Optimal. Leaf size=218 \[ \frac {4 c (d+e x)^{9/2} \left (a B e^2-2 A c d e+5 B c d^2\right )}{9 e^6}+\frac {2 (d+e x)^{5/2} \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6}-\frac {2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2 (B d-A e)}{3 e^6}-\frac {4 c (d+e x)^{7/2} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{7 e^6}-\frac {2 c^2 (d+e x)^{11/2} (5 B d-A e)}{11 e^6}+\frac {2 B c^2 (d+e x)^{13/2}}{13 e^6} \]
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Rubi [A] time = 0.13, antiderivative size = 218, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {4 c (d+e x)^{9/2} \left (a B e^2-2 A c d e+5 B c d^2\right )}{9 e^6}-\frac {4 c (d+e x)^{7/2} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{7 e^6}+\frac {2 (d+e x)^{5/2} \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6}-\frac {2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2 (B d-A e)}{3 e^6}-\frac {2 c^2 (d+e x)^{11/2} (5 B d-A e)}{11 e^6}+\frac {2 B c^2 (d+e x)^{13/2}}{13 e^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 772
Rubi steps
\begin {align*} \int (A+B x) \sqrt {d+e x} \left (a+c x^2\right )^2 \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}{e^5}+\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right ) (d+e x)^{3/2}}{e^5}+\frac {2 c \left (-5 B c d^3+3 A c d^2 e-3 a B d e^2+a A e^3\right ) (d+e x)^{5/2}}{e^5}-\frac {2 c \left (-5 B c d^2+2 A c d e-a B e^2\right ) (d+e x)^{7/2}}{e^5}+\frac {c^2 (-5 B d+A e) (d+e x)^{9/2}}{e^5}+\frac {B c^2 (d+e x)^{11/2}}{e^5}\right ) \, dx\\ &=-\frac {2 (B d-A e) \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}{3 e^6}+\frac {2 \left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right ) (d+e x)^{5/2}}{5 e^6}-\frac {4 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^{7/2}}{7 e^6}+\frac {4 c \left (5 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{9/2}}{9 e^6}-\frac {2 c^2 (5 B d-A e) (d+e x)^{11/2}}{11 e^6}+\frac {2 B c^2 (d+e x)^{13/2}}{13 e^6}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 214, normalized size = 0.98 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (13 A e \left (1155 a^2 e^4+66 a c e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+c^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )+B \left (3003 a^2 e^4 (3 e x-2 d)+286 a c e^2 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )-5 c^2 \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )\right )\right )}{45045 e^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 301, normalized size = 1.38 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (15015 a^2 A e^5+9009 a^2 B e^4 (d+e x)-15015 a^2 B d e^4+30030 a A c d^2 e^3-36036 a A c d e^3 (d+e x)+12870 a A c e^3 (d+e x)^2-30030 a B c d^3 e^2+54054 a B c d^2 e^2 (d+e x)-38610 a B c d e^2 (d+e x)^2+10010 a B c e^2 (d+e x)^3+15015 A c^2 d^4 e-36036 A c^2 d^3 e (d+e x)+38610 A c^2 d^2 e (d+e x)^2-20020 A c^2 d e (d+e x)^3+4095 A c^2 e (d+e x)^4-15015 B c^2 d^5+45045 B c^2 d^4 (d+e x)-64350 B c^2 d^3 (d+e x)^2+50050 B c^2 d^2 (d+e x)^3-20475 B c^2 d (d+e x)^4+3465 B c^2 (d+e x)^5\right )}{45045 e^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 320, normalized size = 1.47 \begin {gather*} \frac {2 \, {\left (3465 \, B c^{2} e^{6} x^{6} - 1280 \, B c^{2} d^{6} + 1664 \, A c^{2} d^{5} e - 4576 \, B a c d^{4} e^{2} + 6864 \, A a c d^{3} e^{3} - 6006 \, B a^{2} d^{2} e^{4} + 15015 \, A a^{2} d e^{5} + 315 \, {\left (B c^{2} d e^{5} + 13 \, A c^{2} e^{6}\right )} x^{5} - 35 \, {\left (10 \, B c^{2} d^{2} e^{4} - 13 \, A c^{2} d e^{5} - 286 \, B a c e^{6}\right )} x^{4} + 10 \, {\left (40 \, B c^{2} d^{3} e^{3} - 52 \, A c^{2} d^{2} e^{4} + 143 \, B a c d e^{5} + 1287 \, A a c e^{6}\right )} x^{3} - 3 \, {\left (160 \, B c^{2} d^{4} e^{2} - 208 \, A c^{2} d^{3} e^{3} + 572 \, B a c d^{2} e^{4} - 858 \, A a c d e^{5} - 3003 \, B a^{2} e^{6}\right )} x^{2} + {\left (640 \, B c^{2} d^{5} e - 832 \, A c^{2} d^{4} e^{2} + 2288 \, B a c d^{3} e^{3} - 3432 \, A a c d^{2} e^{4} + 3003 \, B a^{2} d e^{5} + 15015 \, A a^{2} e^{6}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 670, normalized size = 3.07 \begin {gather*} \frac {2}{45045} \, {\left (15015 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a^{2} d e^{\left (-1\right )} + 6006 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} A a c d e^{\left (-2\right )} + 2574 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} B a c d e^{\left (-3\right )} + 143 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} A c^{2} d e^{\left (-4\right )} + 65 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} B c^{2} d e^{\left (-5\right )} + 3003 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} B a^{2} e^{\left (-1\right )} + 2574 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} A a c e^{\left (-2\right )} + 286 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} B a c e^{\left (-3\right )} + 65 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} A c^{2} e^{\left (-4\right )} + 15 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} B c^{2} e^{\left (-5\right )} + 45045 \, \sqrt {x e + d} A a^{2} d + 15015 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a^{2}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 259, normalized size = 1.19 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3465 B \,c^{2} x^{5} e^{5}+4095 A \,c^{2} e^{5} x^{4}-3150 B \,c^{2} d \,e^{4} x^{4}-3640 A \,c^{2} d \,e^{4} x^{3}+10010 B a c \,e^{5} x^{3}+2800 B \,c^{2} d^{2} e^{3} x^{3}+12870 A a c \,e^{5} x^{2}+3120 A \,c^{2} d^{2} e^{3} x^{2}-8580 B a c d \,e^{4} x^{2}-2400 B \,c^{2} d^{3} e^{2} x^{2}-10296 A a c d \,e^{4} x -2496 A \,c^{2} d^{3} e^{2} x +9009 B \,a^{2} e^{5} x +6864 B a c \,d^{2} e^{3} x +1920 B \,c^{2} d^{4} e x +15015 A \,a^{2} e^{5}+6864 A \,d^{2} a c \,e^{3}+1664 A \,c^{2} d^{4} e -6006 B \,a^{2} d \,e^{4}-4576 B \,d^{3} a c \,e^{2}-1280 B \,c^{2} d^{5}\right )}{45045 e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 248, normalized size = 1.14 \begin {gather*} \frac {2 \, {\left (3465 \, {\left (e x + d\right )}^{\frac {13}{2}} B c^{2} - 4095 \, {\left (5 \, B c^{2} d - A c^{2} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 10010 \, {\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 12870 \, {\left (5 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, B a c d e^{2} - A a c e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 9009 \, {\left (5 \, B c^{2} d^{4} - 4 \, A c^{2} d^{3} e + 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 15015 \, {\left (B c^{2} d^{5} - A c^{2} d^{4} e + 2 \, B a c d^{3} e^{2} - 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} - A a^{2} e^{5}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{45045 \, e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.79, size = 197, normalized size = 0.90 \begin {gather*} \frac {{\left (d+e\,x\right )}^{9/2}\,\left (20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e+4\,B\,a\,c\,e^2\right )}{9\,e^6}+\frac {4\,c\,{\left (d+e\,x\right )}^{7/2}\,\left (-5\,B\,c\,d^3+3\,A\,c\,d^2\,e-3\,B\,a\,d\,e^2+A\,a\,e^3\right )}{7\,e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}+\frac {2\,\left (c\,d^2+a\,e^2\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (5\,B\,c\,d^2-4\,A\,c\,d\,e+B\,a\,e^2\right )}{5\,e^6}+\frac {2\,c^2\,\left (A\,e-5\,B\,d\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^2\,\left (A\,e-B\,d\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.56, size = 308, normalized size = 1.41 \begin {gather*} \frac {2 \left (\frac {B c^{2} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{5}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (A c^{2} e - 5 B c^{2} d\right )}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (- 4 A c^{2} d e + 2 B a c e^{2} + 10 B c^{2} d^{2}\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (2 A a c e^{3} + 6 A c^{2} d^{2} e - 6 B a c d e^{2} - 10 B c^{2} d^{3}\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 4 A a c d e^{3} - 4 A c^{2} d^{3} e + B a^{2} e^{4} + 6 B a c d^{2} e^{2} + 5 B c^{2} d^{4}\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A a^{2} e^{5} + 2 A a c d^{2} e^{3} + A c^{2} d^{4} e - B a^{2} d e^{4} - 2 B a c d^{3} e^{2} - B c^{2} d^{5}\right )}{3 e^{5}}\right )}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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