Optimal. Leaf size=81 \[ \frac {2 \sqrt {e (c+d x)} \left (a+b \sin ^{-1}(c+d x)\right )}{d e}+\frac {4 b \sqrt {e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {-c-d x+1}}{\sqrt {2}}\right )\right |2\right )}{d e \sqrt {c+d x}} \]
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Rubi [A] time = 0.08, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4805, 4627, 320, 318, 424} \[ \frac {2 \sqrt {e (c+d x)} \left (a+b \sin ^{-1}(c+d x)\right )}{d e}+\frac {4 b \sqrt {e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {-c-d x+1}}{\sqrt {2}}\right )\right |2\right )}{d e \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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Rule 318
Rule 320
Rule 424
Rule 4627
Rule 4805
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c+d x)}{\sqrt {c e+d e x}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{\sqrt {e x}} \, dx,x,c+d x\right )}{d}\\ &=\frac {2 \sqrt {e (c+d x)} \left (a+b \sin ^{-1}(c+d x)\right )}{d e}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {\sqrt {e x}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d e}\\ &=\frac {2 \sqrt {e (c+d x)} \left (a+b \sin ^{-1}(c+d x)\right )}{d e}-\frac {\left (2 b \sqrt {e (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d e \sqrt {c+d x}}\\ &=\frac {2 \sqrt {e (c+d x)} \left (a+b \sin ^{-1}(c+d x)\right )}{d e}+\frac {\left (4 b \sqrt {e (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-2 x^2}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c-d x}}{\sqrt {2}}\right )}{d e \sqrt {c+d x}}\\ &=\frac {2 \sqrt {e (c+d x)} \left (a+b \sin ^{-1}(c+d x)\right )}{d e}+\frac {4 b \sqrt {e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {1-c-d x}}{\sqrt {2}}\right )\right |2\right )}{d e \sqrt {c+d x}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 59, normalized size = 0.73 \[ -\frac {2 \sqrt {e (c+d x)} \left (2 b (c+d x) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};(c+d x)^2\right )-3 \left (a+b \sin ^{-1}(c+d x)\right )\right )}{3 d e} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arcsin \left (d x + c\right ) + a}{\sqrt {d e x + c e}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \arcsin \left (d x + c\right ) + a}{\sqrt {d e x + c e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 149, normalized size = 1.84 \[ \frac {2 a \sqrt {d e x +c e}+2 b \left (\sqrt {d e x +c e}\, \arcsin \left (\frac {d e x +c e}{e}\right )+\frac {2 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {\frac {d e x +c e}{e}+1}\, \left (\EllipticF \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\EllipticE \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{\sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (b d \int \frac {\sqrt {d x + c} \sqrt {-d x - c + 1}}{\sqrt {d x + c + 1} d x + \sqrt {d x + c + 1} c - \sqrt {d x + c + 1}}\,{d x} + \sqrt {d x + c} b \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right ) + \sqrt {d x + c} a\right )}}{d \sqrt {e}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asin}\left (c+d\,x\right )}{\sqrt {c\,e+d\,e\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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