Optimal. Leaf size=344 \[ -\frac{2 c (d+e x)^{5/2} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{5 e^8}+\frac{2 c^2 (d+e x)^{9/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{3 e^8}-\frac{2 c^2 (d+e x)^{7/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{7 e^8}+\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8}+\frac{2 \left (a e^2+c d^2\right )^3 (B d-A e)}{e^8 \sqrt{d+e x}}-\frac{2 c (d+e x)^{3/2} \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8}-\frac{2 c^3 (d+e x)^{11/2} (7 B d-A e)}{11 e^8}+\frac{2 B c^3 (d+e x)^{13/2}}{13 e^8} \]
[Out]
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Rubi [A] time = 0.471681, antiderivative size = 344, 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 \[ \frac{2 c (d+e x)^{5/2} \left (3 a^2 B e^4-12 a A c d e^3+30 a B c d^2 e^2-20 A c^2 d^3 e+35 B c^2 d^4\right )}{5 e^8}+\frac{2 c^2 (d+e x)^{9/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{3 e^8}-\frac{2 c^2 (d+e x)^{7/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{7 e^8}+\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8}+\frac{2 \left (a e^2+c d^2\right )^3 (B d-A e)}{e^8 \sqrt{d+e x}}-\frac{2 c (d+e x)^{3/2} \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8}-\frac{2 c^3 (d+e x)^{11/2} (7 B d-A e)}{11 e^8}+\frac{2 B c^3 (d+e x)^{13/2}}{13 e^8} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 106.218, size = 362, normalized size = 1.05 \[ \frac{2 B c^{3} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{8}} + \frac{2 c^{3} \left (d + e x\right )^{\frac{11}{2}} \left (A e - 7 B d\right )}{11 e^{8}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{9}{2}} \left (- 2 A c d e + B a e^{2} + 7 B c d^{2}\right )}{3 e^{8}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{7}{2}} \left (3 A a e^{3} + 15 A c d^{2} e - 15 B a d e^{2} - 35 B c d^{3}\right )}{7 e^{8}} + \frac{2 c \left (d + e x\right )^{\frac{5}{2}} \left (- 12 A a c d e^{3} - 20 A c^{2} d^{3} e + 3 B a^{2} e^{4} + 30 B a c d^{2} e^{2} + 35 B c^{2} d^{4}\right )}{5 e^{8}} + \frac{2 c \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right ) \left (A a e^{3} + 5 A c d^{2} e - 3 B a d e^{2} - 7 B c d^{3}\right )}{e^{8}} + \frac{2 \sqrt{d + e x} \left (a e^{2} + c d^{2}\right )^{2} \left (- 6 A c d e + B a e^{2} + 7 B c d^{2}\right )}{e^{8}} - \frac{2 \left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )^{3}}{e^{8} \sqrt{d + e x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**(3/2),x)
[Out]
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Mathematica [A] time = 0.553868, size = 373, normalized size = 1.08 \[ \frac{2 B \left (15015 a^3 e^6 (2 d+e x)+9009 a^2 c e^4 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+715 a c^2 e^2 \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )+35 c^3 \left (2048 d^7+1024 d^6 e x-256 d^5 e^2 x^2+128 d^4 e^3 x^3-80 d^3 e^4 x^4+56 d^2 e^5 x^5-42 d e^6 x^6+33 e^7 x^7\right )\right )-26 A e \left (1155 a^3 e^6+1155 a^2 c e^4 \left (8 d^2+4 d e x-e^2 x^2\right )+99 a c^2 e^2 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )+5 c^3 \left (1024 d^6+512 d^5 e x-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+28 d e^5 x^5-21 e^6 x^6\right )\right )}{15015 e^8 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(3/2),x]
[Out]
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Maple [A] time = 0.013, size = 489, normalized size = 1.4 \[ -{\frac{-2310\,B{c}^{3}{x}^{7}{e}^{7}-2730\,A{c}^{3}{e}^{7}{x}^{6}+2940\,B{c}^{3}d{e}^{6}{x}^{6}+3640\,A{c}^{3}d{e}^{6}{x}^{5}-10010\,Ba{c}^{2}{e}^{7}{x}^{5}-3920\,B{c}^{3}{d}^{2}{e}^{5}{x}^{5}-12870\,Aa{c}^{2}{e}^{7}{x}^{4}-5200\,A{c}^{3}{d}^{2}{e}^{5}{x}^{4}+14300\,Ba{c}^{2}d{e}^{6}{x}^{4}+5600\,B{c}^{3}{d}^{3}{e}^{4}{x}^{4}+20592\,Aa{c}^{2}d{e}^{6}{x}^{3}+8320\,A{c}^{3}{d}^{3}{e}^{4}{x}^{3}-18018\,B{a}^{2}c{e}^{7}{x}^{3}-22880\,Ba{c}^{2}{d}^{2}{e}^{5}{x}^{3}-8960\,B{c}^{3}{d}^{4}{e}^{3}{x}^{3}-30030\,A{a}^{2}c{e}^{7}{x}^{2}-41184\,Aa{c}^{2}{d}^{2}{e}^{5}{x}^{2}-16640\,A{c}^{3}{d}^{4}{e}^{3}{x}^{2}+36036\,B{a}^{2}cd{e}^{6}{x}^{2}+45760\,Ba{c}^{2}{d}^{3}{e}^{4}{x}^{2}+17920\,B{c}^{3}{d}^{5}{e}^{2}{x}^{2}+120120\,A{a}^{2}cd{e}^{6}x+164736\,Aa{c}^{2}{d}^{3}{e}^{4}x+66560\,A{c}^{3}{d}^{5}{e}^{2}x-30030\,B{a}^{3}{e}^{7}x-144144\,B{a}^{2}c{d}^{2}{e}^{5}x-183040\,Ba{c}^{2}{d}^{4}{e}^{3}x-71680\,B{c}^{3}{d}^{6}ex+30030\,A{a}^{3}{e}^{7}+240240\,A{a}^{2}c{d}^{2}{e}^{5}+329472\,Aa{c}^{2}{d}^{4}{e}^{3}+133120\,A{c}^{3}{d}^{6}e-60060\,B{a}^{3}d{e}^{6}-288288\,B{a}^{2}c{d}^{3}{e}^{4}-366080\,Ba{c}^{2}{d}^{5}{e}^{2}-143360\,B{c}^{3}{d}^{7}}{15015\,{e}^{8}}{\frac{1}{\sqrt{ex+d}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)^3/(e*x+d)^(3/2),x)
[Out]
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Maxima [A] time = 0.691679, size = 622, normalized size = 1.81 \[ \frac{2 \,{\left (\frac{1155 \,{\left (e x + d\right )}^{\frac{13}{2}} B c^{3} - 1365 \,{\left (7 \, B c^{3} d - A c^{3} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 5005 \,{\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 2145 \,{\left (35 \, B c^{3} d^{3} - 15 \, A c^{3} d^{2} e + 15 \, B a c^{2} d e^{2} - 3 \, A a c^{2} e^{3}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 3003 \,{\left (35 \, B c^{3} d^{4} - 20 \, A c^{3} d^{3} e + 30 \, B a c^{2} d^{2} e^{2} - 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 15015 \,{\left (7 \, B c^{3} d^{5} - 5 \, A c^{3} d^{4} e + 10 \, B a c^{2} d^{3} e^{2} - 6 \, A a c^{2} d^{2} e^{3} + 3 \, B a^{2} c d e^{4} - A a^{2} c e^{5}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15015 \,{\left (7 \, B c^{3} d^{6} - 6 \, A c^{3} d^{5} e + 15 \, B a c^{2} d^{4} e^{2} - 12 \, A a c^{2} d^{3} e^{3} + 9 \, B a^{2} c d^{2} e^{4} - 6 \, A a^{2} c d e^{5} + B a^{3} e^{6}\right )} \sqrt{e x + d}}{e^{7}} + \frac{15015 \,{\left (B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} - A a^{3} e^{7}\right )}}{\sqrt{e x + d} e^{7}}\right )}}{15015 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.267926, size = 612, normalized size = 1.78 \[ \frac{2 \,{\left (1155 \, B c^{3} e^{7} x^{7} + 71680 \, B c^{3} d^{7} - 66560 \, A c^{3} d^{6} e + 183040 \, B a c^{2} d^{5} e^{2} - 164736 \, A a c^{2} d^{4} e^{3} + 144144 \, B a^{2} c d^{3} e^{4} - 120120 \, A a^{2} c d^{2} e^{5} + 30030 \, B a^{3} d e^{6} - 15015 \, A a^{3} e^{7} - 105 \,{\left (14 \, B c^{3} d e^{6} - 13 \, A c^{3} e^{7}\right )} x^{6} + 35 \,{\left (56 \, B c^{3} d^{2} e^{5} - 52 \, A c^{3} d e^{6} + 143 \, B a c^{2} e^{7}\right )} x^{5} - 5 \,{\left (560 \, B c^{3} d^{3} e^{4} - 520 \, A c^{3} d^{2} e^{5} + 1430 \, B a c^{2} d e^{6} - 1287 \, A a c^{2} e^{7}\right )} x^{4} +{\left (4480 \, B c^{3} d^{4} e^{3} - 4160 \, A c^{3} d^{3} e^{4} + 11440 \, B a c^{2} d^{2} e^{5} - 10296 \, A a c^{2} d e^{6} + 9009 \, B a^{2} c e^{7}\right )} x^{3} -{\left (8960 \, B c^{3} d^{5} e^{2} - 8320 \, A c^{3} d^{4} e^{3} + 22880 \, B a c^{2} d^{3} e^{4} - 20592 \, A a c^{2} d^{2} e^{5} + 18018 \, B a^{2} c d e^{6} - 15015 \, A a^{2} c e^{7}\right )} x^{2} +{\left (35840 \, B c^{3} d^{6} e - 33280 \, A c^{3} d^{5} e^{2} + 91520 \, B a c^{2} d^{4} e^{3} - 82368 \, A a c^{2} d^{3} e^{4} + 72072 \, B a^{2} c d^{2} e^{5} - 60060 \, A a^{2} c d e^{6} + 15015 \, B a^{3} e^{7}\right )} x\right )}}{15015 \, \sqrt{e x + d} e^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + c x^{2}\right )^{3}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.2947, size = 830, normalized size = 2.41 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^(3/2),x, algorithm="giac")
[Out]