Optimal. Leaf size=427 \[ \frac{\sqrt{\sec (c+d x)} \left (8 a^3 (A+3 C)+48 a^2 b B+a b^2 (16 A+33 C)+12 b^3 B\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )}{12 d \sqrt{a+b \sec (c+d x)}}+\frac{\left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{12 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{b \sqrt{\sec (c+d x)} \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{4 d \sqrt{a+b \sec (c+d x)}}-\frac{b \sin (c+d x) \sqrt{\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt{a+b \sec (c+d x)}}{12 d}-\frac{b (4 A-3 C) \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{6 d}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 1.66229, antiderivative size = 427, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 13, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.289, Rules used = {4094, 4096, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661} \[ \frac{\sqrt{\sec (c+d x)} \left (8 a^3 (A+3 C)+48 a^2 b B+a b^2 (16 A+33 C)+12 b^3 B\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{12 d \sqrt{a+b \sec (c+d x)}}+\frac{\left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{12 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{b \sqrt{\sec (c+d x)} \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{4 d \sqrt{a+b \sec (c+d x)}}-\frac{b \sin (c+d x) \sqrt{\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt{a+b \sec (c+d x)}}{12 d}-\frac{b (4 A-3 C) \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{6 d}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4094
Rule 4096
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 \frac{(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac{3}{2}}(c+d x)} \, dx &=\frac{2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2}{3} \int \frac{(a+b \sec (c+d x))^{3/2} \left (\frac{1}{2} (5 A b+3 a B)+\frac{1}{2} (3 b B+a (A+3 C)) \sec (c+d x)-\frac{1}{2} b (4 A-3 C) \sec ^2(c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx\\ &=-\frac{b (4 A-3 C) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{6 d}+\frac{2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{1}{3} \int \frac{\sqrt{a+b \sec (c+d x)} \left (\frac{3}{4} a (8 A b+4 a B-b C)+\frac{1}{2} \left (12 a b B+3 b^2 (2 A+C)+2 a^2 (A+3 C)\right ) \sec (c+d x)-\frac{1}{4} b (8 a A-12 b B-21 a C) \sec ^2(c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx\\ &=-\frac{b (8 a A-12 b B-21 a C) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{12 d}-\frac{b (4 A-3 C) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{6 d}+\frac{2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{1}{3} \int \frac{\frac{1}{8} a \left (24 a^2 B-12 b^2 B+a b (56 A-27 C)\right )+\frac{1}{4} a \left (36 a b B+3 b^2 (12 A+C)+4 a^2 (A+3 C)\right ) \sec (c+d x)+\frac{3}{8} b \left (8 A b^2+20 a b B+15 a^2 C+4 b^2 C\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx\\ &=-\frac{b (8 a A-12 b B-21 a C) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{12 d}-\frac{b (4 A-3 C) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{6 d}+\frac{2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{1}{3} \int \frac{\frac{1}{8} a \left (24 a^2 B-12 b^2 B+a b (56 A-27 C)\right )+\frac{1}{4} a \left (36 a b B+3 b^2 (12 A+C)+4 a^2 (A+3 C)\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx+\frac{1}{8} \left (b \left (8 A b^2+20 a b B+15 a^2 C+4 b^2 C\right )\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=-\frac{b (8 a A-12 b B-21 a C) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{12 d}-\frac{b (4 A-3 C) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{6 d}+\frac{2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{1}{24} \left (24 a^2 B-12 b^2 B+a b (56 A-27 C)\right ) \int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{24} \left (48 a^2 b B+12 b^3 B+8 a^3 (A+3 C)+a b^2 (16 A+33 C)\right ) \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx+\frac{\left (b \left (8 A b^2+20 a b B+15 a^2 C+4 b^2 C\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}{8 \sqrt{a+b \sec (c+d x)}}\\ &=-\frac{b (8 a A-12 b B-21 a C) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{12 d}-\frac{b (4 A-3 C) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{6 d}+\frac{2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{\left (\left (48 a^2 b B+12 b^3 B+8 a^3 (A+3 C)+a b^2 (16 A+33 C)\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}{24 \sqrt{a+b \sec (c+d x)}}+\frac{\left (b \left (8 A b^2+20 a b B+15 a^2 C+4 b^2 C\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}{8 \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (24 a^2 B-12 b^2 B+a b (56 A-27 C)\right ) \sqrt{a+b \sec (c+d x)}\right ) \int \sqrt{b+a \cos (c+d x)} \, dx}{24 \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)}}\\ &=\frac{b \left (8 A b^2+20 a b B+15 a^2 C+4 b^2 C\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)}}{4 d \sqrt{a+b \sec (c+d x)}}-\frac{b (8 a A-12 b B-21 a C) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{12 d}-\frac{b (4 A-3 C) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{6 d}+\frac{2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{\left (\left (48 a^2 b B+12 b^3 B+8 a^3 (A+3 C)+a b^2 (16 A+33 C)\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}{24 \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (24 a^2 B-12 b^2 B+a b (56 A-27 C)\right ) \sqrt{a+b \sec (c+d x)}\right ) \int \sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}} \, dx}{24 \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}}\\ &=\frac{\left (48 a^2 b B+12 b^3 B+8 a^3 (A+3 C)+a b^2 (16 A+33 C)\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)}}{12 d \sqrt{a+b \sec (c+d x)}}+\frac{b \left (8 A b^2+20 a b B+15 a^2 C+4 b^2 C\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)}}{4 d \sqrt{a+b \sec (c+d x)}}+\frac{\left (24 a^2 B-12 b^2 B+a b (56 A-27 C)\right ) E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{a+b \sec (c+d x)}}{12 d \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}}-\frac{b (8 a A-12 b B-21 a C) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{12 d}-\frac{b (4 A-3 C) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{6 d}+\frac{2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 7.05312, size = 766, normalized size = 1.79 \[ \frac{(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac{4}{3} a^2 A \sin (c+d x)+\frac{1}{2} \sec (c+d x) \left (9 a b C \sin (c+d x)+4 b^2 B \sin (c+d x)\right )+b^2 C \tan (c+d x) \sec (c+d x)\right )}{d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+b)^2 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac{2 \left (16 a^3 A+144 a^2 b B+48 a^3 C+144 a A b^2+12 a b^2 C\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )}{\sqrt{a \cos (c+d x)+b}}+\frac{2 i \sin (c+d x) \cos (2 (c+d x)) \left (56 a^2 A b-27 a^2 b C+24 a^3 B-12 a b^2 B\right ) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left (a \left (2 b \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{1}{a-b}} \sqrt{a \cos (c+d x)+b}\right ),\frac{b-a}{a+b}\right )+a \Pi \left (1-\frac{a}{b};i \sinh ^{-1}\left (\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right )|\frac{b-a}{a+b}\right )\right )-2 b (a+b) E\left (i \sinh ^{-1}\left (\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right )|\frac{b-a}{a+b}\right )\right )}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left (-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right )}+\frac{2 \left (56 a^2 A b+63 a^2 b C+24 a^3 B+108 a b^2 B+48 A b^3+24 b^3 C\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{\sqrt{a \cos (c+d x)+b}}\right )}{24 d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+b)^{5/2} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.642, size = 5629, normalized size = 13.2 \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C b^{2} \sec \left (d x + c\right )^{4} +{\left (2 \, C a b + B b^{2}\right )} \sec \left (d x + c\right )^{3} + A a^{2} +{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \sec \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \sec \left (d x + c\right )\right )} \sqrt{b \sec \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac{3}{2}}}, x\right ) \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 \frac{{\left (C \sec \left (d x + c\right )^{2} + 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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