Optimal. Leaf size=70 \[ -\frac {\sqrt {c+d x}}{b (a+b x)}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2} \sqrt {b c-a d}} \]
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Rubi [A]
time = 0.02, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {43, 65, 214}
\begin {gather*} -\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2} \sqrt {b c-a d}}-\frac {\sqrt {c+d x}}{b (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 214
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x}}{(a+b x)^2} \, dx &=-\frac {\sqrt {c+d x}}{b (a+b x)}+\frac {d \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 b}\\ &=-\frac {\sqrt {c+d x}}{b (a+b x)}+\frac {\text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{b}\\ &=-\frac {\sqrt {c+d x}}{b (a+b x)}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2} \sqrt {b c-a d}}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 69, normalized size = 0.99 \begin {gather*} -\frac {\sqrt {c+d x}}{b (a+b x)}+\frac {d \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{3/2} \sqrt {-b c+a d}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(655\) vs. \(2(70)=140\).
time = 22.41, size = 597, normalized size = 8.53 \begin {gather*} \frac {-a b d \sqrt {\frac {a d-b c}{b}} \sqrt {c+d x}+\frac {a b d^2 \left (a^2 d-a b c+a b d x-b^2 c x\right ) \left (\text {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 ]-\text {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 ]\right ) \sqrt {-\frac {1}{b \left (a d-b c\right )^3}} \sqrt {\frac {a d-b c}{b}}}{2}+b^2 c \sqrt {\frac {a d-b c}{b}} \sqrt {c+d x}+2 d \text {ArcTan}\left [\frac {\sqrt {c+d x}}{\sqrt {\frac {a d-b c}{b}}}\right ] \left (a^2 d-a b c+a b d x-b^2 c x\right )+\frac {b^2 c d \left (a^2 d-a b c+a b d x-b^2 c x\right ) \left (\text {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 ]-\text {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 ]\right ) \sqrt {-\frac {1}{b \left (a d-b c\right )^3}} \sqrt {\frac {a d-b c}{b}}}{2}}{b^2 \sqrt {\frac {a d-b c}{b}} \left (a^2 d-a b c+a b d x-b^2 c x\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 73, normalized size = 1.04
method | result | size |
derivativedivides | \(2 d \left (-\frac {\sqrt {d x +c}}{2 b \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 b \sqrt {\left (a d -b c \right ) b}}\right )\) | \(73\) |
default | \(2 d \left (-\frac {\sqrt {d x +c}}{2 b \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 b \sqrt {\left (a d -b c \right ) b}}\right )\) | \(73\) |
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 = 0.32, size = 232, normalized size = 3.31 \begin {gather*} \left [\frac {\sqrt {b^{2} c - a b d} {\left (b d x + a d\right )} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) - 2 \, {\left (b^{2} c - a b d\right )} \sqrt {d x + c}}{2 \, {\left (a b^{3} c - a^{2} b^{2} d + {\left (b^{4} c - a b^{3} d\right )} x\right )}}, \frac {\sqrt {-b^{2} c + a b d} {\left (b d x + a d\right )} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) - {\left (b^{2} c - a b d\right )} \sqrt {d x + c}}{a b^{3} c - a^{2} b^{2} d + {\left (b^{4} c - a b^{3} d\right )} x}\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. 573 vs.
\(2 (58) = 116\).
time = 21.97, size = 573, normalized size = 8.19 \begin {gather*} - \frac {2 a d^{2} \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^{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 )}}{2 b} - \frac {a 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 )}}{2 b} - \frac {c 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} + \frac {c 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} + \frac {2 c d \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 \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{b^{2} \sqrt {\frac {a d}{b} - c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 82, normalized size = 1.17 \begin {gather*} -\frac {\sqrt {c+d x} d}{b \left (\left (c+d x\right ) b-c b+d a\right )}+\frac {2 d \arctan \left (\frac {b \sqrt {c+d x}}{\sqrt {-b^{2} c+a b d}}\right )}{b\cdot 2 \sqrt {-b^{2} c+a b d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 61, normalized size = 0.87 \begin {gather*} \frac {d\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{b^{3/2}\,\sqrt {a\,d-b\,c}}-\frac {d\,\sqrt {c+d\,x}}{d\,x\,b^2+a\,d\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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