Optimal. Leaf size=196 \[ \frac{2 c^2 (d+e x)^{3/2} \left (a e^2+5 c d^2\right )}{e^7}-\frac{8 c^2 d \sqrt{d+e x} \left (3 a e^2+5 c d^2\right )}{e^7}-\frac{6 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 \sqrt{d+e x}}+\frac{4 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac{2 \left (a e^2+c d^2\right )^3}{5 e^7 (d+e x)^{5/2}}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7}-\frac{12 c^3 d (d+e x)^{5/2}}{5 e^7} \]
[Out]
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Rubi [A] time = 0.225751, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{2 c^2 (d+e x)^{3/2} \left (a e^2+5 c d^2\right )}{e^7}-\frac{8 c^2 d \sqrt{d+e x} \left (3 a e^2+5 c d^2\right )}{e^7}-\frac{6 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 \sqrt{d+e x}}+\frac{4 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac{2 \left (a e^2+c d^2\right )^3}{5 e^7 (d+e x)^{5/2}}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7}-\frac{12 c^3 d (d+e x)^{5/2}}{5 e^7} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)^3/(d + e*x)^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 41.1979, size = 192, normalized size = 0.98 \[ - \frac{12 c^{3} d \left (d + e x\right )^{\frac{5}{2}}}{5 e^{7}} + \frac{2 c^{3} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{7}} - \frac{8 c^{2} d \sqrt{d + e x} \left (3 a e^{2} + 5 c d^{2}\right )}{e^{7}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} + 5 c d^{2}\right )}{e^{7}} + \frac{4 c d \left (a e^{2} + c d^{2}\right )^{2}}{e^{7} \left (d + e x\right )^{\frac{3}{2}}} - \frac{6 c \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{e^{7} \sqrt{d + e x}} - \frac{2 \left (a e^{2} + c d^{2}\right )^{3}}{5 e^{7} \left (d + e x\right )^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)**3/(e*x+d)**(7/2),x)
[Out]
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Mathematica [A] time = 0.181208, size = 170, normalized size = 0.87 \[ -\frac{2 \left (7 a^3 e^6+7 a^2 c e^4 \left (8 d^2+20 d e x+15 e^2 x^2\right )+7 a c^2 e^2 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )+c^3 \left (1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right )\right )}{35 e^7 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)^3/(d + e*x)^(7/2),x]
[Out]
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Maple [A] time = 0.011, size = 205, normalized size = 1.1 \[ -{\frac{-10\,{c}^{3}{x}^{6}{e}^{6}+24\,{c}^{3}d{x}^{5}{e}^{5}-70\,a{c}^{2}{e}^{6}{x}^{4}-80\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}+560\,a{c}^{2}d{e}^{5}{x}^{3}+640\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+210\,{a}^{2}c{e}^{6}{x}^{2}+3360\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}+3840\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+280\,{a}^{2}cd{e}^{5}x+4480\,a{c}^{2}{d}^{3}{e}^{3}x+5120\,{c}^{3}{d}^{5}ex+14\,{a}^{3}{e}^{6}+112\,{a}^{2}c{d}^{2}{e}^{4}+1792\,a{c}^{2}{d}^{4}{e}^{2}+2048\,{c}^{3}{d}^{6}}{35\,{e}^{7}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)^3/(e*x+d)^(7/2),x)
[Out]
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Maxima [A] time = 0.70188, size = 290, normalized size = 1.48 \[ \frac{2 \,{\left (\frac{5 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{3} - 42 \,{\left (e x + d\right )}^{\frac{5}{2}} c^{3} d + 35 \,{\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 140 \,{\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} \sqrt{e x + d}}{e^{6}} - \frac{7 \,{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 15 \,{\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}{\left (e x + d\right )}^{2} - 10 \,{\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{6}}\right )}}{35 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3/(e*x + d)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226654, size = 301, normalized size = 1.54 \[ \frac{2 \,{\left (5 \, c^{3} e^{6} x^{6} - 12 \, c^{3} d e^{5} x^{5} - 1024 \, c^{3} d^{6} - 896 \, a c^{2} d^{4} e^{2} - 56 \, a^{2} c d^{2} e^{4} - 7 \, a^{3} e^{6} + 5 \,{\left (8 \, c^{3} d^{2} e^{4} + 7 \, a c^{2} e^{6}\right )} x^{4} - 40 \,{\left (8 \, c^{3} d^{3} e^{3} + 7 \, a c^{2} d e^{5}\right )} x^{3} - 15 \,{\left (128 \, c^{3} d^{4} e^{2} + 112 \, a c^{2} d^{2} e^{4} + 7 \, a^{2} c e^{6}\right )} x^{2} - 20 \,{\left (128 \, c^{3} d^{5} e + 112 \, a c^{2} d^{3} e^{3} + 7 \, a^{2} c d e^{5}\right )} x\right )}}{35 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3/(e*x + d)^(7/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + c x^{2}\right )^{3}}{\left (d + e x\right )^{\frac{7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)**3/(e*x+d)**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.216361, size = 339, normalized size = 1.73 \[ \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} e^{42} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d e^{42} + 175 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{2} e^{42} - 700 \, \sqrt{x e + d} c^{3} d^{3} e^{42} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a c^{2} e^{44} - 420 \, \sqrt{x e + d} a c^{2} d e^{44}\right )} e^{\left (-49\right )} - \frac{2 \,{\left (75 \,{\left (x e + d\right )}^{2} c^{3} d^{4} - 10 \,{\left (x e + d\right )} c^{3} d^{5} + c^{3} d^{6} + 90 \,{\left (x e + d\right )}^{2} a c^{2} d^{2} e^{2} - 20 \,{\left (x e + d\right )} a c^{2} d^{3} e^{2} + 3 \, a c^{2} d^{4} e^{2} + 15 \,{\left (x e + d\right )}^{2} a^{2} c e^{4} - 10 \,{\left (x e + d\right )} a^{2} c d e^{4} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} e^{\left (-7\right )}}{5 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3/(e*x + d)^(7/2),x, algorithm="giac")
[Out]