Optimal. Leaf size=239 \[ -\frac {3 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{5/2} c^{7/2}}+\frac {3 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^4}{128 a^2 c^3 x}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (b c-a d)^2}{16 c^3 x^3}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^3}{64 a c^3 x^2}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (b c-a d)}{8 c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5} \]
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Rubi [A] time = 0.13, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {94, 93, 208} \[ \frac {3 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^4}{128 a^2 c^3 x}-\frac {3 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{5/2} c^{7/2}}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^3}{64 a c^3 x^2}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (b c-a d)^2}{16 c^3 x^3}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (b c-a d)}{8 c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^6} \, dx &=-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}+\frac {(b c-a d) \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^5} \, dx}{2 c}\\ &=-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}+\frac {\left (3 (b c-a d)^2\right ) \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^4} \, dx}{16 c^2}\\ &=-\frac {(b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/2}}{16 c^3 x^3}-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}+\frac {(b c-a d)^3 \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx}{32 c^3}\\ &=-\frac {(b c-a d)^3 \sqrt {a+b x} (c+d x)^{3/2}}{64 a c^3 x^2}-\frac {(b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/2}}{16 c^3 x^3}-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}-\frac {\left (3 (b c-a d)^4\right ) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx}{128 a c^3}\\ &=\frac {3 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 a^2 c^3 x}-\frac {(b c-a d)^3 \sqrt {a+b x} (c+d x)^{3/2}}{64 a c^3 x^2}-\frac {(b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/2}}{16 c^3 x^3}-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}+\frac {\left (3 (b c-a d)^5\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 a^2 c^3}\\ &=\frac {3 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 a^2 c^3 x}-\frac {(b c-a d)^3 \sqrt {a+b x} (c+d x)^{3/2}}{64 a c^3 x^2}-\frac {(b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/2}}{16 c^3 x^3}-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}+\frac {\left (3 (b c-a d)^5\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{128 a^2 c^3}\\ &=\frac {3 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 a^2 c^3 x}-\frac {(b c-a d)^3 \sqrt {a+b x} (c+d x)^{3/2}}{64 a c^3 x^2}-\frac {(b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/2}}{16 c^3 x^3}-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}-\frac {3 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{5/2} c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.68, size = 208, normalized size = 0.87 \[ -\frac {5 x (b c-a d) \left (x (b c-a d) \left (\frac {x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} (2 a c+5 a d x-3 b c x)\right )}{a^{5/2} \sqrt {c}}+8 \sqrt {a+b x} (c+d x)^{5/2}\right )+16 c (a+b x)^{3/2} (c+d x)^{5/2}\right )+128 c^2 (a+b x)^{5/2} (c+d x)^{5/2}}{640 c^3 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 13.70, size = 730, normalized size = 3.05 \[ \left [-\frac {15 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \sqrt {a c} x^{5} \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 (128 \, a^{5} c^{5} - {\left (15 \, a b^{4} c^{5} - 70 \, a^{2} b^{3} c^{4} d - 128 \, a^{3} b^{2} c^{3} d^{2} + 70 \, a^{4} b c^{2} d^{3} - 15 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (5 \, a^{2} b^{3} c^{5} + 233 \, a^{3} b^{2} c^{4} d + 23 \, a^{4} b c^{3} d^{2} - 5 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (31 \, a^{3} b^{2} c^{5} + 64 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x^{2} + 16 \, {\left (21 \, a^{4} b c^{5} + 11 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2560 \, a^{3} c^{4} x^{5}}, \frac {15 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \sqrt {-a c} x^{5} \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 (128 \, a^{5} c^{5} - {\left (15 \, a b^{4} c^{5} - 70 \, a^{2} b^{3} c^{4} d - 128 \, a^{3} b^{2} c^{3} d^{2} + 70 \, a^{4} b c^{2} d^{3} - 15 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (5 \, a^{2} b^{3} c^{5} + 233 \, a^{3} b^{2} c^{4} d + 23 \, a^{4} b c^{3} d^{2} - 5 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (31 \, a^{3} b^{2} c^{5} + 64 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x^{2} + 16 \, {\left (21 \, a^{4} b c^{5} + 11 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{1280 \, a^{3} c^{4} x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 967, normalized size = 4.05 \[ \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (15 a^{5} d^{5} x^{5} \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 )-75 a^{4} b c \,d^{4} x^{5} \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 )+150 a^{3} b^{2} c^{2} d^{3} x^{5} \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 )-150 a^{2} b^{3} c^{3} d^{2} x^{5} \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 )+75 a \,b^{4} c^{4} d \,x^{5} \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 )-15 b^{5} c^{5} x^{5} \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 )-30 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} d^{4} x^{4}+140 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b c \,d^{3} x^{4}-256 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{2} d^{2} x^{4}-140 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{3} d \,x^{4}+30 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{4} c^{4} x^{4}+20 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c \,d^{3} x^{3}-92 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{2} d^{2} x^{3}-932 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{3} d \,x^{3}-20 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{4} x^{3}-16 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c^{2} d^{2} x^{2}-1024 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{3} d \,x^{2}-496 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{4} x^{2}-352 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c^{3} d x -672 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{4} x -256 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{4} c^{4}\right )}{1280 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{2} c^{3} x^{5}} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{3/2}}{x^6} \,d x \]
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 \[ \int \frac {\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {3}{2}}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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