Optimal. Leaf size=171 \[ \frac {\left (-15 a^2 d^2+6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{7/2}}-\frac {d \sqrt {a+b x} (b c-15 a d)}{4 a c^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x} (b c-5 a d)}{4 a c^2 x \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}} \]
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Rubi [A] time = 0.13, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {\left (-15 a^2 d^2+6 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^{7/2}}-\frac {d \sqrt {a+b x} (b c-15 a d)}{4 a c^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x} (b c-5 a d)}{4 a c^2 x \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}} \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)^{3/2}} \, dx &=-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}+\frac {\int \frac {\frac {1}{2} (b c-5 a d)-2 b d x}{x^2 \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{2 c}\\ &=-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}-\frac {(b c-5 a d) \sqrt {a+b x}}{4 a c^2 x \sqrt {c+d x}}-\frac {\int \frac {\frac {1}{4} \left (b^2 c^2+6 a b c d-15 a^2 d^2\right )+\frac {1}{2} b d (b c-5 a d) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{2 a c^2}\\ &=-\frac {d (b c-15 a d) \sqrt {a+b x}}{4 a c^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}-\frac {(b c-5 a d) \sqrt {a+b x}}{4 a c^2 x \sqrt {c+d x}}+\frac {\int -\frac {(b c-a d) \left (b^2 c^2+6 a b c d-15 a^2 d^2\right )}{8 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{a c^3 (b c-a d)}\\ &=-\frac {d (b c-15 a d) \sqrt {a+b x}}{4 a c^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}-\frac {(b c-5 a d) \sqrt {a+b x}}{4 a c^2 x \sqrt {c+d x}}-\frac {\left (b^2 c^2+6 a b c d-15 a^2 d^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a c^3}\\ &=-\frac {d (b c-15 a d) \sqrt {a+b x}}{4 a c^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}-\frac {(b c-5 a d) \sqrt {a+b x}}{4 a c^2 x \sqrt {c+d x}}-\frac {\left (b^2 c^2+6 a b c d-15 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^3}\\ &=-\frac {d (b c-15 a d) \sqrt {a+b x}}{4 a c^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}-\frac {(b c-5 a d) \sqrt {a+b x}}{4 a c^2 x \sqrt {c+d x}}+\frac {\left (b^2 c^2+6 a b c d-15 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^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 130, normalized size = 0.76 \begin {gather*} \frac {\left (-15 a^2 d^2+6 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^{7/2}}+\frac {\sqrt {a+b x} \left (a \left (-2 c^2+5 c d x+15 d^2 x^2\right )-b c x (c+d x)\right )}{4 a c^3 x^2 \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.28, size = 224, normalized size = 1.31 \begin {gather*} \frac {\left (-15 a^2 d^2+6 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^{7/2}}+\frac {\sqrt {a+b x} \left (15 a^3 d^2-\frac {25 a^2 c d^2 (a+b x)}{c+d x}-6 a^2 b c d-\frac {b^2 c^3 (a+b x)}{c+d x}-a b^2 c^2+\frac {8 a c^2 d^2 (a+b x)^2}{(c+d x)^2}+\frac {10 a b c^2 d (a+b x)}{c+d x}\right )}{4 a c^3 \sqrt {c+d x} \left (a-\frac {c (a+b x)}{c+d x}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 3.28, size = 474, normalized size = 2.77 \begin {gather*} \left [-\frac {{\left ({\left (b^{2} c^{2} d + 6 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{3} + {\left (b^{2} c^{3} + 6 \, a b c^{2} d - 15 \, a^{2} c d^{2}\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 (2 \, a^{2} c^{3} + {\left (a b c^{2} d - 15 \, a^{2} c d^{2}\right )} x^{2} + {\left (a b c^{3} - 5 \, a^{2} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left (a^{2} c^{4} d x^{3} + a^{2} c^{5} x^{2}\right )}}, -\frac {{\left ({\left (b^{2} c^{2} d + 6 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{3} + {\left (b^{2} c^{3} + 6 \, a b c^{2} d - 15 \, a^{2} c d^{2}\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 (2 \, a^{2} c^{3} + {\left (a b c^{2} d - 15 \, a^{2} c d^{2}\right )} x^{2} + {\left (a b c^{3} - 5 \, a^{2} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left (a^{2} c^{4} d x^{3} + a^{2} c^{5} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 15.39, size = 1092, normalized size = 6.39 \begin {gather*} \frac {2 \, \sqrt {b x + a} b^{2} d^{2}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} c^{3} {\left | b \right |}} + \frac {{\left (\sqrt {b d} b^{4} c^{2} + 6 \, \sqrt {b d} a b^{3} c d - 15 \, \sqrt {b d} a^{2} b^{2} d^{2}\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 )}{4 \, \sqrt {-a b c d} a b c^{3} {\left | b \right |}} - \frac {\sqrt {b d} b^{10} c^{5} - 11 \, \sqrt {b d} a b^{9} c^{4} d + 34 \, \sqrt {b d} a^{2} b^{8} c^{3} d^{2} - 46 \, \sqrt {b d} a^{3} b^{7} c^{2} d^{3} + 29 \, \sqrt {b d} a^{4} b^{6} c d^{4} - 7 \, \sqrt {b d} a^{5} b^{5} d^{5} - 3 \, \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} b^{8} c^{4} + 28 \, \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 b^{7} c^{3} d - 26 \, \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^{6} c^{2} d^{2} - 20 \, \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^{5} c d^{3} + 21 \, \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^{4} b^{4} d^{4} + 3 \, \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 )}^{4} b^{6} c^{3} - 19 \, \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 )}^{4} a b^{5} c^{2} d - 11 \, \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 )}^{4} a^{2} b^{4} c d^{2} - 21 \, \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 )}^{4} a^{3} b^{3} d^{3} - \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 )}^{6} b^{4} c^{2} + 2 \, \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 )}^{6} a b^{3} c d + 7 \, \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 )}^{6} a^{2} b^{2} d^{2}}{2 \, {\left (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}\right )}^{2} a c^{3} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 467, normalized size = 2.73 \begin {gather*} -\frac {\sqrt {b x +a}\, \left (15 a^{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 )-6 a b c \,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 )-b^{2} c^{2} 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 )+15 a^{2} c \,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 )-6 a b \,c^{2} 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 )-b^{2} c^{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 )-30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,d^{2} x^{2}+2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c d \,x^{2}-10 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a c d x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b \,c^{2} x +4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,c^{2}\right )}{8 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {d x +c}\, a \,c^{3} 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 = -0.01 \begin {gather*} \int \frac {\sqrt {a+b\,x}}{x^3\,{\left (c+d\,x\right )}^{3/2}} \,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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