Optimal. Leaf size=201 \[ \frac{16 b^2 (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{9009 e (d+e x)^{7/2} (b d-a e)^4}+\frac{8 b (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^3}+\frac{2 (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.364228, 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 \[ \frac{16 b^2 (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{9009 e (d+e x)^{7/2} (b d-a e)^4}+\frac{8 b (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^3}+\frac{2 (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(15/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 36.1463, size = 192, normalized size = 0.96 \[ \frac{16 b^{2} \left (a + b x\right )^{\frac{7}{2}} \left (6 A b e - 13 B a e + 7 B b d\right )}{9009 e \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{4}} - \frac{8 b \left (a + b x\right )^{\frac{7}{2}} \left (6 A b e - 13 B a e + 7 B b d\right )}{1287 e \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{3}} + \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (6 A b e - 13 B a e + 7 B b d\right )}{143 e \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{2}} - \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (A e - B d\right )}{13 e \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(15/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.62586, size = 255, normalized size = 1.27 \[ \frac{2 \sqrt{a+b x} \left (\frac{8 b^5 (d+e x)^6 (-13 a B e+6 A b e+7 b B d)}{(b d-a e)^4}+\frac{4 b^4 (d+e x)^5 (-13 a B e+6 A b e+7 b B d)}{(b d-a e)^3}+\frac{3 b^3 (d+e x)^4 (-13 a B e+6 A b e+7 b B d)}{(b d-a e)^2}-\frac{b^2 (d+e x)^3 (1469 a B e+15 A b e-1484 b B d)}{a e-b d}+7 b (d+e x)^2 (-299 a B e-159 A b e+458 b B d)-63 (d+e x) (a e-b d) (13 a B e+27 A b e-40 b B d)+693 (b d-a e)^2 (B d-A e)\right )}{9009 e^4 (d+e x)^{13/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(15/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.013, size = 322, normalized size = 1.6 \[ -{\frac{-96\,A{b}^{3}{e}^{3}{x}^{3}+208\,Ba{b}^{2}{e}^{3}{x}^{3}-112\,B{b}^{3}d{e}^{2}{x}^{3}+336\,Aa{b}^{2}{e}^{3}{x}^{2}-624\,A{b}^{3}d{e}^{2}{x}^{2}-728\,B{a}^{2}b{e}^{3}{x}^{2}+1744\,Ba{b}^{2}d{e}^{2}{x}^{2}-728\,B{b}^{3}{d}^{2}e{x}^{2}-756\,A{a}^{2}b{e}^{3}x+2184\,Aa{b}^{2}d{e}^{2}x-1716\,A{b}^{3}{d}^{2}ex+1638\,B{a}^{3}{e}^{3}x-5614\,B{a}^{2}bd{e}^{2}x+6266\,Ba{b}^{2}{d}^{2}ex-2002\,B{b}^{3}{d}^{3}x+1386\,A{a}^{3}{e}^{3}-4914\,A{a}^{2}bd{e}^{2}+6006\,Aa{b}^{2}{d}^{2}e-2574\,A{b}^{3}{d}^{3}+252\,B{a}^{3}d{e}^{2}-728\,B{a}^{2}b{d}^{2}e+572\,Ba{b}^{2}{d}^{3}}{9009\,{e}^{4}{a}^{4}-36036\,b{e}^{3}d{a}^{3}+54054\,{b}^{2}{e}^{2}{d}^{2}{a}^{2}-36036\,a{b}^{3}{d}^{3}e+9009\,{b}^{4}{d}^{4}} \left ( bx+a \right ) ^{{\frac{7}{2}}} \left ( ex+d \right ) ^{-{\frac{13}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(15/2),x)
[Out]
_______________________________________________________________________________________
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)^(5/2)/(e*x + d)^(15/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 14.6447, size = 1413, normalized size = 7.03 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(5/2)/(e*x + d)^(15/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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)**(5/2)*(B*x+A)/(e*x+d)**(15/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.572231, 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)^(5/2)/(e*x + d)^(15/2),x, algorithm="giac")
[Out]