Optimal. Leaf size=115 \[ \frac {(b c-a d) \sqrt {c+d x}}{a b (a+b x)}-\frac {2 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2}+\frac {\sqrt {b c-a d} (2 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^2 b^{3/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {100, 162, 65,
214} \begin {gather*} \frac {\sqrt {b c-a d} (a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^2 b^{3/2}}-\frac {2 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2}+\frac {\sqrt {c+d x} (b c-a d)}{a b (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 162
Rule 214
Rubi steps
\begin {align*} \int \frac {(c+d x)^{3/2}}{x (a+b x)^2} \, dx &=\frac {(b c-a d) \sqrt {c+d x}}{a b (a+b x)}+\frac {\int \frac {b c^2+\frac {1}{2} d (b c+a d) x}{x (a+b x) \sqrt {c+d x}} \, dx}{a b}\\ &=\frac {(b c-a d) \sqrt {c+d x}}{a b (a+b x)}+\frac {c^2 \int \frac {1}{x \sqrt {c+d x}} \, dx}{a^2}-\frac {((b c-a d) (2 b c+a d)) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 a^2 b}\\ &=\frac {(b c-a d) \sqrt {c+d x}}{a b (a+b x)}+\frac {\left (2 c^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{a^2 d}-\frac {((b c-a d) (2 b c+a d)) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{a^2 b d}\\ &=\frac {(b c-a d) \sqrt {c+d x}}{a b (a+b x)}-\frac {2 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2}+\frac {\sqrt {b c-a d} (2 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^2 b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.48, size = 111, normalized size = 0.97 \begin {gather*} \frac {\frac {a (b c-a d) \sqrt {c+d x}}{b (a+b x)}+\frac {\sqrt {-b c+a d} (2 b c+a d) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{3/2}}-2 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 124, normalized size = 1.08
method | result | size |
derivativedivides | \(2 d^{2} \left (\frac {\left (a d -b c \right ) \left (-\frac {a d \sqrt {d x +c}}{2 b \left (b \left (d x +c \right )+a d -b c \right )}+\frac {\left (a d +2 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 b \sqrt {\left (a d -b c \right ) b}}\right )}{a^{2} d^{2}}-\frac {c^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{2} d^{2}}\right )\) | \(124\) |
default | \(2 d^{2} \left (\frac {\left (a d -b c \right ) \left (-\frac {a d \sqrt {d x +c}}{2 b \left (b \left (d x +c \right )+a d -b c \right )}+\frac {\left (a d +2 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 b \sqrt {\left (a d -b c \right ) b}}\right )}{a^{2} d^{2}}-\frac {c^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{2} d^{2}}\right )\) | \(124\) |
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 = 1.28, size = 624, normalized size = 5.43 \begin {gather*} \left [\frac {{\left (2 \, a b c + a^{2} d + {\left (2 \, b^{2} c + a b d\right )} x\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (b^{2} c x + a b c\right )} \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (a b c - a^{2} d\right )} \sqrt {d x + c}}{2 \, {\left (a^{2} b^{2} x + a^{3} b\right )}}, \frac {{\left (2 \, a b c + a^{2} d + {\left (2 \, b^{2} c + a b d\right )} x\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left (b^{2} c x + a b c\right )} \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + {\left (a b c - a^{2} d\right )} \sqrt {d x + c}}{a^{2} b^{2} x + a^{3} b}, \frac {4 \, {\left (b^{2} c x + a b c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left (2 \, a b c + a^{2} d + {\left (2 \, b^{2} c + a b d\right )} x\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (a b c - a^{2} d\right )} \sqrt {d x + c}}{2 \, {\left (a^{2} b^{2} x + a^{3} b\right )}}, \frac {{\left (2 \, a b c + a^{2} d + {\left (2 \, b^{2} c + a b d\right )} x\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + 2 \, {\left (b^{2} c x + a b c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left (a b c - a^{2} d\right )} \sqrt {d x + c}}{a^{2} b^{2} x + a^{3} b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 928 vs.
\(2 (100) = 200\).
time = 70.70, size = 928, normalized size = 8.07 \begin {gather*} - \frac {2 a d^{3} \sqrt {c + d x}}{2 a^{2} b d^{2} - 2 a b^{2} c d + 2 a b^{2} d^{2} x - 2 b^{3} c d x} + \frac {a d^{3} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (- a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 b} - \frac {a d^{3} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 b} - \frac {2 b c^{2} d \sqrt {c + d x}}{2 a^{3} d^{2} - 2 a^{2} b c d + 2 a^{2} b d^{2} x - 2 a b^{2} c d x} - c d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (- a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )} + c d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )} + \frac {4 c d^{2} \sqrt {c + d x}}{2 a^{2} d^{2} - 2 a b c d + 2 a b d^{2} x - 2 b^{2} c d x} + \frac {2 d^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{b^{2} \sqrt {\frac {a d}{b} - c}} + \frac {b c^{2} d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (- a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 a} - \frac {b c^{2} d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 a} - \frac {2 c^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{a^{2} \sqrt {\frac {a d}{b} - c}} + \frac {2 c^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{a^{2} \sqrt {- c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 144, normalized size = 1.25 \begin {gather*} \frac {2 \, c^{2} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c}} - \frac {{\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{2} b} + \frac {\sqrt {d x + c} b c d - \sqrt {d x + c} a d^{2}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.60, size = 482, normalized size = 4.19 \begin {gather*} -\frac {2\,\mathrm {atanh}\left (\frac {4\,d^6\,\sqrt {c^3}\,\sqrt {c+d\,x}}{4\,c^2\,d^6+\frac {8\,b\,c^3\,d^5}{a}-\frac {12\,b^2\,c^4\,d^4}{a^2}}+\frac {8\,c\,d^5\,\sqrt {c^3}\,\sqrt {c+d\,x}}{8\,c^3\,d^5+\frac {4\,a\,c^2\,d^6}{b}-\frac {12\,b\,c^4\,d^4}{a}}-\frac {12\,b\,c^2\,d^4\,\sqrt {c^3}\,\sqrt {c+d\,x}}{8\,a\,c^3\,d^5-12\,b\,c^4\,d^4+\frac {4\,a^2\,c^2\,d^6}{b}}\right )\,\sqrt {c^3}}{a^2}-\frac {\mathrm {atanh}\left (\frac {10\,c^2\,d^5\,\sqrt {b^4\,c-a\,b^3\,d}\,\sqrt {c+d\,x}}{2\,a^2\,c\,d^7+2\,b^2\,c^3\,d^5-\frac {12\,b^3\,c^4\,d^4}{a}+8\,a\,b\,c^2\,d^6}+\frac {12\,c^3\,d^4\,\sqrt {b^4\,c-a\,b^3\,d}\,\sqrt {c+d\,x}}{8\,a^2\,c^2\,d^6-12\,b^2\,c^4\,d^4+\frac {2\,a^3\,c\,d^7}{b}+2\,a\,b\,c^3\,d^5}+\frac {2\,c\,d^6\,\sqrt {b^4\,c-a\,b^3\,d}\,\sqrt {c+d\,x}}{8\,b^2\,c^2\,d^6+2\,a\,b\,c\,d^7+\frac {2\,b^3\,c^3\,d^5}{a}-\frac {12\,b^4\,c^4\,d^4}{a^2}}\right )\,\left (a\,d+2\,b\,c\right )\,\sqrt {-b^3\,\left (a\,d-b\,c\right )}}{a^2\,b^3}-\frac {d\,\left (a\,d-b\,c\right )\,\sqrt {c+d\,x}}{a\,b\,\left (a\,d-b\,c+b\,\left (c+d\,x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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