Optimal. Leaf size=234 \[ -\frac {d \sqrt {a+b x} \left (105 a^2 d^2-100 a b c d+3 b^2 c^2\right )}{12 a c^4 \sqrt {c+d x} (b c-a d)}+\frac {\left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{9/2}}-\frac {d \sqrt {a+b x} (3 b c-35 a d)}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x} (b c-7 a d)}{4 a c^2 x (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}} \]
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Rubi [A] time = 0.22, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {99, 151, 152, 12, 93, 208} \begin {gather*} -\frac {d \sqrt {a+b x} \left (105 a^2 d^2-100 a b c d+3 b^2 c^2\right )}{12 a c^4 \sqrt {c+d x} (b c-a d)}+\frac {\left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{9/2}}-\frac {d \sqrt {a+b x} (3 b c-35 a d)}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x} (b c-7 a d)}{4 a c^2 x (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 99
Rule 151
Rule 152
Rule 208
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{5/2}} \, dx &=-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}+\frac {\int \frac {\frac {1}{2} (b c-7 a d)-3 b d x}{x^2 \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{2 c}\\ &=-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(b c-7 a d) \sqrt {a+b x}}{4 a c^2 x (c+d x)^{3/2}}-\frac {\int \frac {\frac {1}{4} \left (b^2 c^2+10 a b c d-35 a^2 d^2\right )+b d (b c-7 a d) x}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{2 a c^2}\\ &=-\frac {d (3 b c-35 a d) \sqrt {a+b x}}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(b c-7 a d) \sqrt {a+b x}}{4 a c^2 x (c+d x)^{3/2}}+\frac {\int \frac {-\frac {3}{8} (b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right )-\frac {1}{4} b d (3 b c-35 a d) (b c-a d) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 a c^3 (b c-a d)}\\ &=-\frac {d (3 b c-35 a d) \sqrt {a+b x}}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(b c-7 a d) \sqrt {a+b x}}{4 a c^2 x (c+d x)^{3/2}}-\frac {d \left (3 b^2 c^2-100 a b c d+105 a^2 d^2\right ) \sqrt {a+b x}}{12 a c^4 (b c-a d) \sqrt {c+d x}}-\frac {2 \int \frac {3 (b c-a d)^2 \left (b^2 c^2+10 a b c d-35 a^2 d^2\right )}{16 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 a c^4 (b c-a d)^2}\\ &=-\frac {d (3 b c-35 a d) \sqrt {a+b x}}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(b c-7 a d) \sqrt {a+b x}}{4 a c^2 x (c+d x)^{3/2}}-\frac {d \left (3 b^2 c^2-100 a b c d+105 a^2 d^2\right ) \sqrt {a+b x}}{12 a c^4 (b c-a d) \sqrt {c+d x}}-\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a c^4}\\ &=-\frac {d (3 b c-35 a d) \sqrt {a+b x}}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(b c-7 a d) \sqrt {a+b x}}{4 a c^2 x (c+d x)^{3/2}}-\frac {d \left (3 b^2 c^2-100 a b c d+105 a^2 d^2\right ) \sqrt {a+b x}}{12 a c^4 (b c-a d) \sqrt {c+d x}}-\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 a c^4}\\ &=-\frac {d (3 b c-35 a d) \sqrt {a+b x}}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(b c-7 a d) \sqrt {a+b x}}{4 a c^2 x (c+d x)^{3/2}}-\frac {d \left (3 b^2 c^2-100 a b c d+105 a^2 d^2\right ) \sqrt {a+b x}}{12 a c^4 (b c-a d) \sqrt {c+d x}}+\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 236, normalized size = 1.01 \begin {gather*} \frac {x^2 \left (c^{3/2} d (a+b x)^{3/2} \left (-35 a^2 d^2+24 a b c d+3 b^2 c^2\right )-3 (c+d x) (b c-a d) \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \left (\sqrt {c} \sqrt {a+b x}-\sqrt {a} \sqrt {c+d x} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )\right )+6 a c^{7/2} (a+b x)^{3/2} (a d-b c)+3 c^{5/2} x (a+b x)^{3/2} (b c-a d) (7 a d+b c)}{12 a^2 c^{9/2} x^2 (c+d x)^{3/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.44, size = 323, normalized size = 1.38 \begin {gather*} \frac {\left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{9/2}}+\frac {\sqrt {a+b x} \left (105 a^4 d^3-\frac {175 a^3 c d^3 (a+b x)}{c+d x}-135 a^3 b c d^2+27 a^2 b^2 c^2 d+\frac {56 a^2 c^2 d^3 (a+b x)^2}{(c+d x)^2}+\frac {225 a^2 b c^2 d^2 (a+b x)}{c+d x}+\frac {3 b^3 c^4 (a+b x)}{c+d x}+3 a b^3 c^3-\frac {45 a b^2 c^3 d (a+b x)}{c+d x}+\frac {8 a c^3 d^3 (a+b x)^3}{(c+d x)^3}-\frac {72 a b c^3 d^2 (a+b x)^2}{(c+d x)^2}\right )}{12 a c^4 \sqrt {c+d x} (a d-b c) \left (a-\frac {c (a+b x)}{c+d x}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 7.83, size = 904, normalized size = 3.86 \begin {gather*} \left [-\frac {3 \, {\left ({\left (b^{3} c^{3} d^{2} + 9 \, a b^{2} c^{2} d^{3} - 45 \, a^{2} b c d^{4} + 35 \, a^{3} d^{5}\right )} x^{4} + 2 \, {\left (b^{3} c^{4} d + 9 \, a b^{2} c^{3} d^{2} - 45 \, a^{2} b c^{2} d^{3} + 35 \, a^{3} c d^{4}\right )} x^{3} + {\left (b^{3} c^{5} + 9 \, a b^{2} c^{4} d - 45 \, a^{2} b c^{3} d^{2} + 35 \, a^{3} c^{2} d^{3}\right )} x^{2}\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 (6 \, a^{2} b c^{5} - 6 \, a^{3} c^{4} d + {\left (3 \, a b^{2} c^{3} d^{2} - 100 \, a^{2} b c^{2} d^{3} + 105 \, a^{3} c d^{4}\right )} x^{3} + 2 \, {\left (3 \, a b^{2} c^{4} d - 69 \, a^{2} b c^{3} d^{2} + 70 \, a^{3} c^{2} d^{3}\right )} x^{2} + 3 \, {\left (a b^{2} c^{5} - 8 \, a^{2} b c^{4} d + 7 \, a^{3} c^{3} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left ({\left (a^{2} b c^{6} d^{2} - a^{3} c^{5} d^{3}\right )} x^{4} + 2 \, {\left (a^{2} b c^{7} d - a^{3} c^{6} d^{2}\right )} x^{3} + {\left (a^{2} b c^{8} - a^{3} c^{7} d\right )} x^{2}\right )}}, -\frac {3 \, {\left ({\left (b^{3} c^{3} d^{2} + 9 \, a b^{2} c^{2} d^{3} - 45 \, a^{2} b c d^{4} + 35 \, a^{3} d^{5}\right )} x^{4} + 2 \, {\left (b^{3} c^{4} d + 9 \, a b^{2} c^{3} d^{2} - 45 \, a^{2} b c^{2} d^{3} + 35 \, a^{3} c d^{4}\right )} x^{3} + {\left (b^{3} c^{5} + 9 \, a b^{2} c^{4} d - 45 \, a^{2} b c^{3} d^{2} + 35 \, a^{3} c^{2} d^{3}\right )} x^{2}\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 (6 \, a^{2} b c^{5} - 6 \, a^{3} c^{4} d + {\left (3 \, a b^{2} c^{3} d^{2} - 100 \, a^{2} b c^{2} d^{3} + 105 \, a^{3} c d^{4}\right )} x^{3} + 2 \, {\left (3 \, a b^{2} c^{4} d - 69 \, a^{2} b c^{3} d^{2} + 70 \, a^{3} c^{2} d^{3}\right )} x^{2} + 3 \, {\left (a b^{2} c^{5} - 8 \, a^{2} b c^{4} d + 7 \, a^{3} c^{3} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left ({\left (a^{2} b c^{6} d^{2} - a^{3} c^{5} d^{3}\right )} x^{4} + 2 \, {\left (a^{2} b c^{7} d - a^{3} c^{6} d^{2}\right )} x^{3} + {\left (a^{2} b c^{8} - a^{3} c^{7} d\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 9.29, size = 1204, normalized size = 5.15
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 988, normalized size = 4.22 \begin {gather*} -\frac {\left (105 a^{3} d^{5} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-135 a^{2} b c \,d^{4} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+27 a \,b^{2} c^{2} d^{3} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+3 b^{3} c^{3} d^{2} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+210 a^{3} c \,d^{4} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-270 a^{2} b \,c^{2} d^{3} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+54 a \,b^{2} c^{3} d^{2} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+6 b^{3} c^{4} d \,x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+105 a^{3} c^{2} d^{3} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-135 a^{2} b \,c^{3} d^{2} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+27 a \,b^{2} c^{4} d \,x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+3 b^{3} c^{5} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-210 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{4} x^{3}+200 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c \,d^{3} x^{3}-6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} d^{2} x^{3}-280 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c \,d^{3} x^{2}+276 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} d^{2} x^{2}-12 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{3} d \,x^{2}-42 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c^{2} d^{2} x +48 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{3} d x -6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{4} x +12 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c^{3} d -12 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{4}\right ) \sqrt {b x +a}}{24 \left (a d -b c \right ) \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \left (d x +c \right )^{\frac {3}{2}} a \,c^{4} x^{2}} \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 {\sqrt {a+b\,x}}{x^3\,{\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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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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