Optimal. Leaf size=110 \[ -\frac {\sqrt {c+d x}}{2 b (a+b x)^2}-\frac {d \sqrt {c+d x}}{4 b (b c-a d) (a+b x)}+\frac {d^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{3/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {43, 44, 65, 214}
\begin {gather*} \frac {d^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{3/2}}-\frac {d \sqrt {c+d x}}{4 b (a+b x) (b c-a d)}-\frac {\sqrt {c+d x}}{2 b (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 44
Rule 65
Rule 214
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x}}{(a+b x)^3} \, dx &=-\frac {\sqrt {c+d x}}{2 b (a+b x)^2}+\frac {d \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx}{4 b}\\ &=-\frac {\sqrt {c+d x}}{2 b (a+b x)^2}-\frac {d \sqrt {c+d x}}{4 b (b c-a d) (a+b x)}-\frac {d^2 \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{8 b (b c-a d)}\\ &=-\frac {\sqrt {c+d x}}{2 b (a+b x)^2}-\frac {d \sqrt {c+d x}}{4 b (b c-a d) (a+b x)}-\frac {d \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{4 b (b c-a d)}\\ &=-\frac {\sqrt {c+d x}}{2 b (a+b x)^2}-\frac {d \sqrt {c+d x}}{4 b (b c-a d) (a+b x)}+\frac {d^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.43, size = 99, normalized size = 0.90 \begin {gather*} \frac {\frac {\sqrt {b} \sqrt {c+d x} (2 b c-a d+b d x)}{(-b c+a d) (a+b x)^2}+\frac {d^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}}{4 b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 106, normalized size = 0.96
method | result | size |
derivativedivides | \(2 d^{2} \left (\frac {\frac {\left (d x +c \right )^{\frac {3}{2}}}{8 a d -8 b c}-\frac {\sqrt {d x +c}}{8 b}}{\left (\left (d x +c \right ) b +a d -b c \right )^{2}}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) b \sqrt {\left (a d -b c \right ) b}}\right )\) | \(106\) |
default | \(2 d^{2} \left (\frac {\frac {\left (d x +c \right )^{\frac {3}{2}}}{8 a d -8 b c}-\frac {\sqrt {d x +c}}{8 b}}{\left (\left (d x +c \right ) b +a d -b c \right )^{2}}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) b \sqrt {\left (a d -b c \right ) b}}\right )\) | \(106\) |
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. 221 vs.
\(2 (90) = 180\).
time = 0.51, size = 456, normalized size = 4.15 \begin {gather*} \left [-\frac {{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt {b^{2} c - a b d} \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 (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2} + {\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {d x + c}}{8 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2} + {\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{2} + 2 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x\right )}}, -\frac {{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) + {\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2} + {\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {d x + c}}{4 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2} + {\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{2} + 2 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x\right )}}\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. 1658 vs.
\(2 (88) = 176\).
time = 93.91, size = 1658, normalized size = 15.07 \begin {gather*} - \frac {10 a^{2} d^{4} \sqrt {c + d x}}{8 a^{4} b d^{4} - 16 a^{3} b^{2} c d^{3} + 16 a^{3} b^{2} d^{4} x - 48 a^{2} b^{3} c d^{3} x + 8 a^{2} b^{3} d^{2} \left (c + d x\right )^{2} + 16 a b^{4} c^{3} d + 48 a b^{4} c^{2} d^{2} x - 16 a b^{4} c d \left (c + d x\right )^{2} - 8 b^{5} c^{4} - 16 b^{5} c^{3} d x + 8 b^{5} c^{2} \left (c + d x\right )^{2}} + \frac {20 a c d^{3} \sqrt {c + d x}}{8 a^{4} d^{4} - 16 a^{3} b c d^{3} + 16 a^{3} b d^{4} x - 48 a^{2} b^{2} c d^{3} x + 8 a^{2} b^{2} d^{2} \left (c + d x\right )^{2} + 16 a b^{3} c^{3} d + 48 a b^{3} c^{2} d^{2} x - 16 a b^{3} c d \left (c + d x\right )^{2} - 8 b^{4} c^{4} - 16 b^{4} c^{3} d x + 8 b^{4} c^{2} \left (c + d x\right )^{2}} - \frac {6 a d^{3} \left (c + d x\right )^{\frac {3}{2}}}{8 a^{4} d^{4} - 16 a^{3} b c d^{3} + 16 a^{3} b d^{4} x - 48 a^{2} b^{2} c d^{3} x + 8 a^{2} b^{2} d^{2} \left (c + d x\right )^{2} + 16 a b^{3} c^{3} d + 48 a b^{3} c^{2} d^{2} x - 16 a b^{3} c d \left (c + d x\right )^{2} - 8 b^{4} c^{4} - 16 b^{4} c^{3} d x + 8 b^{4} c^{2} \left (c + d x\right )^{2}} + \frac {3 a d^{3} \sqrt {- \frac {1}{b \left (a d - b c\right )^{5}}} \log {\left (- a^{3} d^{3} \sqrt {- \frac {1}{b \left (a d - b c\right )^{5}}} + 3 a^{2} b c d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{5}}} - 3 a b^{2} c^{2} d \sqrt {- \frac {1}{b \left (a d - b c\right )^{5}}} + b^{3} c^{3} \sqrt {- \frac {1}{b \left (a d - b c\right )^{5}}} + \sqrt {c + d x} \right )}}{8 b} - \frac {3 a d^{3} \sqrt {- \frac {1}{b \left (a d - b c\right )^{5}}} \log {\left (a^{3} d^{3} \sqrt {- \frac {1}{b \left (a d - b c\right )^{5}}} - 3 a^{2} b c d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{5}}} + 3 a b^{2} c^{2} d \sqrt {- \frac {1}{b \left (a d - b c\right )^{5}}} - b^{3} c^{3} \sqrt {- \frac {1}{b \left (a d - b c\right )^{5}}} + \sqrt {c + d x} \right )}}{8 b} - \frac {10 b c^{2} d^{2} \sqrt {c + d x}}{8 a^{4} d^{4} - 16 a^{3} b c d^{3} + 16 a^{3} b d^{4} x - 48 a^{2} b^{2} c d^{3} x + 8 a^{2} b^{2} d^{2} \left (c + d x\right )^{2} + 16 a b^{3} c^{3} d + 48 a b^{3} c^{2} d^{2} x - 16 a b^{3} c d \left (c + d x\right )^{2} - 8 b^{4} c^{4} - 16 b^{4} c^{3} d x + 8 b^{4} c^{2} \left (c + d x\right )^{2}} + \frac {6 b c d^{2} \left (c + d x\right )^{\frac {3}{2}}}{8 a^{4} d^{4} - 16 a^{3} b c d^{3} + 16 a^{3} b d^{4} x - 48 a^{2} b^{2} c d^{3} x + 8 a^{2} b^{2} d^{2} \left (c + d x\right )^{2} + 16 a b^{3} c^{3} d + 48 a b^{3} c^{2} d^{2} x - 16 a b^{3} c d \left (c + d x\right )^{2} - 8 b^{4} c^{4} - 16 b^{4} c^{3} d x + 8 b^{4} c^{2} \left (c + d x\right )^{2}} - \frac {3 c d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{5}}} \log {\left (- a^{3} d^{3} \sqrt {- \frac {1}{b \left (a d - b c\right )^{5}}} + 3 a^{2} b c d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{5}}} - 3 a b^{2} c^{2} d \sqrt {- \frac {1}{b \left (a d - b c\right )^{5}}} + b^{3} c^{3} \sqrt {- \frac {1}{b \left (a d - b c\right )^{5}}} + \sqrt {c + d x} \right )}}{8} + \frac {3 c d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{5}}} \log {\left (a^{3} d^{3} \sqrt {- \frac {1}{b \left (a d - b c\right )^{5}}} - 3 a^{2} b c d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{5}}} + 3 a b^{2} c^{2} d \sqrt {- \frac {1}{b \left (a d - b c\right )^{5}}} - b^{3} c^{3} \sqrt {- \frac {1}{b \left (a d - b c\right )^{5}}} + \sqrt {c + d x} \right )}}{8} + \frac {2 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 {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 {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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.15, size = 126, normalized size = 1.15 \begin {gather*} -\frac {d^{2} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{4 \, {\left (b^{2} c - a b d\right )} \sqrt {-b^{2} c + a b d}} - \frac {{\left (d x + c\right )}^{\frac {3}{2}} b d^{2} + \sqrt {d x + c} b c d^{2} - \sqrt {d x + c} a d^{3}}{4 \, {\left (b^{2} c - a b d\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.30, size = 135, normalized size = 1.23 \begin {gather*} \frac {d^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{4\,b^{3/2}\,{\left (a\,d-b\,c\right )}^{3/2}}-\frac {\frac {d^2\,\sqrt {c+d\,x}}{4\,b}-\frac {d^2\,{\left (c+d\,x\right )}^{3/2}}{4\,\left (a\,d-b\,c\right )}}{b^2\,{\left (c+d\,x\right )}^2-\left (2\,b^2\,c-2\,a\,b\,d\right )\,\left (c+d\,x\right )+a^2\,d^2+b^2\,c^2-2\,a\,b\,c\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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