Optimal. Leaf size=155 \[ \frac {8 a \sqrt {a+b x}}{(b-c)^3}+\frac {2 (b+3 c) (a+b x)^{3/2}}{3 b (b-c)^3}-\frac {8 a \sqrt {a+c x}}{(b-c)^3}-\frac {2 (3 b+c) (a+c x)^{3/2}}{3 (b-c)^3 c}-\frac {8 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{(b-c)^3}+\frac {8 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )}{(b-c)^3} \]
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Rubi [A]
time = 0.14, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6822, 52, 65,
214} \begin {gather*} -\frac {8 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{(b-c)^3}+\frac {8 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )}{(b-c)^3}+\frac {8 a \sqrt {a+b x}}{(b-c)^3}-\frac {8 a \sqrt {a+c x}}{(b-c)^3}+\frac {2 (b+3 c) (a+b x)^{3/2}}{3 b (b-c)^3}-\frac {2 (3 b+c) (a+c x)^{3/2}}{3 c (b-c)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 214
Rule 6822
Rubi steps
\begin {align*} \int \frac {x^2}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx &=\frac {\int \left (b \left (1+\frac {3 c}{b}\right ) \sqrt {a+b x}+\frac {4 a \sqrt {a+b x}}{x}-3 b \left (1+\frac {c}{3 b}\right ) \sqrt {a+c x}-\frac {4 a \sqrt {a+c x}}{x}\right ) \, dx}{(b-c)^3}\\ &=\frac {2 (b+3 c) (a+b x)^{3/2}}{3 b (b-c)^3}-\frac {2 (3 b+c) (a+c x)^{3/2}}{3 (b-c)^3 c}+\frac {(4 a) \int \frac {\sqrt {a+b x}}{x} \, dx}{(b-c)^3}-\frac {(4 a) \int \frac {\sqrt {a+c x}}{x} \, dx}{(b-c)^3}\\ &=\frac {8 a \sqrt {a+b x}}{(b-c)^3}+\frac {2 (b+3 c) (a+b x)^{3/2}}{3 b (b-c)^3}-\frac {8 a \sqrt {a+c x}}{(b-c)^3}-\frac {2 (3 b+c) (a+c x)^{3/2}}{3 (b-c)^3 c}+\frac {\left (4 a^2\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{(b-c)^3}-\frac {\left (4 a^2\right ) \int \frac {1}{x \sqrt {a+c x}} \, dx}{(b-c)^3}\\ &=\frac {8 a \sqrt {a+b x}}{(b-c)^3}+\frac {2 (b+3 c) (a+b x)^{3/2}}{3 b (b-c)^3}-\frac {8 a \sqrt {a+c x}}{(b-c)^3}-\frac {2 (3 b+c) (a+c x)^{3/2}}{3 (b-c)^3 c}+\frac {\left (8 a^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{b (b-c)^3}-\frac {\left (8 a^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x}\right )}{(b-c)^3 c}\\ &=\frac {8 a \sqrt {a+b x}}{(b-c)^3}+\frac {2 (b+3 c) (a+b x)^{3/2}}{3 b (b-c)^3}-\frac {8 a \sqrt {a+c x}}{(b-c)^3}-\frac {2 (3 b+c) (a+c x)^{3/2}}{3 (b-c)^3 c}-\frac {8 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{(b-c)^3}+\frac {8 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )}{(b-c)^3}\\ \end {align*}
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Mathematica [A]
time = 10.22, size = 127, normalized size = 0.82 \begin {gather*} -\frac {2 \left (b \sqrt {a+c x} (a (3 b+13 c)+c (3 b+c) x)-c \sqrt {a+b x} (a (13 b+3 c)+b (b+3 c) x)+12 a^{3/2} b c \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )-12 a^{3/2} b c \tanh ^{-1}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )\right )}{3 b (b-c)^3 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 148, normalized size = 0.95
method | result | size |
default | \(\frac {2 \left (b x +a \right )^{\frac {3}{2}}}{3 \left (b -c \right )^{3}}+\frac {2 c \left (b x +a \right )^{\frac {3}{2}}}{\left (b -c \right )^{3} b}-\frac {2 b \left (c x +a \right )^{\frac {3}{2}}}{\left (b -c \right )^{3} c}-\frac {2 \left (c x +a \right )^{\frac {3}{2}}}{3 \left (b -c \right )^{3}}+\frac {4 a \left (2 \sqrt {b x +a}-2 \sqrt {a}\, \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )\right )}{\left (b -c \right )^{3}}-\frac {4 a \left (2 \sqrt {c x +a}-2 \sqrt {a}\, \arctanh \left (\frac {\sqrt {c x +a}}{\sqrt {a}}\right )\right )}{\left (b -c \right )^{3}}\) | \(148\) |
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 = 0.36, size = 321, normalized size = 2.07 \begin {gather*} \left [-\frac {2 \, {\left (6 \, a^{\frac {3}{2}} b c \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 6 \, a^{\frac {3}{2}} b c \log \left (\frac {c x - 2 \, \sqrt {c x + a} \sqrt {a} + 2 \, a}{x}\right ) - {\left (13 \, a b c + 3 \, a c^{2} + {\left (b^{2} c + 3 \, b c^{2}\right )} x\right )} \sqrt {b x + a} + {\left (3 \, a b^{2} + 13 \, a b c + {\left (3 \, b^{2} c + b c^{2}\right )} x\right )} \sqrt {c x + a}\right )}}{3 \, {\left (b^{4} c - 3 \, b^{3} c^{2} + 3 \, b^{2} c^{3} - b c^{4}\right )}}, \frac {2 \, {\left (12 \, \sqrt {-a} a b c \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - 12 \, \sqrt {-a} a b c \arctan \left (\frac {\sqrt {c x + a} \sqrt {-a}}{a}\right ) + {\left (13 \, a b c + 3 \, a c^{2} + {\left (b^{2} c + 3 \, b c^{2}\right )} x\right )} \sqrt {b x + a} - {\left (3 \, a b^{2} + 13 \, a b c + {\left (3 \, b^{2} c + b c^{2}\right )} x\right )} \sqrt {c x + a}\right )}}{3 \, {\left (b^{4} c - 3 \, b^{3} c^{2} + 3 \, b^{2} c^{3} - b c^{4}\right )}}\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^{2}}{\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. 2374 vs.
\(2 (131) = 262\).
time = 5.70, size = 2374, normalized size = 15.32 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.02, size = 762, normalized size = 4.92 \begin {gather*} \frac {4\,a^{3/2}\,b^4-\frac {4\,a^{3/2}\,c^4\,\left (\frac {4\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^3}-\frac {15\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^4}+\frac {24\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^5}+\frac {6\,\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {a+c\,x}-\sqrt {a}}\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^6}\right )}{3}-\frac {4\,a^{3/2}\,b^2\,c^2\,\left (\frac {24\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {a+c\,x}-\sqrt {a}}+\frac {12\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}+\frac {12\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^3}-\frac {15\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^4}+\frac {18\,\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {a+c\,x}-\sqrt {a}}\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}-3\right )}{3}+\frac {4\,a^{3/2}\,b\,c^3\,\left (\frac {6\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}-\frac {12\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^3}+\frac {66\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^4}-\frac {24\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^5}+\frac {18\,\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {a+c\,x}-\sqrt {a}}\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^4}\right )}{3}+\frac {4\,a^{3/2}\,b^3\,c\,\left (6\,\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {a+c\,x}-\sqrt {a}}\right )-\frac {24\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {a+c\,x}-\sqrt {a}}+\frac {6\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}-\frac {4\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^3}+26\right )}{3}}{c\,{\left (b-c\right )}^3\,{\left (b-\frac {c\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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