Optimal. Leaf size=128 \[ -\frac{2 b (d+e x)^{13/2} (-2 a B e-A b e+3 b B d)}{13 e^4}+\frac{2 (d+e x)^{11/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{11 e^4}-\frac{2 (d+e x)^{9/2} (b d-a e)^2 (B d-A e)}{9 e^4}+\frac{2 b^2 B (d+e x)^{15/2}}{15 e^4} \]
[Out]
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Rubi [A] time = 0.196146, 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 \[ -\frac{2 b (d+e x)^{13/2} (-2 a B e-A b e+3 b B d)}{13 e^4}+\frac{2 (d+e x)^{11/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{11 e^4}-\frac{2 (d+e x)^{9/2} (b d-a e)^2 (B d-A e)}{9 e^4}+\frac{2 b^2 B (d+e x)^{15/2}}{15 e^4} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^2*(A + B*x)*(d + e*x)^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 32.4907, size = 126, normalized size = 0.98 \[ \frac{2 B b^{2} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{4}} + \frac{2 b \left (d + e x\right )^{\frac{13}{2}} \left (A b e + 2 B a e - 3 B b d\right )}{13 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right ) \left (2 A b e + B a e - 3 B b d\right )}{11 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (A e - B d\right ) \left (a e - b d\right )^{2}}{9 e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**2*(B*x+A)*(e*x+d)**(7/2),x)
[Out]
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Mathematica [A] time = 0.29747, size = 138, normalized size = 1.08 \[ \frac{2 (d+e x)^{9/2} \left (65 a^2 e^2 (11 A e-2 B d+9 B e x)+10 a b e \left (13 A e (9 e x-2 d)+B \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )+b^2 \left (5 A e \left (8 d^2-36 d e x+99 e^2 x^2\right )+B \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )\right )\right )}{6435 e^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^2*(A + B*x)*(d + e*x)^(7/2),x]
[Out]
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Maple [A] time = 0.01, size = 169, normalized size = 1.3 \[{\frac{858\,B{b}^{2}{x}^{3}{e}^{3}+990\,A{b}^{2}{e}^{3}{x}^{2}+1980\,Bab{e}^{3}{x}^{2}-396\,B{b}^{2}d{e}^{2}{x}^{2}+2340\,Aab{e}^{3}x-360\,A{b}^{2}d{e}^{2}x+1170\,B{a}^{2}{e}^{3}x-720\,Babd{e}^{2}x+144\,B{b}^{2}{d}^{2}ex+1430\,{a}^{2}A{e}^{3}-520\,Aabd{e}^{2}+80\,A{b}^{2}{d}^{2}e-260\,B{a}^{2}d{e}^{2}+160\,Bab{d}^{2}e-32\,B{b}^{2}{d}^{3}}{6435\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^2*(B*x+A)*(e*x+d)^(7/2),x)
[Out]
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Maxima [A] time = 1.33885, size = 215, normalized size = 1.68 \[ \frac{2 \,{\left (429 \,{\left (e x + d\right )}^{\frac{15}{2}} B b^{2} - 495 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 585 \,{\left (3 \, B b^{2} d^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 715 \,{\left (B b^{2} d^{3} - A a^{2} e^{3} -{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{6435 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^2*(e*x + d)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224407, size = 572, normalized size = 4.47 \[ \frac{2 \,{\left (429 \, B b^{2} e^{7} x^{7} - 16 \, B b^{2} d^{7} + 715 \, A a^{2} d^{4} e^{3} + 40 \,{\left (2 \, B a b + A b^{2}\right )} d^{6} e - 130 \,{\left (B a^{2} + 2 \, A a b\right )} d^{5} e^{2} + 33 \,{\left (46 \, B b^{2} d e^{6} + 15 \,{\left (2 \, B a b + A b^{2}\right )} e^{7}\right )} x^{6} + 9 \,{\left (206 \, B b^{2} d^{2} e^{5} + 200 \,{\left (2 \, B a b + A b^{2}\right )} d e^{6} + 65 \,{\left (B a^{2} + 2 \, A a b\right )} e^{7}\right )} x^{5} + 5 \,{\left (160 \, B b^{2} d^{3} e^{4} + 143 \, A a^{2} e^{7} + 458 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e^{5} + 442 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{6}\right )} x^{4} + 5 \,{\left (B b^{2} d^{4} e^{3} + 572 \, A a^{2} d e^{6} + 212 \,{\left (2 \, B a b + A b^{2}\right )} d^{3} e^{4} + 598 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{5}\right )} x^{3} - 3 \,{\left (2 \, B b^{2} d^{5} e^{2} - 1430 \, A a^{2} d^{2} e^{5} - 5 \,{\left (2 \, B a b + A b^{2}\right )} d^{4} e^{3} - 520 \,{\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{4}\right )} x^{2} +{\left (8 \, B b^{2} d^{6} e + 2860 \, A a^{2} d^{3} e^{4} - 20 \,{\left (2 \, B a b + A b^{2}\right )} d^{5} e^{2} + 65 \,{\left (B a^{2} + 2 \, A a b\right )} d^{4} e^{3}\right )} x\right )} \sqrt{e x + d}}{6435 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^2*(e*x + d)^(7/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 32.2462, size = 1020, normalized size = 7.97 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**2*(B*x+A)*(e*x+d)**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.239422, 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)^(7/2),x, algorithm="giac")
[Out]