3.2242 \(\int \frac{A+B x}{(a+b x)^{5/2} (d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=186 \[ -\frac{2 (B d-A e)}{3 e (a+b x)^{3/2} (d+e x)^{3/2} (b d-a e)}-\frac{16 e \sqrt{a+b x} (a B e-2 A b e+b B d)}{3 \sqrt{d+e x} (b d-a e)^4}-\frac{8 (a B e-2 A b e+b B d)}{3 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^3}+\frac{2 (a B e-2 A b e+b B d)}{3 e (a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^2} \]

[Out]

(-2*(B*d - A*e))/(3*e*(b*d - a*e)*(a + b*x)^(3/2)*(d + e*x)^(3/2)) + (2*(b*B*d -
2*A*b*e + a*B*e))/(3*e*(b*d - a*e)^2*(a + b*x)^(3/2)*Sqrt[d + e*x]) - (8*(b*B*d
- 2*A*b*e + a*B*e))/(3*(b*d - a*e)^3*Sqrt[a + b*x]*Sqrt[d + e*x]) - (16*e*(b*B*
d - 2*A*b*e + a*B*e)*Sqrt[a + b*x])/(3*(b*d - a*e)^4*Sqrt[d + e*x])

_______________________________________________________________________________________

Rubi [A]  time = 0.351836, antiderivative size = 186, 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 \[ -\frac{2 (B d-A e)}{3 e (a+b x)^{3/2} (d+e x)^{3/2} (b d-a e)}-\frac{16 e \sqrt{a+b x} (a B e-2 A b e+b B d)}{3 \sqrt{d+e x} (b d-a e)^4}-\frac{8 (a B e-2 A b e+b B d)}{3 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^3}+\frac{2 (a B e-2 A b e+b B d)}{3 e (a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(5/2)),x]

[Out]

(-2*(B*d - A*e))/(3*e*(b*d - a*e)*(a + b*x)^(3/2)*(d + e*x)^(3/2)) + (2*(b*B*d -
2*A*b*e + a*B*e))/(3*e*(b*d - a*e)^2*(a + b*x)^(3/2)*Sqrt[d + e*x]) - (8*(b*B*d
- 2*A*b*e + a*B*e))/(3*(b*d - a*e)^3*Sqrt[a + b*x]*Sqrt[d + e*x]) - (16*e*(b*B*
d - 2*A*b*e + a*B*e)*Sqrt[a + b*x])/(3*(b*d - a*e)^4*Sqrt[d + e*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 33.8214, size = 177, normalized size = 0.95 \[ \frac{16 e \sqrt{a + b x} \left (2 A b e - B a e - B b d\right )}{3 \sqrt{d + e x} \left (a e - b d\right )^{4}} - \frac{8 e \sqrt{a + b x} \left (2 A b e - B a e - B b d\right )}{3 b \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3}} - \frac{4 \left (- A b e + \frac{B \left (a e + b d\right )}{2}\right )}{b \sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}} + \frac{2 \left (A b - B a\right )}{3 b \left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b*x+a)**(5/2)/(e*x+d)**(5/2),x)

[Out]

16*e*sqrt(a + b*x)*(2*A*b*e - B*a*e - B*b*d)/(3*sqrt(d + e*x)*(a*e - b*d)**4) -
8*e*sqrt(a + b*x)*(2*A*b*e - B*a*e - B*b*d)/(3*b*(d + e*x)**(3/2)*(a*e - b*d)**3
) - 4*(-A*b*e + B*(a*e + b*d)/2)/(b*sqrt(a + b*x)*(d + e*x)**(3/2)*(a*e - b*d)**
2) + 2*(A*b - B*a)/(3*b*(a + b*x)**(3/2)*(d + e*x)**(3/2)*(a*e - b*d))

_______________________________________________________________________________________

Mathematica [A]  time = 0.593051, size = 135, normalized size = 0.73 \[ \frac{2 \sqrt{a+b x} \sqrt{d+e x} \left (-\frac{b (A b-a B) (b d-a e)}{(a+b x)^2}+\frac{e (b d-a e) (A e-B d)}{(d+e x)^2}-\frac{b (5 a B e-8 A b e+3 b B d)}{a+b x}+\frac{e (-3 a B e+8 A b e-5 b B d)}{d+e x}\right )}{3 (b d-a e)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(5/2)),x]

[Out]

(2*Sqrt[a + b*x]*Sqrt[d + e*x]*(-((b*(A*b - a*B)*(b*d - a*e))/(a + b*x)^2) - (b*
(3*b*B*d - 8*A*b*e + 5*a*B*e))/(a + b*x) + (e*(b*d - a*e)*(-(B*d) + A*e))/(d + e
*x)^2 + (e*(-5*b*B*d + 8*A*b*e - 3*a*B*e))/(d + e*x)))/(3*(b*d - a*e)^4)

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 320, normalized size = 1.7 \[ -{\frac{-32\,A{b}^{3}{e}^{3}{x}^{3}+16\,Ba{b}^{2}{e}^{3}{x}^{3}+16\,B{b}^{3}d{e}^{2}{x}^{3}-48\,Aa{b}^{2}{e}^{3}{x}^{2}-48\,A{b}^{3}d{e}^{2}{x}^{2}+24\,B{a}^{2}b{e}^{3}{x}^{2}+48\,Ba{b}^{2}d{e}^{2}{x}^{2}+24\,B{b}^{3}{d}^{2}e{x}^{2}-12\,A{a}^{2}b{e}^{3}x-72\,Aa{b}^{2}d{e}^{2}x-12\,A{b}^{3}{d}^{2}ex+6\,B{a}^{3}{e}^{3}x+42\,B{a}^{2}bd{e}^{2}x+42\,Ba{b}^{2}{d}^{2}ex+6\,B{b}^{3}{d}^{3}x+2\,A{a}^{3}{e}^{3}-18\,A{a}^{2}bd{e}^{2}-18\,Aa{b}^{2}{d}^{2}e+2\,A{b}^{3}{d}^{3}+4\,B{a}^{3}d{e}^{2}+24\,B{a}^{2}b{d}^{2}e+4\,Ba{b}^{2}{d}^{3}}{3\,{e}^{4}{a}^{4}-12\,b{e}^{3}d{a}^{3}+18\,{b}^{2}{e}^{2}{d}^{2}{a}^{2}-12\,a{b}^{3}{d}^{3}e+3\,{b}^{4}{d}^{4}} \left ( bx+a \right ) ^{-{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)^(5/2)/(e*x+d)^(5/2),x)

[Out]

-2/3*(-16*A*b^3*e^3*x^3+8*B*a*b^2*e^3*x^3+8*B*b^3*d*e^2*x^3-24*A*a*b^2*e^3*x^2-2
4*A*b^3*d*e^2*x^2+12*B*a^2*b*e^3*x^2+24*B*a*b^2*d*e^2*x^2+12*B*b^3*d^2*e*x^2-6*A
*a^2*b*e^3*x-36*A*a*b^2*d*e^2*x-6*A*b^3*d^2*e*x+3*B*a^3*e^3*x+21*B*a^2*b*d*e^2*x
+21*B*a*b^2*d^2*e*x+3*B*b^3*d^3*x+A*a^3*e^3-9*A*a^2*b*d*e^2-9*A*a*b^2*d^2*e+A*b^
3*d^3+2*B*a^3*d*e^2+12*B*a^2*b*d^2*e+2*B*a*b^2*d^3)/(b*x+a)^(3/2)/(e*x+d)^(3/2)/
(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)

_______________________________________________________________________________________

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)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 1.11949, size = 763, normalized size = 4.1 \[ -\frac{2 \,{\left (A a^{3} e^{3} +{\left (2 \, B a b^{2} + A b^{3}\right )} d^{3} + 3 \,{\left (4 \, B a^{2} b - 3 \, A a b^{2}\right )} d^{2} e +{\left (2 \, B a^{3} - 9 \, A a^{2} b\right )} d e^{2} + 8 \,{\left (B b^{3} d e^{2} +{\left (B a b^{2} - 2 \, A b^{3}\right )} e^{3}\right )} x^{3} + 12 \,{\left (B b^{3} d^{2} e + 2 \,{\left (B a b^{2} - A b^{3}\right )} d e^{2} +{\left (B a^{2} b - 2 \, A a b^{2}\right )} e^{3}\right )} x^{2} + 3 \,{\left (B b^{3} d^{3} +{\left (7 \, B a b^{2} - 2 \, A b^{3}\right )} d^{2} e +{\left (7 \, B a^{2} b - 12 \, A a b^{2}\right )} d e^{2} +{\left (B a^{3} - 2 \, A a^{2} b\right )} e^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{3 \,{\left (a^{2} b^{4} d^{6} - 4 \, a^{3} b^{3} d^{5} e + 6 \, a^{4} b^{2} d^{4} e^{2} - 4 \, a^{5} b d^{3} e^{3} + a^{6} d^{2} e^{4} +{\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{4} + 2 \,{\left (b^{6} d^{5} e - 3 \, a b^{5} d^{4} e^{2} + 2 \, a^{2} b^{4} d^{3} e^{3} + 2 \, a^{3} b^{3} d^{2} e^{4} - 3 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x^{3} +{\left (b^{6} d^{6} - 9 \, a^{2} b^{4} d^{4} e^{2} + 16 \, a^{3} b^{3} d^{3} e^{3} - 9 \, a^{4} b^{2} d^{2} e^{4} + a^{6} e^{6}\right )} x^{2} + 2 \,{\left (a b^{5} d^{6} - 3 \, a^{2} b^{4} d^{5} e + 2 \, a^{3} b^{3} d^{4} e^{2} + 2 \, a^{4} b^{2} d^{3} e^{3} - 3 \, a^{5} b d^{2} e^{4} + a^{6} d e^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(5/2)*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

-2/3*(A*a^3*e^3 + (2*B*a*b^2 + A*b^3)*d^3 + 3*(4*B*a^2*b - 3*A*a*b^2)*d^2*e + (2
*B*a^3 - 9*A*a^2*b)*d*e^2 + 8*(B*b^3*d*e^2 + (B*a*b^2 - 2*A*b^3)*e^3)*x^3 + 12*(
B*b^3*d^2*e + 2*(B*a*b^2 - A*b^3)*d*e^2 + (B*a^2*b - 2*A*a*b^2)*e^3)*x^2 + 3*(B*
b^3*d^3 + (7*B*a*b^2 - 2*A*b^3)*d^2*e + (7*B*a^2*b - 12*A*a*b^2)*d*e^2 + (B*a^3
- 2*A*a^2*b)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(a^2*b^4*d^6 - 4*a^3*b^3*d^5*e
+ 6*a^4*b^2*d^4*e^2 - 4*a^5*b*d^3*e^3 + a^6*d^2*e^4 + (b^6*d^4*e^2 - 4*a*b^5*d^3
*e^3 + 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^4 + 2*(b^6*d^5*e - 3
*a*b^5*d^4*e^2 + 2*a^2*b^4*d^3*e^3 + 2*a^3*b^3*d^2*e^4 - 3*a^4*b^2*d*e^5 + a^5*b
*e^6)*x^3 + (b^6*d^6 - 9*a^2*b^4*d^4*e^2 + 16*a^3*b^3*d^3*e^3 - 9*a^4*b^2*d^2*e^
4 + a^6*e^6)*x^2 + 2*(a*b^5*d^6 - 3*a^2*b^4*d^5*e + 2*a^3*b^3*d^4*e^2 + 2*a^4*b^
2*d^3*e^3 - 3*a^5*b*d^2*e^4 + a^6*d*e^5)*x)

_______________________________________________________________________________________

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)/(b*x+a)**(5/2)/(e*x+d)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.70033, 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)^(5/2)*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

Done