Optimal. Leaf size=346 \[ -\frac{2 c \sqrt{d+e x} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{e^8}+\frac{6 c^2 (d+e x)^{5/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{5 e^8}-\frac{2 c^2 (d+e x)^{3/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{3 e^8}+\frac{6 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 \sqrt{d+e x}}-\frac{2 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{3 e^8 (d+e x)^{3/2}}+\frac{2 \left (a e^2+c d^2\right )^3 (B d-A e)}{5 e^8 (d+e x)^{5/2}}-\frac{2 c^3 (d+e x)^{7/2} (7 B d-A e)}{7 e^8}+\frac{2 B c^3 (d+e x)^{9/2}}{9 e^8} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.161639, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {772} \[ -\frac{2 c \sqrt{d+e x} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{e^8}+\frac{6 c^2 (d+e x)^{5/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{5 e^8}-\frac{2 c^2 (d+e x)^{3/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{3 e^8}+\frac{6 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 \sqrt{d+e x}}-\frac{2 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{3 e^8 (d+e x)^{3/2}}+\frac{2 \left (a e^2+c d^2\right )^3 (B d-A e)}{5 e^8 (d+e x)^{5/2}}-\frac{2 c^3 (d+e x)^{7/2} (7 B d-A e)}{7 e^8}+\frac{2 B c^3 (d+e x)^{9/2}}{9 e^8} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 772
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{7/2}} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^3}{e^7 (d+e x)^{7/2}}+\frac{\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{e^7 (d+e x)^{5/2}}+\frac{3 c \left (c d^2+a e^2\right ) \left (-7 B c d^3+5 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^7 (d+e x)^{3/2}}-\frac{c \left (-35 B c^2 d^4+20 A c^2 d^3 e-30 a B c d^2 e^2+12 a A c d e^3-3 a^2 B e^4\right )}{e^7 \sqrt{d+e x}}+\frac{c^2 \left (-35 B c d^3+15 A c d^2 e-15 a B d e^2+3 a A e^3\right ) \sqrt{d+e x}}{e^7}-\frac{3 c^2 \left (-7 B c d^2+2 A c d e-a B e^2\right ) (d+e x)^{3/2}}{e^7}+\frac{c^3 (-7 B d+A e) (d+e x)^{5/2}}{e^7}+\frac{B c^3 (d+e x)^{7/2}}{e^7}\right ) \, dx\\ &=\frac{2 (B d-A e) \left (c d^2+a e^2\right )^3}{5 e^8 (d+e x)^{5/2}}-\frac{2 \left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{3 e^8 (d+e x)^{3/2}}+\frac{6 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right )}{e^8 \sqrt{d+e x}}-\frac{2 c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right ) \sqrt{d+e x}}{e^8}-\frac{2 c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right ) (d+e x)^{3/2}}{3 e^8}+\frac{6 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{5/2}}{5 e^8}-\frac{2 c^3 (7 B d-A e) (d+e x)^{7/2}}{7 e^8}+\frac{2 B c^3 (d+e x)^{9/2}}{9 e^8}\\ \end{align*}
Mathematica [A] time = 0.288793, size = 373, normalized size = 1.08 \[ \frac{2 \left (7 B \left (27 a^2 c e^4 \left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )-3 a^3 e^6 (2 d+5 e x)+9 a c^2 e^2 \left (480 d^3 e^2 x^2+80 d^2 e^3 x^3+640 d^4 e x+256 d^5-10 d e^4 x^4+3 e^5 x^5\right )+c^3 \left (3840 d^5 e^2 x^2+640 d^4 e^3 x^3-80 d^3 e^4 x^4+24 d^2 e^5 x^5+5120 d^6 e x+2048 d^7-10 d e^6 x^6+5 e^7 x^7\right )\right )-9 A e \left (7 a^2 c e^4 \left (8 d^2+20 d e x+15 e^2 x^2\right )+7 a^3 e^6+7 a c^2 e^2 \left (240 d^2 e^2 x^2+320 d^3 e x+128 d^4+40 d e^3 x^3-5 e^4 x^4\right )+c^3 \left (1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+2560 d^5 e x+1024 d^6+12 d e^5 x^5-5 e^6 x^6\right )\right )\right )}{315 e^8 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 489, normalized size = 1.4 \begin{align*} -{\frac{-70\,B{c}^{3}{x}^{7}{e}^{7}-90\,A{c}^{3}{e}^{7}{x}^{6}+140\,B{c}^{3}d{e}^{6}{x}^{6}+216\,A{c}^{3}d{e}^{6}{x}^{5}-378\,Ba{c}^{2}{e}^{7}{x}^{5}-336\,B{c}^{3}{d}^{2}{e}^{5}{x}^{5}-630\,Aa{c}^{2}{e}^{7}{x}^{4}-720\,A{c}^{3}{d}^{2}{e}^{5}{x}^{4}+1260\,Ba{c}^{2}d{e}^{6}{x}^{4}+1120\,B{c}^{3}{d}^{3}{e}^{4}{x}^{4}+5040\,Aa{c}^{2}d{e}^{6}{x}^{3}+5760\,A{c}^{3}{d}^{3}{e}^{4}{x}^{3}-1890\,B{a}^{2}c{e}^{7}{x}^{3}-10080\,Ba{c}^{2}{d}^{2}{e}^{5}{x}^{3}-8960\,B{c}^{3}{d}^{4}{e}^{3}{x}^{3}+1890\,A{a}^{2}c{e}^{7}{x}^{2}+30240\,Aa{c}^{2}{d}^{2}{e}^{5}{x}^{2}+34560\,A{c}^{3}{d}^{4}{e}^{3}{x}^{2}-11340\,B{a}^{2}cd{e}^{6}{x}^{2}-60480\,Ba{c}^{2}{d}^{3}{e}^{4}{x}^{2}-53760\,B{c}^{3}{d}^{5}{e}^{2}{x}^{2}+2520\,A{a}^{2}cd{e}^{6}x+40320\,Aa{c}^{2}{d}^{3}{e}^{4}x+46080\,A{c}^{3}{d}^{5}{e}^{2}x+210\,B{a}^{3}{e}^{7}x-15120\,B{a}^{2}c{d}^{2}{e}^{5}x-80640\,Ba{c}^{2}{d}^{4}{e}^{3}x-71680\,B{c}^{3}{d}^{6}ex+126\,A{a}^{3}{e}^{7}+1008\,A{a}^{2}c{d}^{2}{e}^{5}+16128\,Aa{c}^{2}{d}^{4}{e}^{3}+18432\,A{c}^{3}{d}^{6}e+84\,B{a}^{3}d{e}^{6}-6048\,B{a}^{2}c{d}^{3}{e}^{4}-32256\,Ba{c}^{2}{d}^{5}{e}^{2}-28672\,B{c}^{3}{d}^{7}}{315\,{e}^{8}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00836, size = 622, normalized size = 1.8 \begin{align*} \frac{2 \,{\left (\frac{35 \,{\left (e x + d\right )}^{\frac{9}{2}} B c^{3} - 45 \,{\left (7 \, B c^{3} d - A c^{3} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 189 \,{\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 105 \,{\left (35 \, B c^{3} d^{3} - 15 \, A c^{3} d^{2} e + 15 \, B a c^{2} d e^{2} - 3 \, A a c^{2} e^{3}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 315 \,{\left (35 \, B c^{3} d^{4} - 20 \, A c^{3} d^{3} e + 30 \, B a c^{2} d^{2} e^{2} - 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )} \sqrt{e x + d}}{e^{7}} + \frac{21 \,{\left (3 \, B c^{3} d^{7} - 3 \, A c^{3} d^{6} e + 9 \, B a c^{2} d^{5} e^{2} - 9 \, A a c^{2} d^{4} e^{3} + 9 \, B a^{2} c d^{3} e^{4} - 9 \, A a^{2} c d^{2} e^{5} + 3 \, B a^{3} d e^{6} - 3 \, A a^{3} e^{7} + 45 \,{\left (7 \, B c^{3} d^{5} - 5 \, A c^{3} d^{4} e + 10 \, B a c^{2} d^{3} e^{2} - 6 \, A a c^{2} d^{2} e^{3} + 3 \, B a^{2} c d e^{4} - A a^{2} c e^{5}\right )}{\left (e x + d\right )}^{2} - 5 \,{\left (7 \, B c^{3} d^{6} - 6 \, A c^{3} d^{5} e + 15 \, B a c^{2} d^{4} e^{2} - 12 \, A a c^{2} d^{3} e^{3} + 9 \, B a^{2} c d^{2} e^{4} - 6 \, A a^{2} c d e^{5} + B a^{3} e^{6}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{7}}\right )}}{315 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.62192, size = 1118, normalized size = 3.23 \begin{align*} \frac{2 \,{\left (35 \, B c^{3} e^{7} x^{7} + 14336 \, B c^{3} d^{7} - 9216 \, A c^{3} d^{6} e + 16128 \, B a c^{2} d^{5} e^{2} - 8064 \, A a c^{2} d^{4} e^{3} + 3024 \, B a^{2} c d^{3} e^{4} - 504 \, A a^{2} c d^{2} e^{5} - 42 \, B a^{3} d e^{6} - 63 \, A a^{3} e^{7} - 5 \,{\left (14 \, B c^{3} d e^{6} - 9 \, A c^{3} e^{7}\right )} x^{6} + 3 \,{\left (56 \, B c^{3} d^{2} e^{5} - 36 \, A c^{3} d e^{6} + 63 \, B a c^{2} e^{7}\right )} x^{5} - 5 \,{\left (112 \, B c^{3} d^{3} e^{4} - 72 \, A c^{3} d^{2} e^{5} + 126 \, B a c^{2} d e^{6} - 63 \, A a c^{2} e^{7}\right )} x^{4} + 5 \,{\left (896 \, B c^{3} d^{4} e^{3} - 576 \, A c^{3} d^{3} e^{4} + 1008 \, B a c^{2} d^{2} e^{5} - 504 \, A a c^{2} d e^{6} + 189 \, B a^{2} c e^{7}\right )} x^{3} + 15 \,{\left (1792 \, B c^{3} d^{5} e^{2} - 1152 \, A c^{3} d^{4} e^{3} + 2016 \, B a c^{2} d^{3} e^{4} - 1008 \, A a c^{2} d^{2} e^{5} + 378 \, B a^{2} c d e^{6} - 63 \, A a^{2} c e^{7}\right )} x^{2} + 5 \,{\left (7168 \, B c^{3} d^{6} e - 4608 \, A c^{3} d^{5} e^{2} + 8064 \, B a c^{2} d^{4} e^{3} - 4032 \, A a c^{2} d^{3} e^{4} + 1512 \, B a^{2} c d^{2} e^{5} - 252 \, A a^{2} c d e^{6} - 21 \, B a^{3} e^{7}\right )} x\right )} \sqrt{e x + d}}{315 \,{\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 124.059, size = 377, normalized size = 1.09 \begin{align*} \frac{2 B c^{3} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{8}} + \frac{6 c \left (a e^{2} + c d^{2}\right ) \left (- A a e^{3} - 5 A c d^{2} e + 3 B a d e^{2} + 7 B c d^{3}\right )}{e^{8} \sqrt{d + e x}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (2 A c^{3} e - 14 B c^{3} d\right )}{7 e^{8}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (- 12 A c^{3} d e + 6 B a c^{2} e^{2} + 42 B c^{3} d^{2}\right )}{5 e^{8}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (6 A a c^{2} e^{3} + 30 A c^{3} d^{2} e - 30 B a c^{2} d e^{2} - 70 B c^{3} d^{3}\right )}{3 e^{8}} + \frac{\sqrt{d + e x} \left (- 24 A a c^{2} d e^{3} - 40 A c^{3} d^{3} e + 6 B a^{2} c e^{4} + 60 B a c^{2} d^{2} e^{2} + 70 B c^{3} d^{4}\right )}{e^{8}} - \frac{2 \left (a e^{2} + c d^{2}\right )^{2} \left (- 6 A c d e + B a e^{2} + 7 B c d^{2}\right )}{3 e^{8} \left (d + e x\right )^{\frac{3}{2}}} + \frac{2 \left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )^{3}}{5 e^{8} \left (d + e x\right )^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21446, size = 809, normalized size = 2.34 \begin{align*} \frac{2}{315} \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} B c^{3} e^{64} - 315 \,{\left (x e + d\right )}^{\frac{7}{2}} B c^{3} d e^{64} + 1323 \,{\left (x e + d\right )}^{\frac{5}{2}} B c^{3} d^{2} e^{64} - 3675 \,{\left (x e + d\right )}^{\frac{3}{2}} B c^{3} d^{3} e^{64} + 11025 \, \sqrt{x e + d} B c^{3} d^{4} e^{64} + 45 \,{\left (x e + d\right )}^{\frac{7}{2}} A c^{3} e^{65} - 378 \,{\left (x e + d\right )}^{\frac{5}{2}} A c^{3} d e^{65} + 1575 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{3} d^{2} e^{65} - 6300 \, \sqrt{x e + d} A c^{3} d^{3} e^{65} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} B a c^{2} e^{66} - 1575 \,{\left (x e + d\right )}^{\frac{3}{2}} B a c^{2} d e^{66} + 9450 \, \sqrt{x e + d} B a c^{2} d^{2} e^{66} + 315 \,{\left (x e + d\right )}^{\frac{3}{2}} A a c^{2} e^{67} - 3780 \, \sqrt{x e + d} A a c^{2} d e^{67} + 945 \, \sqrt{x e + d} B a^{2} c e^{68}\right )} e^{\left (-72\right )} + \frac{2 \,{\left (315 \,{\left (x e + d\right )}^{2} B c^{3} d^{5} - 35 \,{\left (x e + d\right )} B c^{3} d^{6} + 3 \, B c^{3} d^{7} - 225 \,{\left (x e + d\right )}^{2} A c^{3} d^{4} e + 30 \,{\left (x e + d\right )} A c^{3} d^{5} e - 3 \, A c^{3} d^{6} e + 450 \,{\left (x e + d\right )}^{2} B a c^{2} d^{3} e^{2} - 75 \,{\left (x e + d\right )} B a c^{2} d^{4} e^{2} + 9 \, B a c^{2} d^{5} e^{2} - 270 \,{\left (x e + d\right )}^{2} A a c^{2} d^{2} e^{3} + 60 \,{\left (x e + d\right )} A a c^{2} d^{3} e^{3} - 9 \, A a c^{2} d^{4} e^{3} + 135 \,{\left (x e + d\right )}^{2} B a^{2} c d e^{4} - 45 \,{\left (x e + d\right )} B a^{2} c d^{2} e^{4} + 9 \, B a^{2} c d^{3} e^{4} - 45 \,{\left (x e + d\right )}^{2} A a^{2} c e^{5} + 30 \,{\left (x e + d\right )} A a^{2} c d e^{5} - 9 \, A a^{2} c d^{2} e^{5} - 5 \,{\left (x e + d\right )} B a^{3} e^{6} + 3 \, B a^{3} d e^{6} - 3 \, A a^{3} e^{7}\right )} e^{\left (-8\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]