Optimal. Leaf size=1041 \[ \text{result too large to display} \]
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Rubi [A] time = 1.47629, antiderivative size = 1041, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {4204, 4191, 3321, 2264, 2190, 2531, 6609, 2282, 6589} \[ \frac{x^4}{4 a}+\frac{2 i b \log \left (\frac{e^{i \left (c+d \sqrt{x}\right )} a}{b-\sqrt{b^2-a^2}}+1\right ) x^{7/2}}{a \sqrt{b^2-a^2} d}-\frac{2 i b \log \left (\frac{e^{i \left (c+d \sqrt{x}\right )} a}{b+\sqrt{b^2-a^2}}+1\right ) x^{7/2}}{a \sqrt{b^2-a^2} d}+\frac{14 b \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) x^3}{a \sqrt{b^2-a^2} d^2}-\frac{14 b \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) x^3}{a \sqrt{b^2-a^2} d^2}+\frac{84 i b \text{PolyLog}\left (3,-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) x^{5/2}}{a \sqrt{b^2-a^2} d^3}-\frac{84 i b \text{PolyLog}\left (3,-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) x^{5/2}}{a \sqrt{b^2-a^2} d^3}-\frac{420 b \text{PolyLog}\left (4,-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) x^2}{a \sqrt{b^2-a^2} d^4}+\frac{420 b \text{PolyLog}\left (4,-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) x^2}{a \sqrt{b^2-a^2} d^4}-\frac{1680 i b \text{PolyLog}\left (5,-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) x^{3/2}}{a \sqrt{b^2-a^2} d^5}+\frac{1680 i b \text{PolyLog}\left (5,-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) x^{3/2}}{a \sqrt{b^2-a^2} d^5}+\frac{5040 b \text{PolyLog}\left (6,-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) x}{a \sqrt{b^2-a^2} d^6}-\frac{5040 b \text{PolyLog}\left (6,-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) x}{a \sqrt{b^2-a^2} d^6}+\frac{10080 i b \text{PolyLog}\left (7,-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) \sqrt{x}}{a \sqrt{b^2-a^2} d^7}-\frac{10080 i b \text{PolyLog}\left (7,-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) \sqrt{x}}{a \sqrt{b^2-a^2} d^7}-\frac{10080 b \text{PolyLog}\left (8,-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right )}{a \sqrt{b^2-a^2} d^8}+\frac{10080 b \text{PolyLog}\left (8,-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right )}{a \sqrt{b^2-a^2} d^8} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4204
Rule 4191
Rule 3321
Rule 2264
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{x^3}{a+b \sec \left (c+d \sqrt{x}\right )} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^7}{a+b \sec (c+d x)} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{x^7}{a}-\frac{b x^7}{a (b+a \cos (c+d x))}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{x^4}{4 a}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{x^7}{b+a \cos (c+d x)} \, dx,x,\sqrt{x}\right )}{a}\\ &=\frac{x^4}{4 a}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^7}{a+2 b e^{i (c+d x)}+a e^{2 i (c+d x)}} \, dx,x,\sqrt{x}\right )}{a}\\ &=\frac{x^4}{4 a}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^7}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,\sqrt{x}\right )}{\sqrt{-a^2+b^2}}+\frac{(4 b) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^7}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,\sqrt{x}\right )}{\sqrt{-a^2+b^2}}\\ &=\frac{x^4}{4 a}+\frac{2 i b x^{7/2} \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{2 i b x^{7/2} \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{(14 i b) \operatorname{Subst}\left (\int x^6 \log \left (1+\frac{2 a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d}+\frac{(14 i b) \operatorname{Subst}\left (\int x^6 \log \left (1+\frac{2 a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d}\\ &=\frac{x^4}{4 a}+\frac{2 i b x^{7/2} \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{2 i b x^{7/2} \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{14 b x^3 \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{14 b x^3 \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{(84 b) \operatorname{Subst}\left (\int x^5 \text{Li}_2\left (-\frac{2 a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{(84 b) \operatorname{Subst}\left (\int x^5 \text{Li}_2\left (-\frac{2 a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^2}\\ &=\frac{x^4}{4 a}+\frac{2 i b x^{7/2} \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{2 i b x^{7/2} \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{14 b x^3 \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{14 b x^3 \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{84 i b x^{5/2} \text{Li}_3\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{84 i b x^{5/2} \text{Li}_3\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{(420 i b) \operatorname{Subst}\left (\int x^4 \text{Li}_3\left (-\frac{2 a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^3}+\frac{(420 i b) \operatorname{Subst}\left (\int x^4 \text{Li}_3\left (-\frac{2 a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^3}\\ &=\frac{x^4}{4 a}+\frac{2 i b x^{7/2} \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{2 i b x^{7/2} \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{14 b x^3 \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{14 b x^3 \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{84 i b x^{5/2} \text{Li}_3\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{84 i b x^{5/2} \text{Li}_3\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{420 b x^2 \text{Li}_4\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}+\frac{420 b x^2 \text{Li}_4\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}+\frac{(1680 b) \operatorname{Subst}\left (\int x^3 \text{Li}_4\left (-\frac{2 a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^4}-\frac{(1680 b) \operatorname{Subst}\left (\int x^3 \text{Li}_4\left (-\frac{2 a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^4}\\ &=\frac{x^4}{4 a}+\frac{2 i b x^{7/2} \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{2 i b x^{7/2} \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{14 b x^3 \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{14 b x^3 \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{84 i b x^{5/2} \text{Li}_3\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{84 i b x^{5/2} \text{Li}_3\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{420 b x^2 \text{Li}_4\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}+\frac{420 b x^2 \text{Li}_4\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}-\frac{1680 i b x^{3/2} \text{Li}_5\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^5}+\frac{1680 i b x^{3/2} \text{Li}_5\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^5}+\frac{(5040 i b) \operatorname{Subst}\left (\int x^2 \text{Li}_5\left (-\frac{2 a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^5}-\frac{(5040 i b) \operatorname{Subst}\left (\int x^2 \text{Li}_5\left (-\frac{2 a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^5}\\ &=\frac{x^4}{4 a}+\frac{2 i b x^{7/2} \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{2 i b x^{7/2} \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{14 b x^3 \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{14 b x^3 \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{84 i b x^{5/2} \text{Li}_3\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{84 i b x^{5/2} \text{Li}_3\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{420 b x^2 \text{Li}_4\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}+\frac{420 b x^2 \text{Li}_4\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}-\frac{1680 i b x^{3/2} \text{Li}_5\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^5}+\frac{1680 i b x^{3/2} \text{Li}_5\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^5}+\frac{5040 b x \text{Li}_6\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^6}-\frac{5040 b x \text{Li}_6\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^6}-\frac{(10080 b) \operatorname{Subst}\left (\int x \text{Li}_6\left (-\frac{2 a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^6}+\frac{(10080 b) \operatorname{Subst}\left (\int x \text{Li}_6\left (-\frac{2 a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^6}\\ &=\frac{x^4}{4 a}+\frac{2 i b x^{7/2} \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{2 i b x^{7/2} \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{14 b x^3 \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{14 b x^3 \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{84 i b x^{5/2} \text{Li}_3\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{84 i b x^{5/2} \text{Li}_3\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{420 b x^2 \text{Li}_4\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}+\frac{420 b x^2 \text{Li}_4\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}-\frac{1680 i b x^{3/2} \text{Li}_5\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^5}+\frac{1680 i b x^{3/2} \text{Li}_5\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^5}+\frac{5040 b x \text{Li}_6\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^6}-\frac{5040 b x \text{Li}_6\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^6}+\frac{10080 i b \sqrt{x} \text{Li}_7\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^7}-\frac{10080 i b \sqrt{x} \text{Li}_7\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^7}-\frac{(10080 i b) \operatorname{Subst}\left (\int \text{Li}_7\left (-\frac{2 a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^7}+\frac{(10080 i b) \operatorname{Subst}\left (\int \text{Li}_7\left (-\frac{2 a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^7}\\ &=\frac{x^4}{4 a}+\frac{2 i b x^{7/2} \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{2 i b x^{7/2} \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{14 b x^3 \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{14 b x^3 \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{84 i b x^{5/2} \text{Li}_3\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{84 i b x^{5/2} \text{Li}_3\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{420 b x^2 \text{Li}_4\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}+\frac{420 b x^2 \text{Li}_4\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}-\frac{1680 i b x^{3/2} \text{Li}_5\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^5}+\frac{1680 i b x^{3/2} \text{Li}_5\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^5}+\frac{5040 b x \text{Li}_6\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^6}-\frac{5040 b x \text{Li}_6\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^6}+\frac{10080 i b \sqrt{x} \text{Li}_7\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^7}-\frac{10080 i b \sqrt{x} \text{Li}_7\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^7}-\frac{(10080 b) \operatorname{Subst}\left (\int \frac{\text{Li}_7\left (\frac{a x}{-b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{a \sqrt{-a^2+b^2} d^8}+\frac{(10080 b) \operatorname{Subst}\left (\int \frac{\text{Li}_7\left (-\frac{a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{a \sqrt{-a^2+b^2} d^8}\\ &=\frac{x^4}{4 a}+\frac{2 i b x^{7/2} \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{2 i b x^{7/2} \log \left (1+\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{14 b x^3 \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{14 b x^3 \text{Li}_2\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{84 i b x^{5/2} \text{Li}_3\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{84 i b x^{5/2} \text{Li}_3\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{420 b x^2 \text{Li}_4\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}+\frac{420 b x^2 \text{Li}_4\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}-\frac{1680 i b x^{3/2} \text{Li}_5\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^5}+\frac{1680 i b x^{3/2} \text{Li}_5\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^5}+\frac{5040 b x \text{Li}_6\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^6}-\frac{5040 b x \text{Li}_6\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^6}+\frac{10080 i b \sqrt{x} \text{Li}_7\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^7}-\frac{10080 i b \sqrt{x} \text{Li}_7\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^7}-\frac{10080 b \text{Li}_8\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^8}+\frac{10080 b \text{Li}_8\left (-\frac{a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^8}\\ \end{align*}
Mathematica [A] time = 2.56465, size = 1122, normalized size = 1.08 \[ \frac{\left (b+a \cos \left (c+d \sqrt{x}\right )\right ) \left (x^4+\frac{8 b e^{i c} \left (7 x^3 \text{PolyLog}\left (2,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{b e^{i c}-\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}\right ) d^6-7 x^3 \text{PolyLog}\left (2,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{e^{i c} b+\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}\right ) d^6+i \left (x^{7/2} \log \left (\frac{e^{i \left (2 c+d \sqrt{x}\right )} a}{b e^{i c}-\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}+1\right ) d^7-x^{7/2} \log \left (\frac{e^{i \left (2 c+d \sqrt{x}\right )} a}{e^{i c} b+\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}+1\right ) d^7+42 x^{5/2} \text{PolyLog}\left (3,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{b e^{i c}-\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}\right ) d^5-42 x^{5/2} \text{PolyLog}\left (3,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{e^{i c} b+\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}\right ) d^5+210 i x^2 \text{PolyLog}\left (4,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{b e^{i c}-\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}\right ) d^4-210 i x^2 \text{PolyLog}\left (4,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{e^{i c} b+\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}\right ) d^4-840 x^{3/2} \text{PolyLog}\left (5,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{b e^{i c}-\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}\right ) d^3+840 x^{3/2} \text{PolyLog}\left (5,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{e^{i c} b+\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}\right ) d^3-2520 i x \text{PolyLog}\left (6,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{b e^{i c}-\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}\right ) d^2+2520 i x \text{PolyLog}\left (6,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{e^{i c} b+\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}\right ) d^2+5040 \sqrt{x} \text{PolyLog}\left (7,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{b e^{i c}-\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}\right ) d-5040 \sqrt{x} \text{PolyLog}\left (7,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{e^{i c} b+\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}\right ) d+5040 i \text{PolyLog}\left (8,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{b e^{i c}-\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}\right )-5040 i \text{PolyLog}\left (8,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{e^{i c} b+\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}\right )\right )\right )}{d^8 \sqrt{\left (b^2-a^2\right ) e^{2 i c}}}\right ) \sec \left (c+d \sqrt{x}\right )}{4 a \left (a+b \sec \left (c+d \sqrt{x}\right )\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.087, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( a+b\sec \left ( c+d\sqrt{x} \right ) \right ) ^{-1}}\, dx \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3}}{b \sec \left (d \sqrt{x} + c\right ) + a}, x\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}}{a + b \sec{\left (c + d \sqrt{x} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{b \sec \left (d \sqrt{x} + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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