Optimal. Leaf size=455 \[ \frac{2 \left (a^2-b^2\right ) \sqrt{\sec (c+d x)} \left (a^2 (39 A b+63 b C)+75 a^3 B-18 a b^2 B+8 A b^3\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )}{315 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{315 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \sin (c+d x) \left (-2 a^2 b (44 A+63 C)-75 a^3 B-9 a b^2 B+4 A b^3\right ) \sqrt{a+b \sec (c+d x)}}{315 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \left (3 a^2 b^2 (11 A+21 C)+21 a^4 (7 A+9 C)+246 a^3 b B-18 a b^3 B+8 A b^4\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{315 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (3 a B+A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{21 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac{7}{2}}(c+d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.70909, antiderivative size = 455, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661} \[ \frac{2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{315 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \sin (c+d x) \left (-2 a^2 b (44 A+63 C)-75 a^3 B-9 a b^2 B+4 A b^3\right ) \sqrt{a+b \sec (c+d x)}}{315 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \left (a^2-b^2\right ) \sqrt{\sec (c+d x)} \left (a^2 (39 A b+63 b C)+75 a^3 B-18 a b^2 B+8 A b^3\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{315 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (3 a^2 b^2 (11 A+21 C)+21 a^4 (7 A+9 C)+246 a^3 b B-18 a b^3 B+8 A b^4\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{315 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (3 a B+A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{21 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4094
Rule 4104
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))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac{9}{2}}(c+d x)} \, dx &=\frac{2 A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2}{9} \int \frac{\sqrt{a+b \sec (c+d x)} \left (\frac{3}{2} (A b+3 a B)+\frac{1}{2} (7 a A+9 b B+9 a C) \sec (c+d x)+\frac{1}{2} b (4 A+9 C) \sec ^2(c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 (A b+3 a B) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{21 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{4}{63} \int \frac{\frac{1}{4} \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right )+\frac{1}{4} \left (92 a A b+45 a^2 B+63 b^2 B+126 a b C\right ) \sec (c+d x)+\frac{1}{4} b (40 A b+36 a B+63 b C) \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{2 (A b+3 a B) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{21 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}-\frac{8 \int \frac{\frac{3}{8} \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right )-\frac{1}{8} a \left (396 a b B+21 a^2 (7 A+9 C)+b^2 (209 A+315 C)\right ) \sec (c+d x)-\frac{1}{4} b \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx}{315 a}\\ &=\frac{2 (A b+3 a B) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{21 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{16 \int \frac{\frac{3}{16} \left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\right )+\frac{3}{16} a \left (2 A b^3+75 a^3 B+153 a b^2 B+6 a^2 b (31 A+42 C)\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx}{945 a^2}\\ &=\frac{2 (A b+3 a B) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{21 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\right ) \int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx}{315 a^3}+\frac{\left (\left (a^2-b^2\right ) \left (8 A b^3+75 a^3 B-18 a b^2 B+a^2 (39 A b+63 b C)\right )\right ) \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx}{315 a^3}\\ &=\frac{2 (A b+3 a B) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{21 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left (\left (a^2-b^2\right ) \left (8 A b^3+75 a^3 B-18 a b^2 B+a^2 (39 A b+63 b 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}{315 a^3 \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\right ) \sqrt{a+b \sec (c+d x)}\right ) \int \sqrt{b+a \cos (c+d x)} \, dx}{315 a^3 \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)}}\\ &=\frac{2 (A b+3 a B) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{21 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left (\left (a^2-b^2\right ) \left (8 A b^3+75 a^3 B-18 a b^2 B+a^2 (39 A b+63 b 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}{315 a^3 \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 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}{315 a^3 \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}}\\ &=\frac{2 \left (a^2-b^2\right ) \left (8 A b^3+75 a^3 B-18 a b^2 B+a^2 (39 A b+63 b 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)}}{315 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\right ) E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{a+b \sec (c+d x)}}{315 a^3 d \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}}+\frac{2 (A b+3 a B) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{21 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}\\ \end{align*}
Mathematica [C] time = 7.03093, size = 5997, normalized size = 13.18 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.868, size = 6526, normalized size = 14.3 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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{3}{2}}}{\sec \left (d x + c\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C b \sec \left (d x + c\right )^{3} +{\left (C a + B b\right )} \sec \left (d x + c\right )^{2} + A a +{\left (B 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{9}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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{3}{2}}}{\sec \left (d x + c\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]