Optimal. Leaf size=214 \[ \frac{4 c (d+e x)^{5/2} \left (a B e^2-2 A c d e+5 B c d^2\right )}{5 e^6}+\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^6}+\frac{2 \left (a e^2+c d^2\right )^2 (B d-A e)}{e^6 \sqrt{d+e x}}-\frac{4 c (d+e x)^{3/2} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6}-\frac{2 c^2 (d+e x)^{7/2} (5 B d-A e)}{7 e^6}+\frac{2 B c^2 (d+e x)^{9/2}}{9 e^6} \]
[Out]
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Rubi [A] time = 0.252145, antiderivative size = 214, 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 \[ \frac{4 c (d+e x)^{5/2} \left (a B e^2-2 A c d e+5 B c d^2\right )}{5 e^6}+\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^6}+\frac{2 \left (a e^2+c d^2\right )^2 (B d-A e)}{e^6 \sqrt{d+e x}}-\frac{4 c (d+e x)^{3/2} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6}-\frac{2 c^2 (d+e x)^{7/2} (5 B d-A e)}{7 e^6}+\frac{2 B c^2 (d+e x)^{9/2}}{9 e^6} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 54.6584, size = 219, normalized size = 1.02 \[ \frac{2 B c^{2} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{6}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{7}{2}} \left (A e - 5 B d\right )}{7 e^{6}} + \frac{4 c \left (d + e x\right )^{\frac{5}{2}} \left (- 2 A c d e + B a e^{2} + 5 B c d^{2}\right )}{5 e^{6}} + \frac{4 c \left (d + e x\right )^{\frac{3}{2}} \left (A a e^{3} + 3 A c d^{2} e - 3 B a d e^{2} - 5 B c d^{3}\right )}{3 e^{6}} + \frac{2 \sqrt{d + e x} \left (a e^{2} + c d^{2}\right ) \left (- 4 A c d e + B a e^{2} + 5 B c d^{2}\right )}{e^{6}} - \frac{2 \left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )^{2}}{e^{6} \sqrt{d + e x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**(3/2),x)
[Out]
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Mathematica [A] time = 0.277771, size = 214, normalized size = 1. \[ \frac{2 B \left (315 a^2 e^4 (2 d+e x)+126 a c e^2 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+5 c^2 \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )\right )-6 A e \left (105 a^2 e^4+70 a c e^2 \left (8 d^2+4 d e x-e^2 x^2\right )+3 c^2 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )\right )}{315 e^6 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^(3/2),x]
[Out]
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Maple [A] time = 0.012, size = 259, normalized size = 1.2 \[ -{\frac{-70\,B{c}^{2}{x}^{5}{e}^{5}-90\,A{c}^{2}{e}^{5}{x}^{4}+100\,B{c}^{2}d{e}^{4}{x}^{4}+144\,A{c}^{2}d{e}^{4}{x}^{3}-252\,Bac{e}^{5}{x}^{3}-160\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}-420\,Aac{e}^{5}{x}^{2}-288\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}+504\,Bacd{e}^{4}{x}^{2}+320\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}+1680\,Aacd{e}^{4}x+1152\,A{c}^{2}{d}^{3}{e}^{2}x-630\,B{a}^{2}{e}^{5}x-2016\,Bac{d}^{2}{e}^{3}x-1280\,B{c}^{2}{d}^{4}ex+630\,A{a}^{2}{e}^{5}+3360\,A{d}^{2}ac{e}^{3}+2304\,A{d}^{4}{c}^{2}e-1260\,Bd{a}^{2}{e}^{4}-4032\,aBc{d}^{3}{e}^{2}-2560\,B{c}^{2}{d}^{5}}{315\,{e}^{6}}{\frac{1}{\sqrt{ex+d}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)^2/(e*x+d)^(3/2),x)
[Out]
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Maxima [A] time = 0.685165, size = 346, normalized size = 1.62 \[ \frac{2 \,{\left (\frac{35 \,{\left (e x + d\right )}^{\frac{9}{2}} B c^{2} - 45 \,{\left (5 \, B c^{2} d - A c^{2} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 126 \,{\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 210 \,{\left (5 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, B a c d e^{2} - A a c e^{3}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 315 \,{\left (5 \, B c^{2} d^{4} - 4 \, A c^{2} d^{3} e + 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )} \sqrt{e x + d}}{e^{5}} + \frac{315 \,{\left (B c^{2} d^{5} - A c^{2} d^{4} e + 2 \, B a c d^{3} e^{2} - 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} - A a^{2} e^{5}\right )}}{\sqrt{e x + d} e^{5}}\right )}}{315 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)/(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.265154, size = 333, normalized size = 1.56 \[ \frac{2 \,{\left (35 \, B c^{2} e^{5} x^{5} + 1280 \, B c^{2} d^{5} - 1152 \, A c^{2} d^{4} e + 2016 \, B a c d^{3} e^{2} - 1680 \, A a c d^{2} e^{3} + 630 \, B a^{2} d e^{4} - 315 \, A a^{2} e^{5} - 5 \,{\left (10 \, B c^{2} d e^{4} - 9 \, A c^{2} e^{5}\right )} x^{4} + 2 \,{\left (40 \, B c^{2} d^{2} e^{3} - 36 \, A c^{2} d e^{4} + 63 \, B a c e^{5}\right )} x^{3} - 2 \,{\left (80 \, B c^{2} d^{3} e^{2} - 72 \, A c^{2} d^{2} e^{3} + 126 \, B a c d e^{4} - 105 \, A a c e^{5}\right )} x^{2} +{\left (640 \, B c^{2} d^{4} e - 576 \, A c^{2} d^{3} e^{2} + 1008 \, B a c d^{2} e^{3} - 840 \, A a c d e^{4} + 315 \, B a^{2} e^{5}\right )} x\right )}}{315 \, \sqrt{e x + d} e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(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. \[ \int \frac{\left (A + B x\right ) \left (a + c x^{2}\right )^{2}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.298125, size = 447, normalized size = 2.09 \[ \frac{2}{315} \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} B c^{2} e^{48} - 225 \,{\left (x e + d\right )}^{\frac{7}{2}} B c^{2} d e^{48} + 630 \,{\left (x e + d\right )}^{\frac{5}{2}} B c^{2} d^{2} e^{48} - 1050 \,{\left (x e + d\right )}^{\frac{3}{2}} B c^{2} d^{3} e^{48} + 1575 \, \sqrt{x e + d} B c^{2} d^{4} e^{48} + 45 \,{\left (x e + d\right )}^{\frac{7}{2}} A c^{2} e^{49} - 252 \,{\left (x e + d\right )}^{\frac{5}{2}} A c^{2} d e^{49} + 630 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{2} d^{2} e^{49} - 1260 \, \sqrt{x e + d} A c^{2} d^{3} e^{49} + 126 \,{\left (x e + d\right )}^{\frac{5}{2}} B a c e^{50} - 630 \,{\left (x e + d\right )}^{\frac{3}{2}} B a c d e^{50} + 1890 \, \sqrt{x e + d} B a c d^{2} e^{50} + 210 \,{\left (x e + d\right )}^{\frac{3}{2}} A a c e^{51} - 1260 \, \sqrt{x e + d} A a c d e^{51} + 315 \, \sqrt{x e + d} B a^{2} e^{52}\right )} e^{\left (-54\right )} + \frac{2 \,{\left (B c^{2} d^{5} - A c^{2} d^{4} e + 2 \, B a c d^{3} e^{2} - 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} - A a^{2} e^{5}\right )} e^{\left (-6\right )}}{\sqrt{x e + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)/(e*x + d)^(3/2),x, algorithm="giac")
[Out]