Optimal. Leaf size=187 \[ \frac{\sqrt{-a-b x+1} (a+b x+1)^{3/2} \left (22 a^2+2 (11-10 a) b x-54 a+29\right )}{8 b^4}+\frac{3 \left (-8 a^3+36 a^2-44 a+17\right ) \sqrt{-a-b x+1} \sqrt{a+b x+1}}{8 b^4}-\frac{3 \left (-8 a^3+36 a^2-44 a+17\right ) \sin ^{-1}(a+b x)}{8 b^4}+\frac{9 x^2 \sqrt{-a-b x+1} (a+b x+1)^{3/2}}{4 b^2}+\frac{2 x^3 (a+b x+1)^{3/2}}{b \sqrt{-a-b x+1}} \]
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Rubi [A] time = 0.182205, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {6163, 97, 153, 147, 50, 53, 619, 216} \[ \frac{\sqrt{-a-b x+1} (a+b x+1)^{3/2} \left (22 a^2+2 (11-10 a) b x-54 a+29\right )}{8 b^4}+\frac{3 \left (-8 a^3+36 a^2-44 a+17\right ) \sqrt{-a-b x+1} \sqrt{a+b x+1}}{8 b^4}-\frac{3 \left (-8 a^3+36 a^2-44 a+17\right ) \sin ^{-1}(a+b x)}{8 b^4}+\frac{9 x^2 \sqrt{-a-b x+1} (a+b x+1)^{3/2}}{4 b^2}+\frac{2 x^3 (a+b x+1)^{3/2}}{b \sqrt{-a-b x+1}} \]
Antiderivative was successfully verified.
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Rule 6163
Rule 97
Rule 153
Rule 147
Rule 50
Rule 53
Rule 619
Rule 216
Rubi steps
\begin{align*} \int e^{3 \tanh ^{-1}(a+b x)} x^3 \, dx &=\int \frac{x^3 (1+a+b x)^{3/2}}{(1-a-b x)^{3/2}} \, dx\\ &=\frac{2 x^3 (1+a+b x)^{3/2}}{b \sqrt{1-a-b x}}-\frac{2 \int \frac{x^2 \sqrt{1+a+b x} \left (3 (1+a)+\frac{9 b x}{2}\right )}{\sqrt{1-a-b x}} \, dx}{b}\\ &=\frac{2 x^3 (1+a+b x)^{3/2}}{b \sqrt{1-a-b x}}+\frac{9 x^2 \sqrt{1-a-b x} (1+a+b x)^{3/2}}{4 b^2}+\frac{\int \frac{x \sqrt{1+a+b x} \left (-9 (1-a) (1+a) b-\frac{3}{2} (11-10 a) b^2 x\right )}{\sqrt{1-a-b x}} \, dx}{2 b^3}\\ &=\frac{2 x^3 (1+a+b x)^{3/2}}{b \sqrt{1-a-b x}}+\frac{9 x^2 \sqrt{1-a-b x} (1+a+b x)^{3/2}}{4 b^2}+\frac{\sqrt{1-a-b x} (1+a+b x)^{3/2} \left (29-54 a+22 a^2+2 (11-10 a) b x\right )}{8 b^4}-\frac{\left (3 \left (17-44 a+36 a^2-8 a^3\right )\right ) \int \frac{\sqrt{1+a+b x}}{\sqrt{1-a-b x}} \, dx}{8 b^3}\\ &=\frac{3 \left (17-44 a+36 a^2-8 a^3\right ) \sqrt{1-a-b x} \sqrt{1+a+b x}}{8 b^4}+\frac{2 x^3 (1+a+b x)^{3/2}}{b \sqrt{1-a-b x}}+\frac{9 x^2 \sqrt{1-a-b x} (1+a+b x)^{3/2}}{4 b^2}+\frac{\sqrt{1-a-b x} (1+a+b x)^{3/2} \left (29-54 a+22 a^2+2 (11-10 a) b x\right )}{8 b^4}-\frac{\left (3 \left (17-44 a+36 a^2-8 a^3\right )\right ) \int \frac{1}{\sqrt{1-a-b x} \sqrt{1+a+b x}} \, dx}{8 b^3}\\ &=\frac{3 \left (17-44 a+36 a^2-8 a^3\right ) \sqrt{1-a-b x} \sqrt{1+a+b x}}{8 b^4}+\frac{2 x^3 (1+a+b x)^{3/2}}{b \sqrt{1-a-b x}}+\frac{9 x^2 \sqrt{1-a-b x} (1+a+b x)^{3/2}}{4 b^2}+\frac{\sqrt{1-a-b x} (1+a+b x)^{3/2} \left (29-54 a+22 a^2+2 (11-10 a) b x\right )}{8 b^4}-\frac{\left (3 \left (17-44 a+36 a^2-8 a^3\right )\right ) \int \frac{1}{\sqrt{(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx}{8 b^3}\\ &=\frac{3 \left (17-44 a+36 a^2-8 a^3\right ) \sqrt{1-a-b x} \sqrt{1+a+b x}}{8 b^4}+\frac{2 x^3 (1+a+b x)^{3/2}}{b \sqrt{1-a-b x}}+\frac{9 x^2 \sqrt{1-a-b x} (1+a+b x)^{3/2}}{4 b^2}+\frac{\sqrt{1-a-b x} (1+a+b x)^{3/2} \left (29-54 a+22 a^2+2 (11-10 a) b x\right )}{8 b^4}+\frac{\left (3 \left (17-44 a+36 a^2-8 a^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{4 b^2}}} \, dx,x,-2 a b-2 b^2 x\right )}{16 b^5}\\ &=\frac{3 \left (17-44 a+36 a^2-8 a^3\right ) \sqrt{1-a-b x} \sqrt{1+a+b x}}{8 b^4}+\frac{2 x^3 (1+a+b x)^{3/2}}{b \sqrt{1-a-b x}}+\frac{9 x^2 \sqrt{1-a-b x} (1+a+b x)^{3/2}}{4 b^2}+\frac{\sqrt{1-a-b x} (1+a+b x)^{3/2} \left (29-54 a+22 a^2+2 (11-10 a) b x\right )}{8 b^4}-\frac{3 \left (17-44 a+36 a^2-8 a^3\right ) \sin ^{-1}(a+b x)}{8 b^4}\\ \end{align*}
Mathematica [A] time = 0.284744, size = 203, normalized size = 1.09 \[ \frac{-\frac{\sqrt{b} \sqrt{a+b x+1} \left (a^2 (22 b x-233)-2 a^4+78 a^3+a \left (-10 b^2 x^2-54 b x+237\right )+2 b^4 x^4+6 b^3 x^3+11 b^2 x^2+29 b x-80\right )}{\sqrt{-a-b x+1}}+24 a \left (2 a^2+11\right ) \sqrt{-b} \sinh ^{-1}\left (\frac{\sqrt{-b} \sqrt{-a-b x+1}}{\sqrt{2} \sqrt{b}}\right )+6 \left (36 a^2+17\right ) \sqrt{-b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{-a-b x+1}}{\sqrt{2} \sqrt{-b}}\right )}{8 b^{9/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.088, size = 756, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.59418, size = 456, normalized size = 2.44 \begin{align*} -\frac{3 \,{\left (8 \, a^{4} - 44 \, a^{3} +{\left (8 \, a^{3} - 36 \, a^{2} + 44 \, a - 17\right )} b x + 80 \, a^{2} - 61 \, a + 17\right )} \arctan \left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) -{\left (2 \, b^{4} x^{4} + 6 \, b^{3} x^{3} -{\left (10 \, a - 11\right )} b^{2} x^{2} - 2 \, a^{4} + 78 \, a^{3} +{\left (22 \, a^{2} - 54 \, a + 29\right )} b x - 233 \, a^{2} + 237 \, a - 80\right )} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{8 \,{\left (b^{5} x +{\left (a - 1\right )} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \left (a + b x + 1\right )^{3}}{\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26128, size = 261, normalized size = 1.4 \begin{align*} \frac{1}{8} \, \sqrt{-{\left (b x + a\right )}^{2} + 1}{\left ({\left (2 \, x{\left (\frac{x}{b} - \frac{a b^{11} - 4 \, b^{11}}{b^{13}}\right )} + \frac{2 \, a^{2} b^{10} - 20 \, a b^{10} + 19 \, b^{10}}{b^{13}}\right )} x - \frac{2 \, a^{3} b^{9} - 44 \, a^{2} b^{9} + 93 \, a b^{9} - 48 \, b^{9}}{b^{13}}\right )} - \frac{3 \,{\left (8 \, a^{3} - 36 \, a^{2} + 44 \, a - 17\right )} \arcsin \left (-b x - a\right ) \mathrm{sgn}\left (b\right )}{8 \, b^{3}{\left | b \right |}} - \frac{8 \,{\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )}}{b^{3}{\left (\frac{\sqrt{-{\left (b x + a\right )}^{2} + 1}{\left | b \right |} + b}{b^{2} x + a b} - 1\right )}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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