Optimal. Leaf size=119 \[ -\frac {2 (b c-a d) \sqrt {a+b x}}{c d \sqrt {c+d x}}-\frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}+\frac {2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{3/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {100, 163, 65,
223, 212, 95, 214} \begin {gather*} -\frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}+\frac {2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{3/2}}-\frac {2 \sqrt {a+b x} (b c-a d)}{c d \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 95
Rule 100
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx &=-\frac {2 (b c-a d) \sqrt {a+b x}}{c d \sqrt {c+d x}}+\frac {2 \int \frac {\frac {a^2 d}{2}+\frac {1}{2} b^2 c x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{c d}\\ &=-\frac {2 (b c-a d) \sqrt {a+b x}}{c d \sqrt {c+d x}}+\frac {a^2 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{c}+\frac {b^2 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{d}\\ &=-\frac {2 (b c-a d) \sqrt {a+b x}}{c d \sqrt {c+d x}}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{c}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{d}\\ &=-\frac {2 (b c-a d) \sqrt {a+b x}}{c d \sqrt {c+d x}}-\frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{d}\\ &=-\frac {2 (b c-a d) \sqrt {a+b x}}{c d \sqrt {c+d x}}-\frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}+\frac {2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.70, size = 165, normalized size = 1.39 \begin {gather*} \frac {2 \left (\frac {(-b c+a d) \sqrt {a+b x}}{c \sqrt {c+d x}}-\frac {a^{3/2} \sqrt {\frac {b}{d}} d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \left (-b x+\sqrt {\frac {b}{d}} \sqrt {a+b x} \sqrt {c+d x}\right )}{\sqrt {a} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} c^{3/2}}-b \sqrt {\frac {b}{d}} \log \left (\sqrt {a+b x}-\sqrt {\frac {b}{d}} \sqrt {c+d x}\right )\right )}{d} \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. \(305\) vs.
\(2(91)=182\).
time = 0.07, size = 306, normalized size = 2.57
method | result | size |
default | \(\frac {\left (-\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} d^{2} x \sqrt {b d}+\ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c d x \sqrt {a c}-\sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} c d +\ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, b^{2} c^{2}+2 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a d -2 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b c \right ) \sqrt {b x +a}}{\sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\, \sqrt {a c}\, \sqrt {d x +c}\, c d}\) | \(306\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 211 vs.
\(2 (91) = 182\).
time = 1.38, size = 956, normalized size = 8.03 \begin {gather*} \left [\frac {{\left (b c d x + b c^{2}\right )} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + {\left (a d^{2} x + a c d\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 (b c - a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (c d^{2} x + c^{2} d\right )}}, -\frac {2 \, {\left (b c d x + b c^{2}\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) - {\left (a d^{2} x + a c d\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 (b c - a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (c d^{2} x + c^{2} d\right )}}, \frac {2 \, {\left (a d^{2} x + a c d\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 ) + {\left (b c d x + b c^{2}\right )} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (b c - a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (c d^{2} x + c^{2} d\right )}}, \frac {{\left (a d^{2} x + a c d\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 ) - {\left (b c d x + b c^{2}\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) - 2 \, {\left (b c - a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{c d^{2} x + c^{2} d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right )^{\frac {3}{2}}}{x \left (c + d x\right )^{\frac {3}{2}}}\, dx \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. 205 vs.
\(2 (91) = 182\).
time = 1.38, size = 205, normalized size = 1.72 \begin {gather*} -\frac {2 \, \sqrt {b d} a^{2} b \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 )}{\sqrt {-a b c d} c {\left | b \right |}} - \frac {\sqrt {b d} b^{2} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{d^{2} {\left | b \right |}} - \frac {2 \, {\left (b^{3} c {\left | b \right |} - a b^{2} d {\left | b \right |}\right )} \sqrt {b x + a}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} b^{2} c d} \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 )}^{3/2}}{x\,{\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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