Optimal. Leaf size=143 \[ \frac{2 (d+e x)^{3/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 e^5}-\frac{2 d^2 (c d-b e)^2}{e^5 \sqrt{d+e x}}-\frac{4 c (d+e x)^{5/2} (2 c d-b e)}{5 e^5}-\frac{4 d \sqrt{d+e x} (c d-b e) (2 c d-b e)}{e^5}+\frac{2 c^2 (d+e x)^{7/2}}{7 e^5} \]
[Out]
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Rubi [A] time = 0.198207, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{2 (d+e x)^{3/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 e^5}-\frac{2 d^2 (c d-b e)^2}{e^5 \sqrt{d+e x}}-\frac{4 c (d+e x)^{5/2} (2 c d-b e)}{5 e^5}-\frac{4 d \sqrt{d+e x} (c d-b e) (2 c d-b e)}{e^5}+\frac{2 c^2 (d+e x)^{7/2}}{7 e^5} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)^2/(d + e*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 32.7221, size = 138, normalized size = 0.97 \[ \frac{2 c^{2} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{5}} + \frac{4 c \left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right )}{5 e^{5}} - \frac{2 d^{2} \left (b e - c d\right )^{2}}{e^{5} \sqrt{d + e x}} - \frac{4 d \sqrt{d + e x} \left (b e - 2 c d\right ) \left (b e - c d\right )}{e^{5}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{3 e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**2/(e*x+d)**(3/2),x)
[Out]
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Mathematica [A] time = 0.118567, size = 123, normalized size = 0.86 \[ \frac{70 b^2 e^2 \left (-8 d^2-4 d e x+e^2 x^2\right )+84 b c e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )-6 c^2 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )}{105 e^5 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)^2/(d + e*x)^(3/2),x]
[Out]
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Maple [A] time = 0.01, size = 141, normalized size = 1. \[ -{\frac{-30\,{c}^{2}{x}^{4}{e}^{4}-84\,bc{e}^{4}{x}^{3}+48\,{c}^{2}d{e}^{3}{x}^{3}-70\,{b}^{2}{e}^{4}{x}^{2}+168\,bcd{e}^{3}{x}^{2}-96\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}+280\,{b}^{2}d{e}^{3}x-672\,bc{d}^{2}{e}^{2}x+384\,{c}^{2}{d}^{3}ex+560\,{b}^{2}{d}^{2}{e}^{2}-1344\,bc{d}^{3}e+768\,{c}^{2}{d}^{4}}{105\,{e}^{5}}{\frac{1}{\sqrt{ex+d}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^2/(e*x+d)^(3/2),x)
[Out]
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Maxima [A] time = 0.696537, size = 198, normalized size = 1.38 \[ \frac{2 \,{\left (\frac{15 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{2} - 42 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 210 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} \sqrt{e x + d}}{e^{4}} - \frac{105 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )}}{\sqrt{e x + d} e^{4}}\right )}}{105 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2/(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.211408, size = 185, normalized size = 1.29 \[ \frac{2 \,{\left (15 \, c^{2} e^{4} x^{4} - 384 \, c^{2} d^{4} + 672 \, b c d^{3} e - 280 \, b^{2} d^{2} e^{2} - 6 \,{\left (4 \, c^{2} d e^{3} - 7 \, b c e^{4}\right )} x^{3} +{\left (48 \, c^{2} d^{2} e^{2} - 84 \, b c d e^{3} + 35 \, b^{2} e^{4}\right )} x^{2} - 4 \,{\left (48 \, c^{2} d^{3} e - 84 \, b c d^{2} e^{2} + 35 \, b^{2} d e^{3}\right )} x\right )}}{105 \, \sqrt{e x + d} e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2/(e*x + d)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \left (b + c x\right )^{2}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**2/(e*x+d)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.210687, size = 254, normalized size = 1.78 \[ \frac{2}{105} \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{2} e^{30} - 84 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{2} d e^{30} + 210 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{2} d^{2} e^{30} - 420 \, \sqrt{x e + d} c^{2} d^{3} e^{30} + 42 \,{\left (x e + d\right )}^{\frac{5}{2}} b c e^{31} - 210 \,{\left (x e + d\right )}^{\frac{3}{2}} b c d e^{31} + 630 \, \sqrt{x e + d} b c d^{2} e^{31} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} e^{32} - 210 \, \sqrt{x e + d} b^{2} d e^{32}\right )} e^{\left (-35\right )} - \frac{2 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )}}{\sqrt{x e + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2/(e*x + d)^(3/2),x, algorithm="giac")
[Out]