} {\left | b \right |} + a^{2} b^{8} c^{7} {\left | b \right |}}{b^{18} c^{3} - 6 \, b^{17} c^{4} + 15 \, b^{16} c^{5} - 20 \, b^{15} c^{6} + 15 \, b^{14} c^{7} - 6 \, b^{13} c^{8} + b^{12} c^{9}}\right )} + \frac {2 \, {\left ({\left (b x + a\right )}^{\frac {5}{2}} b + 5 \, {\left (b x + a\right )}^{\frac {3}{2}} a b + 3 \, {\left (b x + a\right )}^{\frac {5}{2}} c - 5 \, {\left (b x + a\right )}^{\frac {3}{2}} a c\right )}}{5 \, {\left (b^{5} - 3 \, b^{4} c + 3 \, b^{3} c^{2} - b^{2} c^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.36, size = 268, normalized size = 1.64 \begin {gather*} \frac {\left (\frac {8\,a^2}{{\left (b-c\right )}^3}+\frac {2\,a\,\left (\frac {8\,a\,\left (b+3\,c\right )}{5\,{\left (b-c\right )}^3}-\frac {2\,a\,\left (5\,b+3\,c\right )}{{\left (b-c\right )}^3}\right )}{3\,b}\right )\,\sqrt {a+b\,x}}{b}-\frac {\left (\frac {8\,a^2}{{\left (b-c\right )}^3}+\frac {2\,a\,\left (\frac {8\,a\,\left (3\,b+c\right )}{5\,{\left (b-c\right )}^3}-\frac {2\,a\,\left (3\,b+5\,c\right )}{{\left (b-c\right )}^3}\right )}{3\,c}\right )\,\sqrt {a+c\,x}}{c}-\frac {x\,\left (\frac {8\,a\,\left (b+3\,c\right )}{5\,{\left (b-c\right )}^3}-\frac {2\,a\,\left (5\,b+3\,c\right )}{{\left (b-c\right )}^3}\right )\,\sqrt {a+b\,x}}{3\,b}+\frac {x\,\left (\frac {8\,a\,\left (3\,b+c\right )}{5\,{\left (b-c\right )}^3}-\frac {2\,a\,\left (3\,b+5\,c\right )}{{\left (b-c\right )}^3}\right )\,\sqrt {a+c\,x}}{3\,c}+\frac {2\,x^2\,\left (b+3\,c\right )\,\sqrt {a+b\,x}}{5\,{\left (b-c\right )}^3}-\frac {2\,x^2\,\left (3\,b+c\right )\,\sqrt {a+c\,x}}{5\,{\left (b-c\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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