Optimal. Leaf size=313 \[ -\frac {\left (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^2 x}-\frac {(5 b c-a d) (b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\frac {\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{5/2}}+2 b^{5/2} d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \]
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Rubi [A]
time = 0.21, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {99, 154, 163,
65, 223, 212, 95, 214} \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^3 d^3-17 a^2 b c d^2+73 a b^2 c^2 d+5 b^3 c^3\right )}{64 a c^2 x}+\frac {\left (-3 a^4 d^4+20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-60 a b^3 c^3 d+5 b^4 c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{5/2}}+2 b^{5/2} d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 b c-a d) (3 a d+b c)}{32 c^2 x^2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{24 c x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 95
Rule 99
Rule 154
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^5} \, dx &=-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\frac {1}{4} \int \frac {(a+b x)^{3/2} \sqrt {c+d x} \left (\frac {1}{2} (5 b c+3 a d)+4 b d x\right )}{x^4} \, dx\\ &=-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\frac {\int \frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3}{4} (5 b c-a d) (b c+3 a d)+12 b^2 c d x\right )}{x^3} \, dx}{12 c}\\ &=-\frac {(5 b c-a d) (b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\frac {\int \frac {\sqrt {c+d x} \left (\frac {3}{8} \left (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right )+24 b^3 c^2 d x\right )}{x^2 \sqrt {a+b x}} \, dx}{24 c^2}\\ &=-\frac {\left (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^2 x}-\frac {(5 b c-a d) (b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\frac {\int \frac {-\frac {3}{16} \left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right )+24 a b^3 c^2 d^2 x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{24 a c^2}\\ &=-\frac {\left (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^2 x}-\frac {(5 b c-a d) (b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\left (b^3 d^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx-\frac {\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 a c^2}\\ &=-\frac {\left (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^2 x}-\frac {(5 b c-a d) (b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\left (2 b^2 d^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )-\frac {\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{64 a c^2}\\ &=-\frac {\left (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^2 x}-\frac {(5 b c-a d) (b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\frac {\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{5/2}}+\left (2 b^2 d^2\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )\\ &=-\frac {\left (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^2 x}-\frac {(5 b c-a d) (b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\frac {\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{5/2}}+2 b^{5/2} d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.75, size = 260, normalized size = 0.83 \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 b^3 c^3 x^3+a b^2 c^2 x^2 (118 c+337 d x)+a^2 b c x \left (136 c^2+244 c d x+57 d^2 x^2\right )+a^3 \left (48 c^3+72 c^2 d x+6 c d^2 x^2-9 d^3 x^3\right )\right )}{192 a c^2 x^4}-\frac {\left (-5 b^4 c^4+60 a b^3 c^3 d+90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+3 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{5/2}}+2 b^{5/2} d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \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. \(732\) vs.
\(2(263)=526\).
time = 0.08, size = 733, normalized size = 2.34
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (9 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{4} d^{4} x^{4} \sqrt {b d}-60 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} b c \,d^{3} x^{4} \sqrt {b d}+270 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b^{2} c^{2} d^{2} x^{4} \sqrt {b d}+180 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{3} c^{3} d \,x^{4} \sqrt {b d}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{4} c^{4} x^{4} \sqrt {b d}-384 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{3} c^{2} d^{2} x^{4} \sqrt {a c}-18 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} d^{3} x^{3}+114 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b c \,d^{2} x^{3}+674 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{2} c^{2} d \,x^{3}+30 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{3} c^{3} x^{3}+12 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} c \,d^{2} x^{2}+488 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b \,c^{2} d \,x^{2}+236 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{2} c^{3} x^{2}+144 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} c^{2} d x +272 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b \,c^{3} x +96 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} c^{3}\right )}{384 a \,c^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{4} \sqrt {b d}\, \sqrt {a c}}\) | \(733\) |
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 [A]
time = 8.52, size = 1525, normalized size = 4.87 \begin {gather*} \left [\frac {384 \, \sqrt {b d} a^{2} b^{2} c^{3} d x^{4} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 3 \, {\left (5 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d - 90 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4}\right )} \sqrt {a c} x^{4} \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 (48 \, a^{4} c^{4} + {\left (15 \, a b^{3} c^{4} + 337 \, a^{2} b^{2} c^{3} d + 57 \, a^{3} b c^{2} d^{2} - 9 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{2} c^{4} + 122 \, a^{3} b c^{3} d + 3 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (17 \, a^{3} b c^{4} + 9 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, a^{2} c^{3} x^{4}}, -\frac {768 \, \sqrt {-b d} a^{2} b^{2} c^{3} d x^{4} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 3 \, {\left (5 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d - 90 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4}\right )} \sqrt {a c} x^{4} \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 (48 \, a^{4} c^{4} + {\left (15 \, a b^{3} c^{4} + 337 \, a^{2} b^{2} c^{3} d + 57 \, a^{3} b c^{2} d^{2} - 9 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{2} c^{4} + 122 \, a^{3} b c^{3} d + 3 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (17 \, a^{3} b c^{4} + 9 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, a^{2} c^{3} x^{4}}, \frac {192 \, \sqrt {b d} a^{2} b^{2} c^{3} d x^{4} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 3 \, {\left (5 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d - 90 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4}\right )} \sqrt {-a c} x^{4} \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 (48 \, a^{4} c^{4} + {\left (15 \, a b^{3} c^{4} + 337 \, a^{2} b^{2} c^{3} d + 57 \, a^{3} b c^{2} d^{2} - 9 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{2} c^{4} + 122 \, a^{3} b c^{3} d + 3 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (17 \, a^{3} b c^{4} + 9 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, a^{2} c^{3} x^{4}}, -\frac {384 \, \sqrt {-b d} a^{2} b^{2} c^{3} d x^{4} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 3 \, {\left (5 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d - 90 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4}\right )} \sqrt {-a c} x^{4} \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 (48 \, a^{4} c^{4} + {\left (15 \, a b^{3} c^{4} + 337 \, a^{2} b^{2} c^{3} d + 57 \, a^{3} b c^{2} d^{2} - 9 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{2} c^{4} + 122 \, a^{3} b c^{3} d + 3 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (17 \, a^{3} b c^{4} + 9 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, a^{2} c^{3} x^{4}}\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 {\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {3}{2}}}{x^{5}}\, 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. 3887 vs.
\(2 (263) = 526\).
time = 2.93, size = 3887, normalized size = 12.42 \begin {gather*} \text {Too large to display} \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 )}^{3/2}}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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