3.1839 \(\int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=162 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (-a B e-A b e+2 b B d)}{3 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e) (B d-A e)}{e^3 (a+b x)}+\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^3 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.258386, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (-a B e-A b e+2 b B d)}{3 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e) (B d-A e)}{e^3 (a+b x)}+\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.535, size = 167, normalized size = 1.03 \[ \frac{B \left (2 a + 2 b x\right ) \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 b e} + \frac{2 \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (5 A b e - B a e - 4 B b d\right )}{15 b e^{2}} + \frac{4 \sqrt{d + e x} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (5 A b e - B a e - 4 B b d\right )}{15 b e^{3} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0915308, size = 86, normalized size = 0.53 \[ \frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} \left (5 a e (3 A e-2 B d+B e x)+5 A b e (e x-2 d)+b B \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[(a + b*x)^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*(a + b*x))

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Maple [A]  time = 0.007, size = 89, normalized size = 0.6 \[{\frac{6\,B{x}^{2}b{e}^{2}+10\,Ab{e}^{2}x+10\,aB{e}^{2}x-8\,Bbdex+30\,A{e}^{2}a-20\,Abde-20\,aBde+16\,Bb{d}^{2}}{15\, \left ( bx+a \right ){e}^{3}}\sqrt{ex+d}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*((b*x+a)^2)^(1/2)/(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)*((b*x+a)^2)^(1/2)/e^3/(b*x+a)

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Maxima [A]  time = 0.728069, size = 161, normalized size = 0.99 \[ \frac{2 \,{\left (b e^{2} x^{2} - 2 \, b d^{2} + 3 \, a d e -{\left (b d e - 3 \, a e^{2}\right )} x\right )} A}{3 \, \sqrt{e x + d} e^{2}} + \frac{2 \,{\left (3 \, b e^{3} x^{3} + 8 \, b d^{3} - 10 \, a d^{2} e -{\left (b d e^{2} - 5 \, a e^{3}\right )} x^{2} +{\left (4 \, b d^{2} e - 5 \, a d e^{2}\right )} x\right )} B}{15 \, \sqrt{e x + d} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.280959, size = 95, normalized size = 0.59 \[ \frac{2 \,{\left (3 \, B b e^{2} x^{2} + 8 \, B b d^{2} + 15 \, A a e^{2} - 10 \,{\left (B a + A b\right )} d e -{\left (4 \, B b d e - 5 \,{\left (B a + A b\right )} e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(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 [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \sqrt{\left (a + b x\right )^{2}}}{\sqrt{d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)*sqrt((a + b*x)**2)/sqrt(d + e*x), x)

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GIAC/XCAS [A]  time = 0.282385, size = 188, normalized size = 1.16 \[ \frac{2}{15} \,{\left (5 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} B a e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) + 5 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} A b e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) +{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} B b e^{\left (-10\right )}{\rm sign}\left (b x + a\right ) + 15 \, \sqrt{x e + d} A a{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(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)*sign(b*x + a) + 5*((x*e
+ d)^(3/2) - 3*sqrt(x*e + d)*d)*A*b*e^(-1)*sign(b*x + a) + (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)*sign(b*x +
a) + 15*sqrt(x*e + d)*A*a*sign(b*x + a))*e^(-1)