Optimal. Leaf size=200 \[ \frac{2 c^2 (d+e x)^{9/2} \left (a e^2+5 c d^2\right )}{3 e^7}-\frac{8 c^2 d (d+e x)^{7/2} \left (3 a e^2+5 c d^2\right )}{7 e^7}+\frac{6 c (d+e x)^{5/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{5 e^7}-\frac{4 c d (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}{e^7}+\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )^3}{e^7}+\frac{2 c^3 (d+e x)^{13/2}}{13 e^7}-\frac{12 c^3 d (d+e x)^{11/2}}{11 e^7} \]
[Out]
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Rubi [A] time = 0.226237, antiderivative size = 200, 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 \[ \frac{2 c^2 (d+e x)^{9/2} \left (a e^2+5 c d^2\right )}{3 e^7}-\frac{8 c^2 d (d+e x)^{7/2} \left (3 a e^2+5 c d^2\right )}{7 e^7}+\frac{6 c (d+e x)^{5/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{5 e^7}-\frac{4 c d (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}{e^7}+\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )^3}{e^7}+\frac{2 c^3 (d+e x)^{13/2}}{13 e^7}-\frac{12 c^3 d (d+e x)^{11/2}}{11 e^7} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)^3/Sqrt[d + e*x],x]
[Out]
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Rubi in Sympy [A] time = 43.1227, size = 196, normalized size = 0.98 \[ - \frac{12 c^{3} d \left (d + e x\right )^{\frac{11}{2}}}{11 e^{7}} + \frac{2 c^{3} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{7}} - \frac{8 c^{2} d \left (d + e x\right )^{\frac{7}{2}} \left (3 a e^{2} + 5 c d^{2}\right )}{7 e^{7}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{9}{2}} \left (a e^{2} + 5 c d^{2}\right )}{3 e^{7}} - \frac{4 c d \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )^{2}}{e^{7}} + \frac{6 c \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{5 e^{7}} + \frac{2 \sqrt{d + e x} \left (a e^{2} + c d^{2}\right )^{3}}{e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)**3/(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.133057, size = 171, normalized size = 0.86 \[ \frac{2 \sqrt{d+e x} \left (15015 a^3 e^6+3003 a^2 c e^4 \left (8 d^2-4 d e x+3 e^2 x^2\right )+143 a c^2 e^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+5 c^3 \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )\right )}{15015 e^7} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)^3/Sqrt[d + e*x],x]
[Out]
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Maple [A] time = 0.011, size = 205, normalized size = 1. \[{\frac{2310\,{c}^{3}{x}^{6}{e}^{6}-2520\,{c}^{3}d{x}^{5}{e}^{5}+10010\,a{c}^{2}{e}^{6}{x}^{4}+2800\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-11440\,a{c}^{2}d{e}^{5}{x}^{3}-3200\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+18018\,{a}^{2}c{e}^{6}{x}^{2}+13728\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}+3840\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-24024\,{a}^{2}cd{e}^{5}x-18304\,a{c}^{2}{d}^{3}{e}^{3}x-5120\,{c}^{3}{d}^{5}ex+30030\,{a}^{3}{e}^{6}+48048\,{a}^{2}c{d}^{2}{e}^{4}+36608\,a{c}^{2}{d}^{4}{e}^{2}+10240\,{c}^{3}{d}^{6}}{15015\,{e}^{7}}\sqrt{ex+d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)^3/(e*x+d)^(1/2),x)
[Out]
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Maxima [A] time = 0.707755, size = 286, normalized size = 1.43 \[ \frac{2 \,{\left (15015 \, \sqrt{e x + d} a^{3} + \frac{3003 \,{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} a^{2} c}{e^{2}} + \frac{143 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} - 180 \,{\left (e x + d\right )}^{\frac{7}{2}} d + 378 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{e x + d} d^{4}\right )} a c^{2}}{e^{4}} + \frac{5 \,{\left (231 \,{\left (e x + d\right )}^{\frac{13}{2}} - 1638 \,{\left (e x + d\right )}^{\frac{11}{2}} d + 5005 \,{\left (e x + d\right )}^{\frac{9}{2}} d^{2} - 8580 \,{\left (e x + d\right )}^{\frac{7}{2}} d^{3} + 9009 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{4} - 6006 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{5} + 3003 \, \sqrt{e x + d} d^{6}\right )} c^{3}}{e^{6}}\right )}}{15015 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3/sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.223526, size = 273, normalized size = 1.36 \[ \frac{2 \,{\left (1155 \, c^{3} e^{6} x^{6} - 1260 \, c^{3} d e^{5} x^{5} + 5120 \, c^{3} d^{6} + 18304 \, a c^{2} d^{4} e^{2} + 24024 \, a^{2} c d^{2} e^{4} + 15015 \, a^{3} e^{6} + 35 \,{\left (40 \, c^{3} d^{2} e^{4} + 143 \, a c^{2} e^{6}\right )} x^{4} - 40 \,{\left (40 \, c^{3} d^{3} e^{3} + 143 \, a c^{2} d e^{5}\right )} x^{3} + 3 \,{\left (640 \, c^{3} d^{4} e^{2} + 2288 \, a c^{2} d^{2} e^{4} + 3003 \, a^{2} c e^{6}\right )} x^{2} - 4 \,{\left (640 \, c^{3} d^{5} e + 2288 \, a c^{2} d^{3} e^{3} + 3003 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{15015 \, e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3/sqrt(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 46.3686, size = 563, normalized size = 2.82 \[ \begin{cases} - \frac{\frac{2 a^{3} d}{\sqrt{d + e x}} + 2 a^{3} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{6 a^{2} c d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{6 a^{2} c \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{6 a c^{2} d \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{4}} + \frac{6 a c^{2} \left (- \frac{d^{5}}{\sqrt{d + e x}} - 5 d^{4} \sqrt{d + e x} + \frac{10 d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac{5}{2}} + \frac{5 d \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{4}} + \frac{2 c^{3} d \left (\frac{d^{6}}{\sqrt{d + e x}} + 6 d^{5} \sqrt{d + e x} - 5 d^{4} \left (d + e x\right )^{\frac{3}{2}} + 4 d^{3} \left (d + e x\right )^{\frac{5}{2}} - \frac{15 d^{2} \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{2 d \left (d + e x\right )^{\frac{9}{2}}}{3} - \frac{\left (d + e x\right )^{\frac{11}{2}}}{11}\right )}{e^{6}} + \frac{2 c^{3} \left (- \frac{d^{7}}{\sqrt{d + e x}} - 7 d^{6} \sqrt{d + e x} + 7 d^{5} \left (d + e x\right )^{\frac{3}{2}} - 7 d^{4} \left (d + e x\right )^{\frac{5}{2}} + 5 d^{3} \left (d + e x\right )^{\frac{7}{2}} - \frac{7 d^{2} \left (d + e x\right )^{\frac{9}{2}}}{3} + \frac{7 d \left (d + e x\right )^{\frac{11}{2}}}{11} - \frac{\left (d + e x\right )^{\frac{13}{2}}}{13}\right )}{e^{6}}}{e} & \text{for}\: e \neq 0 \\\frac{a^{3} x + a^{2} c x^{3} + \frac{3 a c^{2} x^{5}}{5} + \frac{c^{3} x^{7}}{7}}{\sqrt{d}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)**3/(e*x+d)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.214356, size = 343, normalized size = 1.72 \[ \frac{2}{15015} \,{\left (3003 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} a^{2} c e^{\left (-10\right )} + 143 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{32} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{32} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{32} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{32} + 315 \, \sqrt{x e + d} d^{4} e^{32}\right )} a c^{2} e^{\left (-36\right )} + 5 \,{\left (231 \,{\left (x e + d\right )}^{\frac{13}{2}} e^{72} - 1638 \,{\left (x e + d\right )}^{\frac{11}{2}} d e^{72} + 5005 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} e^{72} - 8580 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} e^{72} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} e^{72} - 6006 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5} e^{72} + 3003 \, \sqrt{x e + d} d^{6} e^{72}\right )} c^{3} e^{\left (-78\right )} + 15015 \, \sqrt{x e + d} a^{3}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3/sqrt(e*x + d),x, algorithm="giac")
[Out]