Optimal. Leaf size=201 \[ \frac{16 b^2 (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{3465 e (d+e x)^{5/2} (b d-a e)^4}+\frac{8 b (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{693 e (d+e x)^{7/2} (b d-a e)^3}+\frac{2 (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \]
[Out]
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Rubi [A] time = 0.366159, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{16 b^2 (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{3465 e (d+e x)^{5/2} (b d-a e)^4}+\frac{8 b (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{693 e (d+e x)^{7/2} (b d-a e)^3}+\frac{2 (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(13/2),x]
[Out]
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Rubi in Sympy [A] time = 37.3252, size = 192, normalized size = 0.96 \[ \frac{16 b^{2} \left (a + b x\right )^{\frac{5}{2}} \left (6 A b e - 11 B a e + 5 B b d\right )}{3465 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{4}} - \frac{8 b \left (a + b x\right )^{\frac{5}{2}} \left (6 A b e - 11 B a e + 5 B b d\right )}{693 e \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{3}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (6 A b e - 11 B a e + 5 B b d\right )}{99 e \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{2}} - \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (A e - B d\right )}{11 e \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(3/2)*(B*x+A)/(e*x+d)**(13/2),x)
[Out]
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Mathematica [A] time = 0.431784, size = 217, normalized size = 1.08 \[ \frac{2 \sqrt{a+b x} \left (\frac{8 b^4 (d+e x)^5 (-11 a B e+6 A b e+5 b B d)}{(b d-a e)^4}+\frac{4 b^3 (d+e x)^4 (-11 a B e+6 A b e+5 b B d)}{(b d-a e)^3}+\frac{3 b^2 (d+e x)^3 (-11 a B e+6 A b e+5 b B d)}{(b d-a e)^2}-\frac{5 b (d+e x)^2 (110 a B e+3 A b e-113 b B d)}{a e-b d}+35 (d+e x) (-11 a B e-12 A b e+23 b B d)-315 (b d-a e) (B d-A e)\right )}{3465 e^3 (d+e x)^{11/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(13/2),x]
[Out]
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Maple [A] time = 0.014, size = 322, normalized size = 1.6 \[ -{\frac{-96\,A{b}^{3}{e}^{3}{x}^{3}+176\,Ba{b}^{2}{e}^{3}{x}^{3}-80\,B{b}^{3}d{e}^{2}{x}^{3}+240\,Aa{b}^{2}{e}^{3}{x}^{2}-528\,A{b}^{3}d{e}^{2}{x}^{2}-440\,B{a}^{2}b{e}^{3}{x}^{2}+1168\,Ba{b}^{2}d{e}^{2}{x}^{2}-440\,B{b}^{3}{d}^{2}e{x}^{2}-420\,A{a}^{2}b{e}^{3}x+1320\,Aa{b}^{2}d{e}^{2}x-1188\,A{b}^{3}{d}^{2}ex+770\,B{a}^{3}{e}^{3}x-2770\,B{a}^{2}bd{e}^{2}x+3278\,Ba{b}^{2}{d}^{2}ex-990\,B{b}^{3}{d}^{3}x+630\,A{a}^{3}{e}^{3}-2310\,A{a}^{2}bd{e}^{2}+2970\,Aa{b}^{2}{d}^{2}e-1386\,A{b}^{3}{d}^{3}+140\,B{a}^{3}d{e}^{2}-440\,B{a}^{2}b{d}^{2}e+396\,Ba{b}^{2}{d}^{3}}{3465\,{e}^{4}{a}^{4}-13860\,b{e}^{3}d{a}^{3}+20790\,{b}^{2}{e}^{2}{d}^{2}{a}^{2}-13860\,a{b}^{3}{d}^{3}e+3465\,{b}^{4}{d}^{4}} \left ( bx+a \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{11}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(13/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(3/2)/(e*x + d)^(13/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 6.09255, size = 1188, normalized size = 5.91 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(3/2)/(e*x + d)^(13/2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(3/2)*(B*x+A)/(e*x+d)**(13/2),x)
[Out]
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GIAC/XCAS [A] time = 0.410804, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(3/2)/(e*x + d)^(13/2),x, algorithm="giac")
[Out]