Optimal. Leaf size=547 \[ \frac {35 c \sqrt [4]{1-a x} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{64 a \sqrt {1-a^2 x^2}}+\frac {7 c (1-a x)^{5/4} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{32 a \sqrt {1-a^2 x^2}}-\frac {7 c (1-a x)^{9/4} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{24 a \sqrt {1-a^2 x^2}}-\frac {c (1-a x)^{9/4} (1+a x)^{7/4} \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}}+\frac {35 c \sqrt {c-a^2 c x^2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a \sqrt {1-a^2 x^2}}-\frac {35 c \sqrt {c-a^2 c x^2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a \sqrt {1-a^2 x^2}}+\frac {35 c \sqrt {c-a^2 c x^2} \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a \sqrt {1-a^2 x^2}}-\frac {35 c \sqrt {c-a^2 c x^2} \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.23, antiderivative size = 547, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {6278, 6275,
52, 65, 246, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {35 c \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right ) \sqrt {c-a^2 c x^2}}{64 \sqrt {2} a \sqrt {1-a^2 x^2}}-\frac {35 c \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right ) \sqrt {c-a^2 c x^2}}{64 \sqrt {2} a \sqrt {1-a^2 x^2}}-\frac {c (a x+1)^{7/4} (1-a x)^{9/4} \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}}-\frac {7 c (a x+1)^{3/4} (1-a x)^{9/4} \sqrt {c-a^2 c x^2}}{24 a \sqrt {1-a^2 x^2}}+\frac {7 c (a x+1)^{3/4} (1-a x)^{5/4} \sqrt {c-a^2 c x^2}}{32 a \sqrt {1-a^2 x^2}}+\frac {35 c (a x+1)^{3/4} \sqrt [4]{1-a x} \sqrt {c-a^2 c x^2}}{64 a \sqrt {1-a^2 x^2}}+\frac {35 c \sqrt {c-a^2 c x^2} \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt {2} a \sqrt {1-a^2 x^2}}-\frac {35 c \sqrt {c-a^2 c x^2} \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt {2} a \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 210
Rule 217
Rule 246
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 6275
Rule 6278
Rubi steps
\begin {align*} \int e^{\frac {1}{2} \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx &=\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int e^{\frac {1}{2} \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{3/2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int (1-a x)^{5/4} (1+a x)^{7/4} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {c (1-a x)^{9/4} (1+a x)^{7/4} \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}}+\frac {\left (7 c \sqrt {c-a^2 c x^2}\right ) \int (1-a x)^{5/4} (1+a x)^{3/4} \, dx}{8 \sqrt {1-a^2 x^2}}\\ &=-\frac {7 c (1-a x)^{9/4} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{24 a \sqrt {1-a^2 x^2}}-\frac {c (1-a x)^{9/4} (1+a x)^{7/4} \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}}+\frac {\left (7 c \sqrt {c-a^2 c x^2}\right ) \int \frac {(1-a x)^{5/4}}{\sqrt [4]{1+a x}} \, dx}{16 \sqrt {1-a^2 x^2}}\\ &=\frac {7 c (1-a x)^{5/4} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{32 a \sqrt {1-a^2 x^2}}-\frac {7 c (1-a x)^{9/4} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{24 a \sqrt {1-a^2 x^2}}-\frac {c (1-a x)^{9/4} (1+a x)^{7/4} \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}}+\frac {\left (35 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}} \, dx}{64 \sqrt {1-a^2 x^2}}\\ &=\frac {35 c \sqrt [4]{1-a x} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{64 a \sqrt {1-a^2 x^2}}+\frac {7 c (1-a x)^{5/4} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{32 a \sqrt {1-a^2 x^2}}-\frac {7 c (1-a x)^{9/4} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{24 a \sqrt {1-a^2 x^2}}-\frac {c (1-a x)^{9/4} (1+a x)^{7/4} \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}}+\frac {\left (35 c \sqrt {c-a^2 c x^2}\right ) \int \frac {1}{(1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx}{128 \sqrt {1-a^2 x^2}}\\ &=\frac {35 c \sqrt [4]{1-a x} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{64 a \sqrt {1-a^2 x^2}}+\frac {7 c (1-a x)^{5/4} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{32 a \sqrt {1-a^2 x^2}}-\frac {7 c (1-a x)^{9/4} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{24 a \sqrt {1-a^2 x^2}}-\frac {c (1-a x)^{9/4} (1+a x)^{7/4} \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}}-\frac {\left (35 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-a x}\right )}{32 a \sqrt {1-a^2 x^2}}\\ &=\frac {35 c \sqrt [4]{1-a x} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{64 a \sqrt {1-a^2 x^2}}+\frac {7 c (1-a x)^{5/4} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{32 a \sqrt {1-a^2 x^2}}-\frac {7 c (1-a x)^{9/4} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{24 a \sqrt {1-a^2 x^2}}-\frac {c (1-a x)^{9/4} (1+a x)^{7/4} \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}}-\frac {\left (35 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{32 a \sqrt {1-a^2 x^2}}\\ &=\frac {35 c \sqrt [4]{1-a x} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{64 a \sqrt {1-a^2 x^2}}+\frac {7 c (1-a x)^{5/4} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{32 a \sqrt {1-a^2 x^2}}-\frac {7 c (1-a x)^{9/4} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{24 a \sqrt {1-a^2 x^2}}-\frac {c (1-a x)^{9/4} (1+a x)^{7/4} \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}}-\frac {\left (35 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 a \sqrt {1-a^2 x^2}}-\frac {\left (35 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 a \sqrt {1-a^2 x^2}}\\ &=\frac {35 c \sqrt [4]{1-a x} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{64 a \sqrt {1-a^2 x^2}}+\frac {7 c (1-a x)^{5/4} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{32 a \sqrt {1-a^2 x^2}}-\frac {7 c (1-a x)^{9/4} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{24 a \sqrt {1-a^2 x^2}}-\frac {c (1-a x)^{9/4} (1+a x)^{7/4} \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}}-\frac {\left (35 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 a \sqrt {1-a^2 x^2}}-\frac {\left (35 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 a \sqrt {1-a^2 x^2}}+\frac {\left (35 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a \sqrt {1-a^2 x^2}}+\frac {\left (35 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a \sqrt {1-a^2 x^2}}\\ &=\frac {35 c \sqrt [4]{1-a x} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{64 a \sqrt {1-a^2 x^2}}+\frac {7 c (1-a x)^{5/4} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{32 a \sqrt {1-a^2 x^2}}-\frac {7 c (1-a x)^{9/4} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{24 a \sqrt {1-a^2 x^2}}-\frac {c (1-a x)^{9/4} (1+a x)^{7/4} \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}}+\frac {35 c \sqrt {c-a^2 c x^2} \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a \sqrt {1-a^2 x^2}}-\frac {35 c \sqrt {c-a^2 c x^2} \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a \sqrt {1-a^2 x^2}}-\frac {\left (35 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a \sqrt {1-a^2 x^2}}+\frac {\left (35 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a \sqrt {1-a^2 x^2}}\\ &=\frac {35 c \sqrt [4]{1-a x} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{64 a \sqrt {1-a^2 x^2}}+\frac {7 c (1-a x)^{5/4} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{32 a \sqrt {1-a^2 x^2}}-\frac {7 c (1-a x)^{9/4} (1+a x)^{3/4} \sqrt {c-a^2 c x^2}}{24 a \sqrt {1-a^2 x^2}}-\frac {c (1-a x)^{9/4} (1+a x)^{7/4} \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}}+\frac {35 c \sqrt {c-a^2 c x^2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a \sqrt {1-a^2 x^2}}-\frac {35 c \sqrt {c-a^2 c x^2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a \sqrt {1-a^2 x^2}}+\frac {35 c \sqrt {c-a^2 c x^2} \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a \sqrt {1-a^2 x^2}}-\frac {35 c \sqrt {c-a^2 c x^2} \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 72, normalized size = 0.13 \begin {gather*} -\frac {8\ 2^{3/4} c (1-a x)^{9/4} \sqrt {c-a^2 c x^2} \, _2F_1\left (-\frac {7}{4},\frac {9}{4};\frac {13}{4};\frac {1}{2} (1-a x)\right )}{9 a \sqrt {1-a^2 x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}\, \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int {\left (c-a^2\,c\,x^2\right )}^{3/2}\,\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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