Optimal. Leaf size=83 \[ -\frac{2 (d+e x)^{9/2} (-a B e-A b e+2 b B d)}{9 e^3}+\frac{2 (d+e x)^{7/2} (b d-a e) (B d-A e)}{7 e^3}+\frac{2 b B (d+e x)^{11/2}}{11 e^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.102462, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{2 (d+e x)^{9/2} (-a B e-A b e+2 b B d)}{9 e^3}+\frac{2 (d+e x)^{7/2} (b d-a e) (B d-A e)}{7 e^3}+\frac{2 b B (d+e x)^{11/2}}{11 e^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)*(A + B*x)*(d + e*x)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 17.2187, size = 78, normalized size = 0.94 \[ \frac{2 B b \left (d + e x\right )^{\frac{11}{2}}}{11 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (A b e + B a e - 2 B b d\right )}{9 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (A e - B d\right ) \left (a e - b d\right )}{7 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(B*x+A)*(e*x+d)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.123406, size = 70, normalized size = 0.84 \[ \frac{2 (d+e x)^{7/2} \left (11 a e (9 A e-2 B d+7 B e x)+11 A b e (7 e x-2 d)+b B \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)*(A + B*x)*(d + e*x)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.006, size = 73, normalized size = 0.9 \[{\frac{126\,bB{x}^{2}{e}^{2}+154\,Ab{e}^{2}x+154\,Ba{e}^{2}x-56\,Bbdex+198\,aA{e}^{2}-44\,Abde-44\,Bade+16\,bB{d}^{2}}{693\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(B*x+A)*(e*x+d)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.34416, size = 101, normalized size = 1.22 \[ \frac{2 \,{\left (63 \,{\left (e x + d\right )}^{\frac{11}{2}} B b - 77 \,{\left (2 \, B b d -{\left (B a + A b\right )} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 99 \,{\left (B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{693 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)*(e*x + d)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.222415, size = 255, normalized size = 3.07 \[ \frac{2 \,{\left (63 \, B b e^{5} x^{5} + 8 \, B b d^{5} + 99 \, A a d^{3} e^{2} - 22 \,{\left (B a + A b\right )} d^{4} e + 7 \,{\left (23 \, B b d e^{4} + 11 \,{\left (B a + A b\right )} e^{5}\right )} x^{4} +{\left (113 \, B b d^{2} e^{3} + 99 \, A a e^{5} + 209 \,{\left (B a + A b\right )} d e^{4}\right )} x^{3} + 3 \,{\left (B b d^{3} e^{2} + 99 \, A a d e^{4} + 55 \,{\left (B a + A b\right )} d^{2} e^{3}\right )} x^{2} -{\left (4 \, B b d^{4} e - 297 \, A a d^{2} e^{3} - 11 \,{\left (B a + A b\right )} d^{3} e^{2}\right )} x\right )} \sqrt{e x + d}}{693 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)*(e*x + d)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 10.362, size = 476, normalized size = 5.73 \[ \begin{cases} \frac{2 A a d^{3} \sqrt{d + e x}}{7 e} + \frac{6 A a d^{2} x \sqrt{d + e x}}{7} + \frac{6 A a d e x^{2} \sqrt{d + e x}}{7} + \frac{2 A a e^{2} x^{3} \sqrt{d + e x}}{7} - \frac{4 A b d^{4} \sqrt{d + e x}}{63 e^{2}} + \frac{2 A b d^{3} x \sqrt{d + e x}}{63 e} + \frac{10 A b d^{2} x^{2} \sqrt{d + e x}}{21} + \frac{38 A b d e x^{3} \sqrt{d + e x}}{63} + \frac{2 A b e^{2} x^{4} \sqrt{d + e x}}{9} - \frac{4 B a d^{4} \sqrt{d + e x}}{63 e^{2}} + \frac{2 B a d^{3} x \sqrt{d + e x}}{63 e} + \frac{10 B a d^{2} x^{2} \sqrt{d + e x}}{21} + \frac{38 B a d e x^{3} \sqrt{d + e x}}{63} + \frac{2 B a e^{2} x^{4} \sqrt{d + e x}}{9} + \frac{16 B b d^{5} \sqrt{d + e x}}{693 e^{3}} - \frac{8 B b d^{4} x \sqrt{d + e x}}{693 e^{2}} + \frac{2 B b d^{3} x^{2} \sqrt{d + e x}}{231 e} + \frac{226 B b d^{2} x^{3} \sqrt{d + e x}}{693} + \frac{46 B b d e x^{4} \sqrt{d + e x}}{99} + \frac{2 B b e^{2} x^{5} \sqrt{d + e x}}{11} & \text{for}\: e \neq 0 \\d^{\frac{5}{2}} \left (A a x + \frac{A b x^{2}}{2} + \frac{B a x^{2}}{2} + \frac{B b x^{3}}{3}\right ) & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(B*x+A)*(e*x+d)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.218383, 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)*(e*x + d)^(5/2),x, algorithm="giac")
[Out]