Optimal. Leaf size=334 \[ -5 a^{3/2} c^{3/2} (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {5 \sqrt {a+b x} (c+d x)^{3/2} \left (-31 a^2 d^2-18 a b c d+b^2 c^2\right )}{96 d}-\frac {5 \sqrt {a+b x} \sqrt {c+d x} \left (-a^3 d^3-45 a^2 b c d^2-19 a b^2 c^2 d+b^3 c^3\right )}{64 b d}-\frac {5 \left (a^4 d^4-20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+b^4 c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{3/2}}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}+\frac {5 b \sqrt {a+b x} (c+d x)^{5/2} (7 a d+b c)}{24 d} \]
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Rubi [A] time = 0.39, antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {97, 154, 157, 63, 217, 206, 93, 208} \begin {gather*} -\frac {5 \sqrt {a+b x} (c+d x)^{3/2} \left (-31 a^2 d^2-18 a b c d+b^2 c^2\right )}{96 d}-\frac {5 \sqrt {a+b x} \sqrt {c+d x} \left (-45 a^2 b c d^2-a^3 d^3-19 a b^2 c^2 d+b^3 c^3\right )}{64 b d}-\frac {5 \left (-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4-20 a b^3 c^3 d+b^4 c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{3/2}}-5 a^{3/2} c^{3/2} (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}+\frac {5 b \sqrt {a+b x} (c+d x)^{5/2} (7 a d+b c)}{24 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 97
Rule 154
Rule 157
Rule 206
Rule 208
Rule 217
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^2} \, dx &=-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\int \frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (\frac {5}{2} (b c+a d)+5 b d x\right )}{x} \, dx\\ &=\frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac {\int \frac {\sqrt {a+b x} (c+d x)^{3/2} \left (10 a d (b c+a d)+\frac {5}{2} b d (b c+7 a d) x\right )}{x} \, dx}{4 d}\\ &=\frac {5 b (b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac {\int \frac {(c+d x)^{3/2} \left (30 a^2 d^2 (b c+a d)-\frac {5}{4} b d \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) x\right )}{x \sqrt {a+b x}} \, dx}{12 d^2}\\ &=-\frac {5 \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 d}+\frac {5 b (b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac {\int \frac {\sqrt {c+d x} \left (60 a^2 b c d^2 (b c+a d)-\frac {15}{8} b d \left (b^3 c^3-19 a b^2 c^2 d-45 a^2 b c d^2-a^3 d^3\right ) x\right )}{x \sqrt {a+b x}} \, dx}{24 b d^2}\\ &=-\frac {5 \left (b^3 c^3-19 a b^2 c^2 d-45 a^2 b c d^2-a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d}-\frac {5 \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 d}+\frac {5 b (b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac {\int \frac {60 a^2 b^2 c^2 d^2 (b c+a d)-\frac {15}{16} b d \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{24 b^2 d^2}\\ &=-\frac {5 \left (b^3 c^3-19 a b^2 c^2 d-45 a^2 b c d^2-a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d}-\frac {5 \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 d}+\frac {5 b (b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac {1}{2} \left (5 a^2 c^2 (b c+a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx-\frac {\left (5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 b d}\\ &=-\frac {5 \left (b^3 c^3-19 a b^2 c^2 d-45 a^2 b c d^2-a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d}-\frac {5 \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 d}+\frac {5 b (b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\left (5 a^2 c^2 (b c+a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )-\frac {\left (5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{64 b^2 d}\\ &=-\frac {5 \left (b^3 c^3-19 a b^2 c^2 d-45 a^2 b c d^2-a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d}-\frac {5 \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 d}+\frac {5 b (b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}-5 a^{3/2} c^{3/2} (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\left (5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{64 b^2 d}\\ &=-\frac {5 \left (b^3 c^3-19 a b^2 c^2 d-45 a^2 b c d^2-a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d}-\frac {5 \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 d}+\frac {5 b (b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}-5 a^{3/2} c^{3/2} (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{3/2}}\\ \end {align*}
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Mathematica [B] time = 4.88, size = 1077, normalized size = 3.22 \begin {gather*} \frac {-15 b^4 x (c+d x)^2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) c^4+15 b \sqrt {d} (b c-a d)^{5/2} x \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c^3+300 a b^3 d x (c+d x)^2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) c^3-960 a^{3/2} b d^{3/2} (b c-a d)^{3/2} x \sqrt {c+d x} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right ) c^{5/2}-\frac {192 a^2 d^{3/2} (b c-a d)^{5/2} \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c^2}{b}+118 b d^{3/2} (b c-a d)^{5/2} x^2 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c^2+601 a d^{3/2} (b c-a d)^{5/2} x \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c^2+1350 a^2 b^2 d^2 x (c+d x)^2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) c^2-960 a^{5/2} d^{5/2} (b c-a d)^{3/2} x \sqrt {c+d x} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right ) c^{3/2}+136 b d^{5/2} (b c-a d)^{5/2} x^3 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c+452 a d^{5/2} (b c-a d)^{5/2} x^2 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c+\frac {601 a^2 d^{5/2} (b c-a d)^{5/2} x \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c}{b}+300 a^3 b d^3 x (c+d x)^2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) c+48 b d^{7/2} (b c-a d)^{5/2} x^4 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2}+136 a d^{7/2} (b c-a d)^{5/2} x^3 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2}+\frac {118 a^2 d^{7/2} (b c-a d)^{5/2} x^2 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2}}{b}-15 a^4 d^4 x (c+d x)^2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )+15 a^3 d^{7/2} \sqrt {b c-a d} x \sqrt {a+b x} (c+d x)^2 \sqrt {\frac {b (c+d x)}{b c-a d}}}{192 d^{3/2} (b c-a d)^{3/2} x \sqrt {c+d x} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.95, size = 1029, normalized size = 3.08 \begin {gather*} \frac {-\frac {15 a c^4 (c+d x)^{9/2} b^7}{(a+b x)^{9/2}}+\frac {15 c^5 (c+d x)^{7/2} b^7}{(a+b x)^{7/2}}-\frac {660 a^2 c^3 d (c+d x)^{9/2} b^6}{(a+b x)^{9/2}}+\frac {395 a c^4 d (c+d x)^{7/2} b^6}{(a+b x)^{7/2}}+\frac {73 c^5 d (c+d x)^{5/2} b^6}{(a+b x)^{5/2}}+\frac {390 a^3 c^2 d^2 (c+d x)^{9/2} b^5}{(a+b x)^{9/2}}+\frac {2030 a^2 c^3 d^2 (c+d x)^{7/2} b^5}{(a+b x)^{7/2}}-\frac {1405 a c^4 d^2 (c+d x)^{5/2} b^5}{(a+b x)^{5/2}}-\frac {55 c^5 d^2 (c+d x)^{3/2} b^5}{(a+b x)^{3/2}}+\frac {300 a^4 c d^3 (c+d x)^{9/2} b^4}{(a+b x)^{9/2}}-\frac {1410 a^3 c^2 d^3 (c+d x)^{7/2} b^4}{(a+b x)^{7/2}}-\frac {1910 a^2 c^3 d^3 (c+d x)^{5/2} b^4}{(a+b x)^{5/2}}+\frac {1085 a c^4 d^3 (c+d x)^{3/2} b^4}{(a+b x)^{3/2}}+\frac {15 c^5 d^3 \sqrt {c+d x} b^4}{\sqrt {a+b x}}-\frac {15 a^5 d^4 (c+d x)^{9/2} b^3}{(a+b x)^{9/2}}-\frac {1085 a^4 c d^4 (c+d x)^{7/2} b^3}{(a+b x)^{7/2}}+\frac {1910 a^3 c^2 d^4 (c+d x)^{5/2} b^3}{(a+b x)^{5/2}}+\frac {1410 a^2 c^3 d^4 (c+d x)^{3/2} b^3}{(a+b x)^{3/2}}-\frac {300 a c^4 d^4 \sqrt {c+d x} b^3}{\sqrt {a+b x}}+\frac {55 a^5 d^5 (c+d x)^{7/2} b^2}{(a+b x)^{7/2}}+\frac {1405 a^4 c d^5 (c+d x)^{5/2} b^2}{(a+b x)^{5/2}}-\frac {2030 a^3 c^2 d^5 (c+d x)^{3/2} b^2}{(a+b x)^{3/2}}-\frac {390 a^2 c^3 d^5 \sqrt {c+d x} b^2}{\sqrt {a+b x}}-\frac {73 a^5 d^6 (c+d x)^{5/2} b}{(a+b x)^{5/2}}-\frac {395 a^4 c d^6 (c+d x)^{3/2} b}{(a+b x)^{3/2}}+\frac {660 a^3 c^2 d^6 \sqrt {c+d x} b}{\sqrt {a+b x}}-\frac {15 a^5 d^7 (c+d x)^{3/2}}{(a+b x)^{3/2}}+\frac {15 a^4 c d^7 \sqrt {c+d x}}{\sqrt {a+b x}}}{192 b d \left (c-\frac {a (c+d x)}{a+b x}\right ) \left (\frac {b (c+d x)}{a+b x}-d\right )^4}-5 \left (c^{3/2} d a^{5/2}+b c^{5/2} a^{3/2}\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )-\frac {5 \left (b^4 c^4-20 a b^3 d c^3-90 a^2 b^2 d^2 c^2-20 a^3 b d^3 c+a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{64 b^{3/2} d^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 39.50, size = 1613, normalized size = 4.83
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 7.68, size = 779, normalized size = 2.33 \begin {gather*} \frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )} d^{2} {\left | b \right |}}{b^{2}} + \frac {17 \, b^{3} c d^{7} {\left | b \right |} - a b^{2} d^{8} {\left | b \right |}}{b^{4} d^{6}}\right )} + \frac {59 \, b^{4} c^{2} d^{6} {\left | b \right |} + 90 \, a b^{3} c d^{7} {\left | b \right |} - 5 \, a^{2} b^{2} d^{8} {\left | b \right |}}{b^{4} d^{6}}\right )} {\left (b x + a\right )} + \frac {3 \, {\left (5 \, b^{5} c^{3} d^{5} {\left | b \right |} + 161 \, a b^{4} c^{2} d^{6} {\left | b \right |} + 95 \, a^{2} b^{3} c d^{7} {\left | b \right |} - 5 \, a^{3} b^{2} d^{8} {\left | b \right |}\right )}}{b^{4} d^{6}}\right )} \sqrt {b x + a} - \frac {1920 \, {\left (\sqrt {b d} a^{2} b^{2} c^{3} {\left | b \right |} + \sqrt {b d} a^{3} b c^{2} d {\left | b \right |}\right )} \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} b} - \frac {768 \, {\left (\sqrt {b d} a^{2} b^{4} c^{4} {\left | b \right |} - 2 \, \sqrt {b d} a^{3} b^{3} c^{3} d {\left | b \right |} + \sqrt {b d} a^{4} b^{2} c^{2} d^{2} {\left | b \right |} - \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^{2} c^{3} {\left | b \right |} - \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^{3} b c^{2} d {\left | b \right |}\right )}}{b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\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^{2} c - 2 \, {\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 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}} + \frac {15 \, {\left (\sqrt {b d} b^{4} c^{4} {\left | b \right |} - 20 \, \sqrt {b d} a b^{3} c^{3} d {\left | b \right |} - 90 \, \sqrt {b d} a^{2} b^{2} c^{2} d^{2} {\left | b \right |} - 20 \, \sqrt {b d} a^{3} b c d^{3} {\left | b \right |} + \sqrt {b d} a^{4} d^{4} {\left | b \right |}\right )} \log \left ({\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 )}{b^{2} d^{2}}}{384 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 950, normalized size = 2.84 \begin {gather*} -\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (15 \sqrt {a c}\, a^{4} d^{4} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+960 \sqrt {b d}\, a^{3} b \,c^{2} d^{2} x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-300 \sqrt {a c}\, a^{3} b c \,d^{3} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+960 \sqrt {b d}\, a^{2} b^{2} c^{3} d x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-1350 \sqrt {a c}\, a^{2} b^{2} c^{2} d^{2} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-300 \sqrt {a c}\, a \,b^{3} c^{3} d x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+15 \sqrt {a c}\, b^{4} c^{4} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-96 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{3} d^{3} x^{4}-272 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{2} d^{3} x^{3}-272 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{3} c \,d^{2} x^{3}-236 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b \,d^{3} x^{2}-904 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{2} c \,d^{2} x^{2}-236 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{3} c^{2} d \,x^{2}-30 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} d^{3} x -1202 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b c \,d^{2} x -1202 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{2} c^{2} d x -30 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{3} c^{3} x +384 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b \,c^{2} d \right )}{384 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b d x} \end {gather*}
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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}}{x^2} \,d x \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 {\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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