Optimal. Leaf size=204 \[ -\frac {d (23 b c-35 a d) \sqrt {a+b x}}{12 c^3 (c+d x)^{3/2}}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(5 b c-7 a d) \sqrt {a+b x}}{4 c^2 x (c+d x)^{3/2}}-\frac {5 d (11 b c-21 a d) \sqrt {a+b x}}{12 c^4 \sqrt {c+d x}}-\frac {\left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 \sqrt {a} c^{9/2}} \]
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Rubi [A]
time = 0.16, antiderivative size = 204, 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 = {100, 156, 157,
12, 95, 214} \begin {gather*} -\frac {\left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 \sqrt {a} c^{9/2}}-\frac {5 d \sqrt {a+b x} (11 b c-21 a d)}{12 c^4 \sqrt {c+d x}}-\frac {d \sqrt {a+b x} (23 b c-35 a d)}{12 c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x} (5 b c-7 a d)}{4 c^2 x (c+d x)^{3/2}}-\frac {a \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 95
Rule 100
Rule 156
Rule 157
Rule 214
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2}}{x^3 (c+d x)^{5/2}} \, dx &=-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {\int \frac {-\frac {1}{2} a (5 b c-7 a d)-b (2 b c-3 a d) x}{x^2 \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{2 c}\\ &=-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(5 b c-7 a d) \sqrt {a+b x}}{4 c^2 x (c+d x)^{3/2}}+\frac {\int \frac {\frac {1}{4} a \left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right )-a b d (5 b c-7 a d) x}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{2 a c^2}\\ &=-\frac {d (23 b c-35 a d) \sqrt {a+b x}}{12 c^3 (c+d x)^{3/2}}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(5 b c-7 a d) \sqrt {a+b x}}{4 c^2 x (c+d x)^{3/2}}-\frac {\int \frac {-\frac {3}{8} a (b c-a d) \left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right )+\frac {1}{4} a b d (23 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 (23 b c-35 a d) \sqrt {a+b x}}{12 c^3 (c+d x)^{3/2}}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(5 b c-7 a d) \sqrt {a+b x}}{4 c^2 x (c+d x)^{3/2}}-\frac {5 d (11 b c-21 a d) \sqrt {a+b x}}{12 c^4 \sqrt {c+d x}}+\frac {2 \int \frac {3 a (b c-a d)^2 \left (3 b^2 c^2-30 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 (23 b c-35 a d) \sqrt {a+b x}}{12 c^3 (c+d x)^{3/2}}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(5 b c-7 a d) \sqrt {a+b x}}{4 c^2 x (c+d x)^{3/2}}-\frac {5 d (11 b c-21 a d) \sqrt {a+b x}}{12 c^4 \sqrt {c+d x}}+\frac {\left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 c^4}\\ &=-\frac {d (23 b c-35 a d) \sqrt {a+b x}}{12 c^3 (c+d x)^{3/2}}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(5 b c-7 a d) \sqrt {a+b x}}{4 c^2 x (c+d x)^{3/2}}-\frac {5 d (11 b c-21 a d) \sqrt {a+b x}}{12 c^4 \sqrt {c+d x}}+\frac {\left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 c^4}\\ &=-\frac {d (23 b c-35 a d) \sqrt {a+b x}}{12 c^3 (c+d x)^{3/2}}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(5 b c-7 a d) \sqrt {a+b x}}{4 c^2 x (c+d x)^{3/2}}-\frac {5 d (11 b c-21 a d) \sqrt {a+b x}}{12 c^4 \sqrt {c+d x}}-\frac {\left (3 b^2 c^2-30 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 \sqrt {a} c^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 153, normalized size = 0.75 \begin {gather*} \frac {\sqrt {a+b x} \left (-b c x \left (15 c^2+78 c d x+55 d^2 x^2\right )+a \left (-6 c^3+21 c^2 d x+140 c d^2 x^2+105 d^3 x^3\right )\right )}{12 c^4 x^2 (c+d x)^{3/2}}-\frac {\left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{4 \sqrt {a} c^{9/2}} \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. \(678\) vs.
\(2(166)=332\).
time = 0.07, size = 679, normalized size = 3.33
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \left (105 \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} d^{4} x^{4}-90 \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 c \,d^{3} x^{4}+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 ) b^{2} c^{2} d^{2} x^{4}+210 \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} c \,d^{3} x^{3}-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 \,c^{2} d^{2} x^{3}+18 \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^{2} c^{3} d \,x^{3}+105 \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} c^{2} d^{2} x^{2}-90 \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 \,c^{3} d \,x^{2}+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 ) b^{2} c^{4} x^{2}-210 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a \,d^{3} x^{3}+110 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, b c \,d^{2} x^{3}-280 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a c \,d^{2} x^{2}+156 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, b \,c^{2} d \,x^{2}-42 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a \,c^{2} d x +30 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, b \,c^{3} x +12 a \,c^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\right )}{24 c^{4} \sqrt {a c}\, x^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \left (d x +c \right )^{\frac {3}{2}}}\) | \(679\) |
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 = 3.27, size = 634, normalized size = 3.11 \begin {gather*} \left [\frac {3 \, {\left ({\left (3 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (3 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{3} + {\left (3 \, b^{2} c^{4} - 30 \, a b c^{3} d + 35 \, a^{2} c^{2} 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 (6 \, a^{2} c^{4} + 5 \, {\left (11 \, a b c^{2} d^{2} - 21 \, a^{2} c d^{3}\right )} x^{3} + 2 \, {\left (39 \, a b c^{3} d - 70 \, a^{2} c^{2} d^{2}\right )} x^{2} + 3 \, {\left (5 \, a b c^{4} - 7 \, a^{2} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (a c^{5} d^{2} x^{4} + 2 \, a c^{6} d x^{3} + a c^{7} x^{2}\right )}}, \frac {3 \, {\left ({\left (3 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (3 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{3} + {\left (3 \, b^{2} c^{4} - 30 \, a b c^{3} d + 35 \, a^{2} c^{2} 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 (6 \, a^{2} c^{4} + 5 \, {\left (11 \, a b c^{2} d^{2} - 21 \, a^{2} c d^{3}\right )} x^{3} + 2 \, {\left (39 \, a b c^{3} d - 70 \, a^{2} c^{2} d^{2}\right )} x^{2} + 3 \, {\left (5 \, a b c^{4} - 7 \, a^{2} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left (a c^{5} d^{2} x^{4} + 2 \, a c^{6} d x^{3} + a c^{7} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1234 vs.
\(2 (166) = 332\).
time = 2.39, size = 1234, normalized size = 6.05 \begin {gather*} -\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (5 \, b^{5} c^{6} d^{3} {\left | b \right |} - 14 \, a b^{4} c^{5} d^{4} {\left | b \right |} + 9 \, a^{2} b^{3} c^{4} d^{5} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{3} c^{9} d - a b^{2} c^{8} d^{2}} + \frac {3 \, {\left (2 \, b^{6} c^{7} d^{2} {\left | b \right |} - 7 \, a b^{5} c^{6} d^{3} {\left | b \right |} + 8 \, a^{2} b^{4} c^{5} d^{4} {\left | b \right |} - 3 \, a^{3} b^{3} c^{4} d^{5} {\left | b \right |}\right )}}{b^{3} c^{9} d - a b^{2} c^{8} d^{2}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {{\left (3 \, \sqrt {b d} b^{4} c^{2} - 30 \, \sqrt {b d} a b^{3} c d + 35 \, \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} b c^{4} {\left | b \right |}} - \frac {5 \, \sqrt {b d} b^{10} c^{5} - 31 \, \sqrt {b d} a b^{9} c^{4} d + 74 \, \sqrt {b d} a^{2} b^{8} c^{3} d^{2} - 86 \, \sqrt {b d} a^{3} b^{7} c^{2} d^{3} + 49 \, \sqrt {b d} a^{4} b^{6} c d^{4} - 11 \, \sqrt {b d} a^{5} b^{5} d^{5} - 15 \, \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} + 52 \, \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} - 44 \, \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} + 33 \, \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} + 15 \, \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} - 23 \, \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 - 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 )}^{4} a^{2} b^{4} c d^{2} - 33 \, \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} - 5 \, \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 + 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 )}^{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} c^{4} {\left | b \right |}} \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 )}^{3/2}}{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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