Optimal. Leaf size=252 \[ \frac {2 b}{3 a (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 b (b c-3 a d)}{a^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d \left (3 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c (b c-a d)^3 (c+d x)^{3/2}}+\frac {2 d (b c+a d) \left (3 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^4 \sqrt {c+d x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{5/2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {106, 157, 12,
95, 214} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{5/2}}+\frac {2 d \sqrt {a+b x} (a d+b c) \left (3 a^2 d^2-14 a b c d+3 b^2 c^2\right )}{3 a^2 c^2 \sqrt {c+d x} (b c-a d)^4}+\frac {2 d \sqrt {a+b x} \left (-a^2 d^2-10 a b c d+3 b^2 c^2\right )}{3 a^2 c (c+d x)^{3/2} (b c-a d)^3}+\frac {2 b (b c-3 a d)}{a^2 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac {2 b}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 106
Rule 157
Rule 214
Rubi steps
\begin {align*} \int \frac {1}{x (a+b x)^{5/2} (c+d x)^{5/2}} \, dx &=\frac {2 b}{3 a (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 \int \frac {\frac {3}{2} (b c-a d)+3 b d x}{x (a+b x)^{3/2} (c+d x)^{5/2}} \, dx}{3 a (b c-a d)}\\ &=\frac {2 b}{3 a (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 b (b c-3 a d)}{a^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {4 \int \frac {\frac {3}{4} (b c-a d)^2+3 b d (b c-3 a d) x}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{3 a^2 (b c-a d)^2}\\ &=\frac {2 b}{3 a (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 b (b c-3 a d)}{a^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d \left (3 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c (b c-a d)^3 (c+d x)^{3/2}}-\frac {8 \int \frac {-\frac {9}{8} (b c-a d)^3-\frac {3}{4} b d \left (3 b^2 c^2-10 a b c d-a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{9 a^2 c (b c-a d)^3}\\ &=\frac {2 b}{3 a (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 b (b c-3 a d)}{a^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d \left (3 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c (b c-a d)^3 (c+d x)^{3/2}}+\frac {2 d (b c+a d) \left (3 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^4 \sqrt {c+d x}}+\frac {16 \int \frac {9 (b c-a d)^4}{16 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{9 a^2 c^2 (b c-a d)^4}\\ &=\frac {2 b}{3 a (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 b (b c-3 a d)}{a^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d \left (3 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c (b c-a d)^3 (c+d x)^{3/2}}+\frac {2 d (b c+a d) \left (3 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^4 \sqrt {c+d x}}+\frac {\int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{a^2 c^2}\\ &=\frac {2 b}{3 a (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 b (b c-3 a d)}{a^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d \left (3 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c (b c-a d)^3 (c+d x)^{3/2}}+\frac {2 d (b c+a d) \left (3 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^4 \sqrt {c+d x}}+\frac {2 \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{a^2 c^2}\\ &=\frac {2 b}{3 a (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 b (b c-3 a d)}{a^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d \left (3 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c (b c-a d)^3 (c+d x)^{3/2}}+\frac {2 d (b c+a d) \left (3 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^4 \sqrt {c+d x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 200, normalized size = 0.79 \begin {gather*} \frac {2 (a+b x)^{3/2} \left (a^2 c d^4-\frac {12 a^2 b c d^3 (c+d x)}{a+b x}+\frac {3 a^3 d^4 (c+d x)}{a+b x}+\frac {3 b^4 c^3 (c+d x)^2}{(a+b x)^2}-\frac {12 a b^3 c^2 d (c+d x)^2}{(a+b x)^2}+\frac {a b^4 c^2 (c+d x)^3}{(a+b x)^3}\right )}{3 a^2 c^2 (-b c+a d)^4 (c+d x)^{3/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{a^{5/2} c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2032\) vs.
\(2(218)=436\).
time = 0.08, size = 2033, normalized size = 8.07
method | result | size |
default | \(\text {Expression too large to display}\) | \(2033\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1029 vs.
\(2 (218) = 436\).
time = 3.74, size = 2078, normalized size = 8.25 \begin {gather*} \left [\frac {3 \, {\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} + {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \, {\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} + {\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (4 \, a^{2} b^{4} c^{6} - 12 \, a^{3} b^{3} c^{5} d - 12 \, a^{5} b c^{3} d^{3} + 4 \, a^{6} c^{2} d^{4} + {\left (3 \, a b^{5} c^{4} d^{2} - 11 \, a^{2} b^{4} c^{3} d^{3} - 11 \, a^{3} b^{3} c^{2} d^{4} + 3 \, a^{4} b^{2} c d^{5}\right )} x^{3} + 6 \, {\left (a b^{5} c^{5} d - 3 \, a^{2} b^{4} c^{4} d^{2} - 4 \, a^{3} b^{3} c^{3} d^{3} - 3 \, a^{4} b^{2} c^{2} d^{4} + a^{5} b c d^{5}\right )} x^{2} + 3 \, {\left (a b^{5} c^{6} - a^{2} b^{4} c^{5} d - 8 \, a^{3} b^{3} c^{4} d^{2} - 8 \, a^{4} b^{2} c^{3} d^{3} - a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (a^{5} b^{4} c^{9} - 4 \, a^{6} b^{3} c^{8} d + 6 \, a^{7} b^{2} c^{7} d^{2} - 4 \, a^{8} b c^{6} d^{3} + a^{9} c^{5} d^{4} + {\left (a^{3} b^{6} c^{7} d^{2} - 4 \, a^{4} b^{5} c^{6} d^{3} + 6 \, a^{5} b^{4} c^{5} d^{4} - 4 \, a^{6} b^{3} c^{4} d^{5} + a^{7} b^{2} c^{3} d^{6}\right )} x^{4} + 2 \, {\left (a^{3} b^{6} c^{8} d - 3 \, a^{4} b^{5} c^{7} d^{2} + 2 \, a^{5} b^{4} c^{6} d^{3} + 2 \, a^{6} b^{3} c^{5} d^{4} - 3 \, a^{7} b^{2} c^{4} d^{5} + a^{8} b c^{3} d^{6}\right )} x^{3} + {\left (a^{3} b^{6} c^{9} - 9 \, a^{5} b^{4} c^{7} d^{2} + 16 \, a^{6} b^{3} c^{6} d^{3} - 9 \, a^{7} b^{2} c^{5} d^{4} + a^{9} c^{3} d^{6}\right )} x^{2} + 2 \, {\left (a^{4} b^{5} c^{9} - 3 \, a^{5} b^{4} c^{8} d + 2 \, a^{6} b^{3} c^{7} d^{2} + 2 \, a^{7} b^{2} c^{6} d^{3} - 3 \, a^{8} b c^{5} d^{4} + a^{9} c^{4} d^{5}\right )} x\right )}}, \frac {3 \, {\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} + {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \, {\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} + {\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (4 \, a^{2} b^{4} c^{6} - 12 \, a^{3} b^{3} c^{5} d - 12 \, a^{5} b c^{3} d^{3} + 4 \, a^{6} c^{2} d^{4} + {\left (3 \, a b^{5} c^{4} d^{2} - 11 \, a^{2} b^{4} c^{3} d^{3} - 11 \, a^{3} b^{3} c^{2} d^{4} + 3 \, a^{4} b^{2} c d^{5}\right )} x^{3} + 6 \, {\left (a b^{5} c^{5} d - 3 \, a^{2} b^{4} c^{4} d^{2} - 4 \, a^{3} b^{3} c^{3} d^{3} - 3 \, a^{4} b^{2} c^{2} d^{4} + a^{5} b c d^{5}\right )} x^{2} + 3 \, {\left (a b^{5} c^{6} - a^{2} b^{4} c^{5} d - 8 \, a^{3} b^{3} c^{4} d^{2} - 8 \, a^{4} b^{2} c^{3} d^{3} - a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a^{5} b^{4} c^{9} - 4 \, a^{6} b^{3} c^{8} d + 6 \, a^{7} b^{2} c^{7} d^{2} - 4 \, a^{8} b c^{6} d^{3} + a^{9} c^{5} d^{4} + {\left (a^{3} b^{6} c^{7} d^{2} - 4 \, a^{4} b^{5} c^{6} d^{3} + 6 \, a^{5} b^{4} c^{5} d^{4} - 4 \, a^{6} b^{3} c^{4} d^{5} + a^{7} b^{2} c^{3} d^{6}\right )} x^{4} + 2 \, {\left (a^{3} b^{6} c^{8} d - 3 \, a^{4} b^{5} c^{7} d^{2} + 2 \, a^{5} b^{4} c^{6} d^{3} + 2 \, a^{6} b^{3} c^{5} d^{4} - 3 \, a^{7} b^{2} c^{4} d^{5} + a^{8} b c^{3} d^{6}\right )} x^{3} + {\left (a^{3} b^{6} c^{9} - 9 \, a^{5} b^{4} c^{7} d^{2} + 16 \, a^{6} b^{3} c^{6} d^{3} - 9 \, a^{7} b^{2} c^{5} d^{4} + a^{9} c^{3} d^{6}\right )} x^{2} + 2 \, {\left (a^{4} b^{5} c^{9} - 3 \, a^{5} b^{4} c^{8} d + 2 \, a^{6} b^{3} c^{7} d^{2} + 2 \, a^{7} b^{2} c^{6} d^{3} - 3 \, a^{8} b c^{5} d^{4} + a^{9} c^{4} d^{5}\right )} x\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 {1}{x \left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, 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. 942 vs.
\(2 (218) = 436\).
time = 3.82, size = 942, normalized size = 3.74 \begin {gather*} -\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (11 \, b^{7} c^{6} d^{5} {\left | b \right |} - 36 \, a b^{6} c^{5} d^{6} {\left | b \right |} + 42 \, a^{2} b^{5} c^{4} d^{7} {\left | b \right |} - 20 \, a^{3} b^{4} c^{3} d^{8} {\left | b \right |} + 3 \, a^{4} b^{3} c^{2} d^{9} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{9} c^{11} d - 7 \, a b^{8} c^{10} d^{2} + 21 \, a^{2} b^{7} c^{9} d^{3} - 35 \, a^{3} b^{6} c^{8} d^{4} + 35 \, a^{4} b^{5} c^{7} d^{5} - 21 \, a^{5} b^{4} c^{6} d^{6} + 7 \, a^{6} b^{3} c^{5} d^{7} - a^{7} b^{2} c^{4} d^{8}} + \frac {3 \, {\left (4 \, b^{8} c^{7} d^{4} {\left | b \right |} - 17 \, a b^{7} c^{6} d^{5} {\left | b \right |} + 28 \, a^{2} b^{6} c^{5} d^{6} {\left | b \right |} - 22 \, a^{3} b^{5} c^{4} d^{7} {\left | b \right |} + 8 \, a^{4} b^{4} c^{3} d^{8} {\left | b \right |} - a^{5} b^{3} c^{2} d^{9} {\left | b \right |}\right )}}{b^{9} c^{11} d - 7 \, a b^{8} c^{10} d^{2} + 21 \, a^{2} b^{7} c^{9} d^{3} - 35 \, a^{3} b^{6} c^{8} d^{4} + 35 \, a^{4} b^{5} c^{7} d^{5} - 21 \, a^{5} b^{4} c^{6} d^{6} + 7 \, a^{6} b^{3} c^{5} d^{7} - a^{7} b^{2} c^{4} d^{8}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {4 \, {\left (3 \, \sqrt {b d} b^{9} c^{3} - 17 \, \sqrt {b d} a b^{8} c^{2} d + 25 \, \sqrt {b d} a^{2} b^{7} c d^{2} - 11 \, \sqrt {b d} a^{3} b^{6} d^{3} - 6 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{7} c^{2} + 30 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{6} c d - 24 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{5} d^{2} + 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{5} c - 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{4} d\right )}}{3 \, {\left (a^{2} b^{3} c^{3} {\left | b \right |} - 3 \, a^{3} b^{2} c^{2} d {\left | b \right |} + 3 \, a^{4} b c d^{2} {\left | b \right |} - a^{5} d^{3} {\left | b \right |}\right )} {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3}} - \frac {2 \, \sqrt {b d} b \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a^{2} c^{2} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x\,{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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