Optimal. Leaf size=446 \[ \frac{\sqrt{\sec (c+d x)} \left (a^2 (48 A+17 C)+42 a b B+8 b^2 (3 A+2 C)\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )}{24 d \sqrt{a+b \sec (c+d x)}}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left (3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right ) \sqrt{a+b \sec (c+d x)}}{24 b d}-\frac{\left (3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{24 b d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\sqrt{\sec (c+d x)} \left (6 a^2 b B+a^3 (-C)+12 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{8 b d \sqrt{a+b \sec (c+d x)}}+\frac{(a C+2 b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d} \]
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Rubi [A] time = 1.65448, antiderivative size = 446, 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 = {4096, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661} \[ \frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left (3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right ) \sqrt{a+b \sec (c+d x)}}{24 b d}+\frac{\sqrt{\sec (c+d x)} \left (a^2 (48 A+17 C)+42 a b B+8 b^2 (3 A+2 C)\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{24 d \sqrt{a+b \sec (c+d x)}}-\frac{\left (3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{24 b d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\sqrt{\sec (c+d x)} \left (6 a^2 b B+a^3 (-C)+12 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{8 b d \sqrt{a+b \sec (c+d x)}}+\frac{(a C+2 b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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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 \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac{1}{3} \int \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \left (\frac{1}{2} a (6 A+C)+(3 A b+3 a B+2 b C) \sec (c+d x)+\frac{3}{2} (2 b B+a C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{(2 b B+a C) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac{C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac{1}{6} \int \frac{\sqrt{\sec (c+d x)} \left (\frac{1}{4} a (24 a A+6 b B+7 a C)+\frac{1}{2} \left (12 a^2 B+6 b^2 B+a b (24 A+13 C)\right ) \sec (c+d x)+\frac{1}{4} \left (24 A b^2+30 a b B+3 a^2 C+16 b^2 C\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{\left (24 A b^2+30 a b B+3 a^2 C+16 b^2 C\right ) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 b d}+\frac{(2 b B+a C) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac{C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac{\int \frac{-\frac{1}{8} a \left (24 A b^2+30 a b B+3 a^2 C+16 b^2 C\right )+\frac{1}{4} a b (24 a A+6 b B+7 a C) \sec (c+d x)+\frac{3}{8} \left (6 a^2 b B+8 b^3 B-a^3 C+12 a b^2 (2 A+C)\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx}{6 b}\\ &=\frac{\left (24 A b^2+30 a b B+3 a^2 C+16 b^2 C\right ) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 b d}+\frac{(2 b B+a C) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac{C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac{\int \frac{-\frac{1}{8} a \left (24 A b^2+30 a b B+3 a^2 C+16 b^2 C\right )+\frac{1}{4} a b (24 a A+6 b B+7 a C) \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx}{6 b}+\frac{\left (6 a^2 b B+8 b^3 B-a^3 C+12 a b^2 (2 A+C)\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{16 b}\\ &=\frac{\left (24 A b^2+30 a b B+3 a^2 C+16 b^2 C\right ) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 b d}+\frac{(2 b B+a C) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac{C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac{\left (-24 A b^2-30 a b B-3 a^2 C-16 b^2 C\right ) \int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx}{48 b}+\frac{1}{48} \left (42 a b B+8 b^2 (3 A+2 C)+a^2 (48 A+17 C)\right ) \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx+\frac{\left (\left (6 a^2 b B+8 b^3 B-a^3 C+12 a b^2 (2 A+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}{16 b \sqrt{a+b \sec (c+d x)}}\\ &=\frac{\left (24 A b^2+30 a b B+3 a^2 C+16 b^2 C\right ) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 b d}+\frac{(2 b B+a C) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac{C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac{\left (\left (42 a b B+8 b^2 (3 A+2 C)+a^2 (48 A+17 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}{48 \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (6 a^2 b B+8 b^3 B-a^3 C+12 a b^2 (2 A+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}{16 b \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (-24 A b^2-30 a b B-3 a^2 C-16 b^2 C\right ) \sqrt{a+b \sec (c+d x)}\right ) \int \sqrt{b+a \cos (c+d x)} \, dx}{48 b \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)}}\\ &=\frac{\left (6 a^2 b B+8 b^3 B-a^3 C+12 a b^2 (2 A+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)}}{8 b d \sqrt{a+b \sec (c+d x)}}+\frac{\left (24 A b^2+30 a b B+3 a^2 C+16 b^2 C\right ) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 b d}+\frac{(2 b B+a C) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac{C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac{\left (\left (42 a b B+8 b^2 (3 A+2 C)+a^2 (48 A+17 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}{48 \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (-24 A b^2-30 a b B-3 a^2 C-16 b^2 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}{48 b \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}}\\ &=\frac{\left (42 a b B+8 b^2 (3 A+2 C)+a^2 (48 A+17 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)}}{24 d \sqrt{a+b \sec (c+d x)}}+\frac{\left (6 a^2 b B+8 b^3 B-a^3 C+12 a b^2 (2 A+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)}}{8 b d \sqrt{a+b \sec (c+d x)}}-\frac{\left (24 A b^2+30 a b B+3 a^2 C+16 b^2 C\right ) E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{a+b \sec (c+d x)}}{24 b d \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}}+\frac{\left (24 A b^2+30 a b B+3 a^2 C+16 b^2 C\right ) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 b d}+\frac{(2 b B+a C) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac{C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [C] time = 6.95525, size = 800, normalized size = 1.79 \[ \frac{(a+b \sec (c+d x))^{3/2} \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (\frac{1}{6} (6 b B \sin (c+d x)+7 a C \sin (c+d x)) \sec ^2(c+d x)+\frac{2}{3} b C \tan (c+d x) \sec ^2(c+d x)+\frac{\left (3 C \sin (c+d x) a^2+30 b B \sin (c+d x) a+24 A b^2 \sin (c+d x)+16 b^2 C \sin (c+d x)\right ) \sec (c+d x)}{12 b}\right )}{d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}-\frac{(a+b \sec (c+d x))^{3/2} \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (\frac{2 \left (-96 A b a^2-28 b C a^2-24 b^2 B a\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )}{\sqrt{b+a \cos (c+d x)}}+\frac{2 \left (9 C a^3-6 b B a^2-120 A b^2 a-56 b^2 C a-48 b^3 B\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{\sqrt{b+a \cos (c+d x)}}+\frac{2 i \left (3 C a^3+30 b B a^2+24 A b^2 a+16 b^2 C a\right ) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{\cos (c+d x) a+a}{a-b}} \cos (2 (c+d x)) \left (a \left (2 b \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right ),\frac{b-a}{a+b}\right )+a \Pi \left (1-\frac{a}{b};i \sinh ^{-1}\left (\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right )|\frac{b-a}{a+b}\right )\right )-2 b (a+b) E\left (i \sinh ^{-1}\left (\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right )|\frac{b-a}{a+b}\right )\right ) \sin (c+d x)}{\sqrt{\frac{1}{a-b}} b \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left (-a^2+2 b^2+2 (b+a \cos (c+d x))^2-4 b (b+a \cos (c+d x))\right )}\right )}{48 b d (b+a \cos (c+d x))^{3/2} (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.583, size = 5245, normalized size = 11.8 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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{3}{2}} \sqrt{\sec \left (d x + c\right )}\,{d x} \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 (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{3}{2}} \sqrt{\sec \left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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