Optimal. Leaf size=340 \[ -\frac {\left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{7/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-15 a^3 d^3+9 a^2 b c d^2-61 a b^2 c^2 d+35 b^3 c^3\right )}{960 a^3 c^2 x^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-45 a^4 d^4+30 a^3 b c d^3+36 a^2 b^2 c^2 d^2-190 a b^3 c^3 d+105 b^4 c^4\right )}{1920 a^4 c^3 x}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {7 b^2 c}{a}-\frac {3 a d^2}{c}-12 b d\right )}{240 a x^3}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{5 x^5}-\frac {\sqrt {a+b x} \sqrt {c+d x} (3 a d+b c)}{40 a x^4} \]
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Rubi [A] time = 0.33, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {97, 149, 151, 12, 93, 208} \[ -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (9 a^2 b c d^2-15 a^3 d^3-61 a b^2 c^2 d+35 b^3 c^3\right )}{960 a^3 c^2 x^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4-190 a b^3 c^3 d+105 b^4 c^4\right )}{1920 a^4 c^3 x}-\frac {\left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{7/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {7 b^2 c}{a}-\frac {3 a d^2}{c}-12 b d\right )}{240 a x^3}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{5 x^5}-\frac {\sqrt {a+b x} \sqrt {c+d x} (3 a d+b c)}{40 a x^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 97
Rule 149
Rule 151
Rule 208
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^6} \, dx &=-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{5 x^5}+\frac {1}{5} \int \frac {\sqrt {c+d x} \left (\frac {1}{2} (b c+3 a d)+2 b d x\right )}{x^5 \sqrt {a+b x}} \, dx\\ &=-\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a x^4}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{5 x^5}+\frac {\int \frac {\frac {1}{4} \left (-7 b^2 c^2+12 a b c d+3 a^2 d^2\right )-\frac {1}{2} b d (3 b c-7 a d) x}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{20 a}\\ &=-\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a x^4}+\frac {\left (\frac {7 b^2 c}{a}-12 b d-\frac {3 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a x^3}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{5 x^5}-\frac {\int \frac {\frac {1}{8} \left (-35 b^3 c^3+61 a b^2 c^2 d-9 a^2 b c d^2+15 a^3 d^3\right )-\frac {1}{2} b d \left (7 b^2 c^2-12 a b c d-3 a^2 d^2\right ) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{60 a^2 c}\\ &=-\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a x^4}+\frac {\left (\frac {7 b^2 c}{a}-12 b d-\frac {3 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a x^3}-\frac {\left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^2 x^2}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{5 x^5}+\frac {\int \frac {\frac {1}{16} \left (-105 b^4 c^4+190 a b^3 c^3 d-36 a^2 b^2 c^2 d^2-30 a^3 b c d^3+45 a^4 d^4\right )-\frac {1}{8} b d \left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{120 a^3 c^2}\\ &=-\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a x^4}+\frac {\left (\frac {7 b^2 c}{a}-12 b d-\frac {3 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a x^3}-\frac {\left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^2 x^2}+\frac {\left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^4 c^3 x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{5 x^5}-\frac {\int -\frac {15 (b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )}{32 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{120 a^4 c^3}\\ &=-\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a x^4}+\frac {\left (\frac {7 b^2 c}{a}-12 b d-\frac {3 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a x^3}-\frac {\left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^2 x^2}+\frac {\left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^4 c^3 x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{5 x^5}+\frac {\left ((b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 a^4 c^3}\\ &=-\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a x^4}+\frac {\left (\frac {7 b^2 c}{a}-12 b d-\frac {3 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a x^3}-\frac {\left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^2 x^2}+\frac {\left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^4 c^3 x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{5 x^5}+\frac {\left ((b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )\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^4 c^3}\\ &=-\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a x^4}+\frac {\left (\frac {7 b^2 c}{a}-12 b d-\frac {3 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a x^3}-\frac {\left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^2 x^2}+\frac {\left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^4 c^3 x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{5 x^5}-\frac {(b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 232, normalized size = 0.68 \[ \frac {-\frac {5 \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) \left (\frac {x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} (2 a c+5 a d x-3 b c x)\right )}{a^{5/2} \sqrt {c}}+8 \sqrt {a+b x} (c+d x)^{5/2}\right )}{24 c x^3}-\frac {16 a c (a+b x)^{3/2} (c+d x)^{5/2}}{x^5}+\frac {2 (a+b x)^{3/2} (c+d x)^{5/2} (5 a d+7 b c)}{x^4}}{80 a^2 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 10.76, size = 732, normalized size = 2.15 \[ \left [-\frac {15 \, {\left (7 \, b^{5} c^{5} - 15 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + 3 \, a^{4} b c d^{4} - 3 \, 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 (384 \, a^{5} c^{5} - {\left (105 \, a b^{4} c^{5} - 190 \, a^{2} b^{3} c^{4} d + 36 \, a^{3} b^{2} c^{3} d^{2} + 30 \, a^{4} b c^{2} d^{3} - 45 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (35 \, a^{2} b^{3} c^{5} - 61 \, a^{3} b^{2} c^{4} d + 9 \, a^{4} b c^{3} d^{2} - 15 \, a^{5} c^{2} d^{3}\right )} x^{3} - 8 \, {\left (7 \, a^{3} b^{2} c^{5} - 12 \, a^{4} b c^{4} d - 3 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (a^{4} b c^{5} + 11 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, a^{5} c^{4} x^{5}}, \frac {15 \, {\left (7 \, b^{5} c^{5} - 15 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + 3 \, a^{4} b c d^{4} - 3 \, 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 (384 \, a^{5} c^{5} - {\left (105 \, a b^{4} c^{5} - 190 \, a^{2} b^{3} c^{4} d + 36 \, a^{3} b^{2} c^{3} d^{2} + 30 \, a^{4} b c^{2} d^{3} - 45 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (35 \, a^{2} b^{3} c^{5} - 61 \, a^{3} b^{2} c^{4} d + 9 \, a^{4} b c^{3} d^{2} - 15 \, a^{5} c^{2} d^{3}\right )} x^{3} - 8 \, {\left (7 \, a^{3} b^{2} c^{5} - 12 \, a^{4} b c^{4} d - 3 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (a^{4} b c^{5} + 11 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, a^{5} 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 = 2.84 \[ \frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (45 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 )-45 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 )-30 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 )-90 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 )+225 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 )-105 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 )-90 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} d^{4} x^{4}+60 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b c \,d^{3} x^{4}+72 \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}-380 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{3} d \,x^{4}+210 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{4} c^{4} x^{4}+60 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c \,d^{3} x^{3}-36 \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}+244 \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}-140 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{4} x^{3}-48 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c^{2} d^{2} x^{2}-192 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{3} d \,x^{2}+112 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{4} x^{2}-1056 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c^{3} d x -96 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{4} x -768 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{4} c^{4}\right )}{3840 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{4} 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 \[ \int \frac {\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{3/2}}{x^6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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