Optimal. Leaf size=128 \[ -\frac{2 b (d+e x)^{9/2} (-2 a B e-A b e+3 b B d)}{9 e^4}+\frac{2 (d+e x)^{7/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{7 e^4}-\frac{2 (d+e x)^{5/2} (b d-a e)^2 (B d-A e)}{5 e^4}+\frac{2 b^2 B (d+e x)^{11/2}}{11 e^4} \]
[Out]
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Rubi [A] time = 0.158135, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{2 b (d+e x)^{9/2} (-2 a B e-A b e+3 b B d)}{9 e^4}+\frac{2 (d+e x)^{7/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{7 e^4}-\frac{2 (d+e x)^{5/2} (b d-a e)^2 (B d-A e)}{5 e^4}+\frac{2 b^2 B (d+e x)^{11/2}}{11 e^4} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^2*(A + B*x)*(d + e*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 30.7594, size = 126, normalized size = 0.98 \[ \frac{2 B b^{2} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{4}} + \frac{2 b \left (d + e x\right )^{\frac{9}{2}} \left (A b e + 2 B a e - 3 B b d\right )}{9 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right ) \left (2 A b e + B a e - 3 B b d\right )}{7 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (A e - B d\right ) \left (a e - b d\right )^{2}}{5 e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**2*(B*x+A)*(e*x+d)**(3/2),x)
[Out]
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Mathematica [A] time = 0.230435, size = 139, normalized size = 1.09 \[ \frac{2 (d+e x)^{5/2} \left (99 a^2 e^2 (7 A e-2 B d+5 B e x)+22 a b e \left (9 A e (5 e x-2 d)+B \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )+b^2 \left (11 A e \left (8 d^2-20 d e x+35 e^2 x^2\right )-3 B \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )\right )\right )}{3465 e^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^2*(A + B*x)*(d + e*x)^(3/2),x]
[Out]
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Maple [A] time = 0.01, size = 169, normalized size = 1.3 \[{\frac{630\,B{b}^{2}{x}^{3}{e}^{3}+770\,A{b}^{2}{e}^{3}{x}^{2}+1540\,Bab{e}^{3}{x}^{2}-420\,B{b}^{2}d{e}^{2}{x}^{2}+1980\,Aab{e}^{3}x-440\,A{b}^{2}d{e}^{2}x+990\,B{a}^{2}{e}^{3}x-880\,Babd{e}^{2}x+240\,B{b}^{2}{d}^{2}ex+1386\,{a}^{2}A{e}^{3}-792\,Aabd{e}^{2}+176\,A{b}^{2}{d}^{2}e-396\,B{a}^{2}d{e}^{2}+352\,Bab{d}^{2}e-96\,B{b}^{2}{d}^{3}}{3465\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^2*(B*x+A)*(e*x+d)^(3/2),x)
[Out]
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Maxima [A] time = 1.35708, size = 215, normalized size = 1.68 \[ \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} B b^{2} - 385 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 495 \,{\left (3 \, B b^{2} d^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 693 \,{\left (B b^{2} d^{3} - A a^{2} e^{3} -{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{3465 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^2*(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224787, size = 390, normalized size = 3.05 \[ \frac{2 \,{\left (315 \, B b^{2} e^{5} x^{5} - 48 \, B b^{2} d^{5} + 693 \, A a^{2} d^{2} e^{3} + 88 \,{\left (2 \, B a b + A b^{2}\right )} d^{4} e - 198 \,{\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{2} + 35 \,{\left (12 \, B b^{2} d e^{4} + 11 \,{\left (2 \, B a b + A b^{2}\right )} e^{5}\right )} x^{4} + 5 \,{\left (3 \, B b^{2} d^{2} e^{3} + 110 \,{\left (2 \, B a b + A b^{2}\right )} d e^{4} + 99 \,{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x^{3} - 3 \,{\left (6 \, B b^{2} d^{3} e^{2} - 231 \, A a^{2} e^{5} - 11 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e^{3} - 264 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{4}\right )} x^{2} +{\left (24 \, B b^{2} d^{4} e + 1386 \, A a^{2} d e^{4} - 44 \,{\left (2 \, B a b + A b^{2}\right )} d^{3} e^{2} + 99 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^2*(e*x + d)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.6279, size = 586, normalized size = 4.58 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**2*(B*x+A)*(e*x+d)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.221369, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^2*(e*x + d)^(3/2),x, algorithm="giac")
[Out]