Optimal. Leaf size=99 \[ -\frac {3 d}{(b c-a d)^2 \sqrt {c+d x}}-\frac {1}{(b c-a d) (a+b x) \sqrt {c+d x}}+\frac {3 \sqrt {b} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{5/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 99, 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 = {44, 53, 65, 214}
\begin {gather*} -\frac {3 d}{\sqrt {c+d x} (b c-a d)^2}-\frac {1}{(a+b x) \sqrt {c+d x} (b c-a d)}+\frac {3 \sqrt {b} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 53
Rule 65
Rule 214
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^2 (c+d x)^{3/2}} \, dx &=-\frac {1}{(b c-a d) (a+b x) \sqrt {c+d x}}-\frac {(3 d) \int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx}{2 (b c-a d)}\\ &=-\frac {3 d}{(b c-a d)^2 \sqrt {c+d x}}-\frac {1}{(b c-a d) (a+b x) \sqrt {c+d x}}-\frac {(3 b d) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 (b c-a d)^2}\\ &=-\frac {3 d}{(b c-a d)^2 \sqrt {c+d x}}-\frac {1}{(b c-a d) (a+b x) \sqrt {c+d x}}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{(b c-a d)^2}\\ &=-\frac {3 d}{(b c-a d)^2 \sqrt {c+d x}}-\frac {1}{(b c-a d) (a+b x) \sqrt {c+d x}}+\frac {3 \sqrt {b} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 90, normalized size = 0.91 \begin {gather*} -\frac {2 a d+b (c+3 d x)}{(b c-a d)^2 (a+b x) \sqrt {c+d x}}-\frac {3 \sqrt {b} d \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 100, normalized size = 1.01
method | result | size |
derivativedivides | \(2 d \left (-\frac {b \left (\frac {\sqrt {d x +c}}{2 \left (d x +c \right ) b +2 a d -2 b c}+\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{2}}-\frac {1}{\left (a d -b c \right )^{2} \sqrt {d x +c}}\right )\) | \(100\) |
default | \(2 d \left (-\frac {b \left (\frac {\sqrt {d x +c}}{2 \left (d x +c \right ) b +2 a d -2 b c}+\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{2}}-\frac {1}{\left (a d -b c \right )^{2} \sqrt {d x +c}}\right )\) | \(100\) |
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. 206 vs.
\(2 (85) = 170\).
time = 0.74, size = 423, normalized size = 4.27 \begin {gather*} \left [\frac {3 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, {\left (b c - a d\right )} \sqrt {d x + c} \sqrt {\frac {b}{b c - a d}}}{b x + a}\right ) - 2 \, {\left (3 \, b d x + b c + 2 \, a d\right )} \sqrt {d x + c}}{2 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x\right )}}, \frac {3 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {{\left (b c - a d\right )} \sqrt {d x + c} \sqrt {-\frac {b}{b c - a d}}}{b d x + b c}\right ) - {\left (3 \, b d x + b c + 2 \, a d\right )} \sqrt {d x + c}}{a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x}\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 {1}{\left (a + b x\right )^{2} \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 [A]
time = 1.20, size = 143, normalized size = 1.44 \begin {gather*} -\frac {3 \, b d \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b^{2} c + a b d}} - \frac {3 \, {\left (d x + c\right )} b d - 2 \, b c d + 2 \, a d^{2}}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left ({\left (d x + c\right )}^{\frac {3}{2}} b - \sqrt {d x + c} b c + \sqrt {d x + c} a d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 123, normalized size = 1.24 \begin {gather*} -\frac {\frac {2\,d}{a\,d-b\,c}+\frac {3\,b\,d\,\left (c+d\,x\right )}{{\left (a\,d-b\,c\right )}^2}}{b\,{\left (c+d\,x\right )}^{3/2}+\left (a\,d-b\,c\right )\,\sqrt {c+d\,x}}-\frac {3\,\sqrt {b}\,d\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^{5/2}}\right )}{{\left (a\,d-b\,c\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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