3.1705 \(\int \frac{(a+b x) (A+B x)}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=81 \[ -\frac{2 (d+e x)^{3/2} (-a B e-A b e+2 b B d)}{3 e^3}+\frac{2 \sqrt{d+e x} (b d-a e) (B d-A e)}{e^3}+\frac{2 b B (d+e x)^{5/2}}{5 e^3} \]

[Out]

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

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Rubi [A]  time = 0.10138, antiderivative size = 81, 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 \[ -\frac{2 (d+e x)^{3/2} (-a B e-A b e+2 b B d)}{3 e^3}+\frac{2 \sqrt{d+e x} (b d-a e) (B d-A e)}{e^3}+\frac{2 b B (d+e x)^{5/2}}{5 e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.4003, size = 76, normalized size = 0.94 \[ \frac{2 B b \left (d + e x\right )^{\frac{5}{2}}}{5 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (A b e + B a e - 2 B b d\right )}{3 e^{3}} + \frac{2 \sqrt{d + e x} \left (A e - B d\right ) \left (a e - b d\right )}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(B*x+A)/(e*x+d)**(1/2),x)

[Out]

2*B*b*(d + e*x)**(5/2)/(5*e**3) + 2*(d + e*x)**(3/2)*(A*b*e + B*a*e - 2*B*b*d)/(
3*e**3) + 2*sqrt(d + e*x)*(A*e - B*d)*(a*e - b*d)/e**3

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Mathematica [A]  time = 0.0747272, size = 68, normalized size = 0.84 \[ \frac{2 \sqrt{d+e x} \left (5 a e (3 A e-2 B d+B e x)+5 A b e (e x-2 d)+b B \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(5*A*b*e*(-2*d + e*x) + 5*a*e*(-2*B*d + 3*A*e + B*e*x) + b*B*(8
*d^2 - 4*d*e*x + 3*e^2*x^2)))/(15*e^3)

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Maple [A]  time = 0.005, size = 73, normalized size = 0.9 \[{\frac{6\,bB{x}^{2}{e}^{2}+10\,Ab{e}^{2}x+10\,Ba{e}^{2}x-8\,Bbdex+30\,aA{e}^{2}-20\,Abde-20\,Bade+16\,bB{d}^{2}}{15\,{e}^{3}}\sqrt{ex+d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(B*x+A)/(e*x+d)^(1/2),x)

[Out]

2/15*(e*x+d)^(1/2)*(3*B*b*e^2*x^2+5*A*b*e^2*x+5*B*a*e^2*x-4*B*b*d*e*x+15*A*a*e^2
-10*A*b*d*e-10*B*a*d*e+8*B*b*d^2)/e^3

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Maxima [A]  time = 1.34608, size = 101, normalized size = 1.25 \[ \frac{2 \,{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} B b - 5 \,{\left (2 \, B b d -{\left (B a + A b\right )} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15 \,{\left (B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e\right )} \sqrt{e x + d}\right )}}{15 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/sqrt(e*x + d),x, algorithm="maxima")

[Out]

2/15*(3*(e*x + d)^(5/2)*B*b - 5*(2*B*b*d - (B*a + A*b)*e)*(e*x + d)^(3/2) + 15*(
B*b*d^2 + A*a*e^2 - (B*a + A*b)*d*e)*sqrt(e*x + d))/e^3

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Fricas [A]  time = 0.231713, size = 95, normalized size = 1.17 \[ \frac{2 \,{\left (3 \, B b e^{2} x^{2} + 8 \, B b d^{2} + 15 \, A a e^{2} - 10 \,{\left (B a + A b\right )} d e -{\left (4 \, B b d e - 5 \,{\left (B a + A b\right )} e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*B*b*e^2*x^2 + 8*B*b*d^2 + 15*A*a*e^2 - 10*(B*a + A*b)*d*e - (4*B*b*d*e -
5*(B*a + A*b)*e^2)*x)*sqrt(e*x + d)/e^3

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Sympy [A]  time = 16.9335, size = 311, normalized size = 3.84 \[ \begin{cases} - \frac{\frac{2 A a d}{\sqrt{d + e x}} + 2 A a \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{2 A b d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{2 A b \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{2 B a d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{2 B a \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{2 B b d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{2 B b \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}}}{e} & \text{for}\: e \neq 0 \\\frac{A a x + \frac{B b x^{3}}{3} + \frac{x^{2} \left (A b + B a\right )}{2}}{\sqrt{d}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(B*x+A)/(e*x+d)**(1/2),x)

[Out]

Piecewise((-(2*A*a*d/sqrt(d + e*x) + 2*A*a*(-d/sqrt(d + e*x) - sqrt(d + e*x)) +
2*A*b*d*(-d/sqrt(d + e*x) - sqrt(d + e*x))/e + 2*A*b*(d**2/sqrt(d + e*x) + 2*d*s
qrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 2*B*a*d*(-d/sqrt(d + e*x) - sqrt(d + e*x)
)/e + 2*B*a*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 2*
B*b*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 2*B*b
*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(
5/2)/5)/e**2)/e, Ne(e, 0)), ((A*a*x + B*b*x**3/3 + x**2*(A*b + B*a)/2)/sqrt(d),
True))

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GIAC/XCAS [A]  time = 0.207103, size = 155, normalized size = 1.91 \[ \frac{2}{15} \,{\left (5 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} B a e^{\left (-1\right )} + 5 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} A b e^{\left (-1\right )} +{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} B b e^{\left (-10\right )} + 15 \, \sqrt{x e + d} A a\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/sqrt(e*x + d),x, algorithm="giac")

[Out]

2/15*(5*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a*e^(-1) + 5*((x*e + d)^(3/2) -
3*sqrt(x*e + d)*d)*A*b*e^(-1) + (3*(x*e + d)^(5/2)*e^8 - 10*(x*e + d)^(3/2)*d*e^
8 + 15*sqrt(x*e + d)*d^2*e^8)*B*b*e^(-10) + 15*sqrt(x*e + d)*A*a)*e^(-1)