Optimal. Leaf size=61 \[ -\frac {b \sqrt {1-(c+d x)^2}}{2 d e^3 (c+d x)}-\frac {a+b \text {ArcSin}(c+d x)}{2 d e^3 (c+d x)^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4889, 12, 4723,
270} \begin {gather*} -\frac {a+b \text {ArcSin}(c+d x)}{2 d e^3 (c+d x)^2}-\frac {b \sqrt {1-(c+d x)^2}}{2 d e^3 (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 4723
Rule 4889
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c+d x)}{(c e+d e x)^3} \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{e^3 x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{x^3} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac {a+b \sin ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}+\frac {b \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-x^2}} \, dx,x,c+d x\right )}{2 d e^3}\\ &=-\frac {b \sqrt {1-(c+d x)^2}}{2 d e^3 (c+d x)}-\frac {a+b \sin ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 49, normalized size = 0.80 \begin {gather*} -\frac {a+b (c+d x) \sqrt {1-(c+d x)^2}+b \text {ArcSin}(c+d x)}{2 d e^3 (c+d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 62, normalized size = 1.02
method | result | size |
derivativedivides | \(\frac {-\frac {a}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b \left (-\frac {\arcsin \left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3}}}{d}\) | \(62\) |
default | \(\frac {-\frac {a}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b \left (-\frac {\arcsin \left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3}}}{d}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs.
\(2 (53) = 106\).
time = 0.48, size = 112, normalized size = 1.84 \begin {gather*} -\frac {1}{2} \, b {\left (\frac {\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} d}{d^{3} x e^{3} + c d^{2} e^{3}} + \frac {\arcsin \left (d x + c\right )}{d^{3} x^{2} e^{3} + 2 \, c d^{2} x e^{3} + c^{2} d e^{3}}\right )} - \frac {a}{2 \, {\left (d^{3} x^{2} e^{3} + 2 \, c d^{2} x e^{3} + c^{2} d e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.57, size = 95, normalized size = 1.56 \begin {gather*} \frac {{\left (a d^{2} x^{2} + 2 \, a c d x - b c^{2} \arcsin \left (d x + c\right ) - {\left (b c^{2} d x + b c^{3}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}\right )} e^{\left (-3\right )}}{2 \, {\left (c^{2} d^{3} x^{2} + 2 \, c^{3} d^{2} x + c^{4} 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*} \frac {\int \frac {a}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {b \operatorname {asin}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \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. 231 vs.
\(2 (55) = 110\).
time = 0.41, size = 231, normalized size = 3.79 \begin {gather*} -\frac {b \arcsin \left (d x + c\right )}{4 \, d e^{3}} - \frac {{\left (d x + c\right )}^{2} b \arcsin \left (d x + c\right )}{8 \, d e^{3} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}} - \frac {b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2} \arcsin \left (d x + c\right )}{8 \, {\left (d x + c\right )}^{2} d e^{3}} - \frac {a}{4 \, d e^{3}} - \frac {{\left (d x + c\right )}^{2} a}{8 \, d e^{3} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}} + \frac {{\left (d x + c\right )} b}{4 \, d e^{3} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}} - \frac {b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}}{4 \, {\left (d x + c\right )} d e^{3}} - \frac {a {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}}{8 \, {\left (d x + c\right )}^{2} d e^{3}} \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 = -0.02 \begin {gather*} \int \frac {a+b\,\mathrm {asin}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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