Optimal. Leaf size=513 \[ \frac{\left (472 a^2 A b+133 a^3 B+356 a b^2 B+128 A b^3\right ) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )}{192 d \sqrt{a+b \sec (c+d x)}}+\frac{\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{96 d}+\frac{\left (264 a^2 A b+15 a^3 B+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{192 b d}-\frac{\left (264 a^2 A b+15 a^3 B+284 a b^2 B+128 A b^3\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{192 b d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\left (40 a^3 A b+120 a^2 b^2 B-5 a^4 B+160 a A b^3+48 b^4 B\right ) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{64 b d \sqrt{a+b \sec (c+d x)}}+\frac{b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{b B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d} \]
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Rubi [A] time = 1.99846, antiderivative size = 513, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 14, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4026, 4096, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661} \[ \frac{\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{96 d}+\frac{\left (264 a^2 A b+15 a^3 B+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{192 b d}+\frac{\left (472 a^2 A b+133 a^3 B+356 a b^2 B+128 A b^3\right ) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{192 d \sqrt{a+b \sec (c+d x)}}-\frac{\left (264 a^2 A b+15 a^3 B+284 a b^2 B+128 A b^3\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{192 b d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\left (40 a^3 A b+120 a^2 b^2 B-5 a^4 B+160 a A b^3+48 b^4 B\right ) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{64 b d \sqrt{a+b \sec (c+d x)}}+\frac{b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{b B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d} \]
Antiderivative was successfully verified.
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Rule 4026
Rule 4096
Rule 4102
Rule 4108
Rule 3859
Rule 2807
Rule 2805
Rule 4035
Rule 3856
Rule 2655
Rule 2653
Rule 3858
Rule 2663
Rule 2661
Rubi steps
\begin{align*} \int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx &=\frac{b B \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{1}{4} \int \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \left (\frac{1}{2} a (8 a A+3 b B)+\left (8 a A b+4 a^2 B+3 b^2 B\right ) \sec (c+d x)+\frac{1}{2} b (8 A b+11 a B) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{b (8 A b+11 a B) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{b B \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{1}{12} \int \frac{\sec ^{\frac{3}{2}}(c+d x) \left (\frac{3}{4} a \left (16 a^2 A+8 A b^2+17 a b B\right )+\frac{1}{2} \left (72 a^2 A b+16 A b^3+24 a^3 B+49 a b^2 B\right ) \sec (c+d x)+\frac{1}{4} b \left (104 a A b+59 a^2 B+36 b^2 B\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{\left (104 a A b+59 a^2 B+36 b^2 B\right ) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac{b (8 A b+11 a B) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{b B \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{\int \frac{\sqrt{\sec (c+d x)} \left (\frac{1}{8} a b \left (104 a A b+59 a^2 B+36 b^2 B\right )+\frac{1}{4} b \left (96 a^3 A+152 a A b^2+161 a^2 b B+36 b^3 B\right ) \sec (c+d x)+\frac{1}{8} b \left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{24 b}\\ &=\frac{\left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 b d}+\frac{\left (104 a A b+59 a^2 B+36 b^2 B\right ) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac{b (8 A b+11 a B) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{b B \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{\int \frac{-\frac{1}{16} a b \left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right )+\frac{1}{8} a b^2 \left (104 a A b+59 a^2 B+36 b^2 B\right ) \sec (c+d x)+\frac{3}{16} b \left (40 a^3 A b+160 a A b^3-5 a^4 B+120 a^2 b^2 B+48 b^4 B\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx}{24 b^2}\\ &=\frac{\left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 b d}+\frac{\left (104 a A b+59 a^2 B+36 b^2 B\right ) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac{b (8 A b+11 a B) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{b B \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{\int \frac{-\frac{1}{16} a b \left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right )+\frac{1}{8} a b^2 \left (104 a A b+59 a^2 B+36 b^2 B\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx}{24 b^2}+\frac{\left (40 a^3 A b+160 a A b^3-5 a^4 B+120 a^2 b^2 B+48 b^4 B\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{128 b}\\ &=\frac{\left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 b d}+\frac{\left (104 a A b+59 a^2 B+36 b^2 B\right ) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac{b (8 A b+11 a B) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{b B \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}-\frac{\left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) \int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx}{384 b}+\frac{1}{384} \left (472 a^2 A b+128 A b^3+133 a^3 B+356 a b^2 B\right ) \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx+\frac{\left (\left (40 a^3 A b+160 a A b^3-5 a^4 B+120 a^2 b^2 B+48 b^4 B\right ) \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec (c+d x)}{\sqrt{b+a \cos (c+d x)}} \, dx}{128 b \sqrt{a+b \sec (c+d x)}}\\ &=\frac{\left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 b d}+\frac{\left (104 a A b+59 a^2 B+36 b^2 B\right ) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac{b (8 A b+11 a B) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{b B \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{\left (\left (472 a^2 A b+128 A b^3+133 a^3 B+356 a b^2 B\right ) \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{b+a \cos (c+d x)}} \, dx}{384 \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (40 a^3 A b+160 a A b^3-5 a^4 B+120 a^2 b^2 B+48 b^4 B\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec (c+d x)}{\sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}}} \, dx}{128 b \sqrt{a+b \sec (c+d x)}}-\frac{\left (\left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) \sqrt{a+b \sec (c+d x)}\right ) \int \sqrt{b+a \cos (c+d x)} \, dx}{384 b \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)}}\\ &=\frac{\left (40 a^3 A b+160 a A b^3-5 a^4 B+120 a^2 b^2 B+48 b^4 B\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{\sec (c+d x)}}{64 b d \sqrt{a+b \sec (c+d x)}}+\frac{\left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 b d}+\frac{\left (104 a A b+59 a^2 B+36 b^2 B\right ) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac{b (8 A b+11 a B) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{b B \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{\left (\left (472 a^2 A b+128 A b^3+133 a^3 B+356 a b^2 B\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}}} \, dx}{384 \sqrt{a+b \sec (c+d x)}}-\frac{\left (\left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) \sqrt{a+b \sec (c+d x)}\right ) \int \sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}} \, dx}{384 b \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}}\\ &=\frac{\left (472 a^2 A b+128 A b^3+133 a^3 B+356 a b^2 B\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{\sec (c+d x)}}{192 d \sqrt{a+b \sec (c+d x)}}+\frac{\left (40 a^3 A b+160 a A b^3-5 a^4 B+120 a^2 b^2 B+48 b^4 B\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{\sec (c+d x)}}{64 b d \sqrt{a+b \sec (c+d x)}}-\frac{\left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{a+b \sec (c+d x)}}{192 b d \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}}+\frac{\left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 b d}+\frac{\left (104 a A b+59 a^2 B+36 b^2 B\right ) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac{b (8 A b+11 a B) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{b B \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}\\ \end{align*}
Mathematica [C] time = 6.90957, size = 768, normalized size = 1.5 \[ \frac{(a+b \sec (c+d x))^{5/2} \left (\frac{1}{96} \sec ^2(c+d x) \left (59 a^2 B \sin (c+d x)+104 a A b \sin (c+d x)+36 b^2 B \sin (c+d x)\right )+\frac{\sec (c+d x) \left (264 a^2 A b \sin (c+d x)+15 a^3 B \sin (c+d x)+284 a b^2 B \sin (c+d x)+128 A b^3 \sin (c+d x)\right )}{192 b}+\frac{1}{24} \sec ^3(c+d x) \left (17 a b B \sin (c+d x)+8 A b^2 \sin (c+d x)\right )+\frac{1}{4} b^2 B \tan (c+d x) \sec ^3(c+d x)\right )}{d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^2}-\frac{(a+b \sec (c+d x))^{5/2} \left (\frac{2 \left (-416 a^2 A b^2-236 a^3 b B-144 a b^3 B\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )}{\sqrt{a \cos (c+d x)+b}}+\frac{2 i \left (264 a^3 A b+284 a^2 b^2 B+15 a^4 B+128 a A b^3\right ) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left (a \left (2 b \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{1}{a-b}} \sqrt{a \cos (c+d x)+b}\right ),\frac{b-a}{a+b}\right )+a \Pi \left (1-\frac{a}{b};i \sinh ^{-1}\left (\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right )|\frac{b-a}{a+b}\right )\right )-2 b (a+b) E\left (i \sinh ^{-1}\left (\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right )|\frac{b-a}{a+b}\right )\right )}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left (-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right )}+\frac{2 \left (24 a^3 A b-436 a^2 b^2 B+45 a^4 B-832 a A b^3-288 b^4 B\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{\sqrt{a \cos (c+d x)+b}}\right )}{768 b d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.791, size = 5392, normalized size = 10.5 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (B \sec \left (d x + c\right ) + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sec \left (d x + c\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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