Optimal. Leaf size=154 \[ \frac {15 (b c-a d)^2 \sqrt {a+b x}}{4 c^3 \sqrt {c+d x}}-\frac {5 (b c-a d) (a+b x)^{3/2}}{4 c^2 x \sqrt {c+d x}}-\frac {(a+b x)^{5/2}}{2 c x^2 \sqrt {c+d x}}-\frac {15 \sqrt {a} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 c^{7/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {96, 95, 214}
\begin {gather*} -\frac {15 \sqrt {a} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 c^{7/2}}+\frac {15 \sqrt {a+b x} (b c-a d)^2}{4 c^3 \sqrt {c+d x}}-\frac {5 (a+b x)^{3/2} (b c-a d)}{4 c^2 x \sqrt {c+d x}}-\frac {(a+b x)^{5/2}}{2 c x^2 \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 214
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2}}{x^3 (c+d x)^{3/2}} \, dx &=-\frac {(a+b x)^{5/2}}{2 c x^2 \sqrt {c+d x}}+\frac {(5 (b c-a d)) \int \frac {(a+b x)^{3/2}}{x^2 (c+d x)^{3/2}} \, dx}{4 c}\\ &=-\frac {5 (b c-a d) (a+b x)^{3/2}}{4 c^2 x \sqrt {c+d x}}-\frac {(a+b x)^{5/2}}{2 c x^2 \sqrt {c+d x}}+\frac {\left (15 (b c-a d)^2\right ) \int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}} \, dx}{8 c^2}\\ &=\frac {15 (b c-a d)^2 \sqrt {a+b x}}{4 c^3 \sqrt {c+d x}}-\frac {5 (b c-a d) (a+b x)^{3/2}}{4 c^2 x \sqrt {c+d x}}-\frac {(a+b x)^{5/2}}{2 c x^2 \sqrt {c+d x}}+\frac {\left (15 a (b c-a d)^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 c^3}\\ &=\frac {15 (b c-a d)^2 \sqrt {a+b x}}{4 c^3 \sqrt {c+d x}}-\frac {5 (b c-a d) (a+b x)^{3/2}}{4 c^2 x \sqrt {c+d x}}-\frac {(a+b x)^{5/2}}{2 c x^2 \sqrt {c+d x}}+\frac {\left (15 a (b c-a 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^3}\\ &=\frac {15 (b c-a d)^2 \sqrt {a+b x}}{4 c^3 \sqrt {c+d x}}-\frac {5 (b c-a d) (a+b x)^{3/2}}{4 c^2 x \sqrt {c+d x}}-\frac {(a+b x)^{5/2}}{2 c x^2 \sqrt {c+d x}}-\frac {15 \sqrt {a} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 c^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 132, normalized size = 0.86 \begin {gather*} \frac {\sqrt {a+b x} \left (8 b^2 c^2 x^2-a b c x (9 c+25 d x)+a^2 \left (-2 c^2+5 c d x+15 d^2 x^2\right )\right )}{4 c^3 x^2 \sqrt {c+d x}}-\frac {15 \sqrt {a} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 c^{7/2}} \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. \(506\) vs.
\(2(122)=244\).
time = 0.09, size = 507, normalized size = 3.29
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} d^{3} x^{3}-30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x^{3}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d \,x^{3}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} c \,d^{2} x^{2}-30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b \,c^{2} d \,x^{2}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{2} c^{3} x^{2}-30 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} d^{2} x^{2}+50 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b c d \,x^{2}-16 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{2} c^{2} x^{2}-10 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c d x +18 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b \,c^{2} x +4 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c^{2} \sqrt {a c}\right )}{8 c^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{2} \sqrt {a c}\, \sqrt {d x +c}}\) | \(507\) |
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 = 2.00, size = 479, normalized size = 3.11 \begin {gather*} \left [\frac {15 \, {\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {\frac {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^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (2 \, a^{2} c^{2} - {\left (8 \, b^{2} c^{2} - 25 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{2} + {\left (9 \, a b c^{2} - 5 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left (c^{3} d x^{3} + c^{4} x^{2}\right )}}, \frac {15 \, {\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{2} c^{2} - {\left (8 \, b^{2} c^{2} - 25 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{2} + {\left (9 \, a b c^{2} - 5 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left (c^{3} d x^{3} + c^{4} 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. 1121 vs.
\(2 (122) = 244\).
time = 4.13, size = 1121, normalized size = 7.28 \begin {gather*} \frac {2 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \sqrt {b x + a}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} c^{3} {\left | b \right |}} - \frac {15 \, {\left (\sqrt {b d} a b^{4} c^{2} - 2 \, \sqrt {b d} a^{2} b^{3} c d + \sqrt {b d} a^{3} 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^{3} {\left | b \right |}} - \frac {9 \, \sqrt {b d} a b^{10} c^{5} - 43 \, \sqrt {b d} a^{2} b^{9} c^{4} d + 82 \, \sqrt {b d} a^{3} b^{8} c^{3} d^{2} - 78 \, \sqrt {b d} a^{4} b^{7} c^{2} d^{3} + 37 \, \sqrt {b d} a^{5} b^{6} c d^{4} - 7 \, \sqrt {b d} a^{6} b^{5} d^{5} - 27 \, \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^{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^{2} b^{7} c^{3} d - 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 )}^{2} a^{3} 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^{4} 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^{5} b^{4} d^{4} + 27 \, \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^{6} c^{3} - 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} a^{2} b^{5} c^{2} d + 13 \, \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^{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^{4} b^{3} d^{3} - 9 \, \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^{4} c^{2} - 6 \, \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^{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^{3} 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^{3} {\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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/2}}{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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