Optimal. Leaf size=255 \[ \frac{32 b^3 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{15015 e (d+e x)^{5/2} (b d-a e)^5}+\frac{16 b^2 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{3003 e (d+e x)^{7/2} (b d-a e)^4}+\frac{4 b (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{429 e (d+e x)^{9/2} (b d-a e)^3}+\frac{2 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]
[Out]
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Rubi [A] time = 0.449307, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{32 b^3 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{15015 e (d+e x)^{5/2} (b d-a e)^5}+\frac{16 b^2 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{3003 e (d+e x)^{7/2} (b d-a e)^4}+\frac{4 b (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{429 e (d+e x)^{9/2} (b d-a e)^3}+\frac{2 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(15/2),x]
[Out]
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Rubi in Sympy [A] time = 50.9976, size = 246, normalized size = 0.96 \[ - \frac{32 b^{3} \left (a + b x\right )^{\frac{5}{2}} \left (8 A b e - 13 B a e + 5 B b d\right )}{15015 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{5}} + \frac{16 b^{2} \left (a + b x\right )^{\frac{5}{2}} \left (8 A b e - 13 B a e + 5 B b d\right )}{3003 e \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{4}} - \frac{4 b \left (a + b x\right )^{\frac{5}{2}} \left (8 A b e - 13 B a e + 5 B b d\right )}{429 e \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{3}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (8 A b e - 13 B a e + 5 B b d\right )}{143 e \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{2}} - \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (A e - B d\right )}{13 e \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(3/2)*(B*x+A)/(e*x+d)**(15/2),x)
[Out]
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Mathematica [A] time = 0.54001, size = 254, normalized size = 1. \[ \frac{2 \sqrt{a+b x} \left (\frac{16 b^5 (d+e x)^6 (-13 a B e+8 A b e+5 b B d)}{(b d-a e)^5}+\frac{8 b^4 (d+e x)^5 (-13 a B e+8 A b e+5 b B d)}{(b d-a e)^4}+\frac{6 b^3 (d+e x)^4 (-13 a B e+8 A b e+5 b B d)}{(b d-a e)^3}+\frac{5 b^2 (d+e x)^3 (-13 a B e+8 A b e+5 b B d)}{(b d-a e)^2}-\frac{35 b (d+e x)^2 (52 a B e+A b e-53 b B d)}{a e-b d}+105 (d+e x) (-13 a B e-14 A b e+27 b B d)-1155 (b d-a e) (B d-A e)\right )}{15015 e^3 (d+e x)^{13/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(15/2),x]
[Out]
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Maple [B] time = 0.017, size = 505, normalized size = 2. \[ -{\frac{256\,A{b}^{4}{e}^{4}{x}^{4}-416\,Ba{b}^{3}{e}^{4}{x}^{4}+160\,B{b}^{4}d{e}^{3}{x}^{4}-640\,Aa{b}^{3}{e}^{4}{x}^{3}+1664\,A{b}^{4}d{e}^{3}{x}^{3}+1040\,B{a}^{2}{b}^{2}{e}^{4}{x}^{3}-3104\,Ba{b}^{3}d{e}^{3}{x}^{3}+1040\,B{b}^{4}{d}^{2}{e}^{2}{x}^{3}+1120\,A{a}^{2}{b}^{2}{e}^{4}{x}^{2}-4160\,Aa{b}^{3}d{e}^{3}{x}^{2}+4576\,A{b}^{4}{d}^{2}{e}^{2}{x}^{2}-1820\,B{a}^{3}b{e}^{4}{x}^{2}+7460\,B{a}^{2}{b}^{2}d{e}^{3}{x}^{2}-10036\,Ba{b}^{3}{d}^{2}{e}^{2}{x}^{2}+2860\,B{b}^{4}{d}^{3}e{x}^{2}-1680\,A{a}^{3}b{e}^{4}x+7280\,A{a}^{2}{b}^{2}d{e}^{3}x-11440\,Aa{b}^{3}{d}^{2}{e}^{2}x+6864\,A{b}^{4}{d}^{3}ex+2730\,B{a}^{4}{e}^{4}x-12880\,B{a}^{3}bd{e}^{3}x+23140\,B{a}^{2}{b}^{2}{d}^{2}{e}^{2}x-18304\,Ba{b}^{3}{d}^{3}ex+4290\,B{b}^{4}{d}^{4}x+2310\,A{a}^{4}{e}^{4}-10920\,A{a}^{3}bd{e}^{3}+20020\,A{a}^{2}{b}^{2}{d}^{2}{e}^{2}-17160\,Aa{b}^{3}{d}^{3}e+6006\,A{b}^{4}{d}^{4}+420\,B{a}^{4}d{e}^{3}-1820\,B{a}^{3}b{d}^{2}{e}^{2}+2860\,B{a}^{2}{b}^{2}{d}^{3}e-1716\,Ba{b}^{3}{d}^{4}}{15015\,{a}^{5}{e}^{5}-75075\,{a}^{4}bd{e}^{4}+150150\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-150150\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+75075\,a{b}^{4}{d}^{4}e-15015\,{b}^{5}{d}^{5}} \left ( bx+a \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{13}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(15/2),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)^(15/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 14.3208, size = 1690, normalized size = 6.63 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(3/2)/(e*x + d)^(15/2),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)**(15/2),x)
[Out]
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GIAC/XCAS [A] time = 0.535647, 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)^(15/2),x, algorithm="giac")
[Out]