3.1840 \(\int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=160 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (-a B e-A b e+2 b B d)}{e^3 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{e^3 (a+b x) \sqrt{d+e x}}+\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^3 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.256505, antiderivative size = 160, 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 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (-a B e-A b e+2 b B d)}{e^3 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{e^3 (a+b x) \sqrt{d+e x}}+\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 36.8681, size = 172, normalized size = 1.08 \[ - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e \sqrt{d + e x} \left (a e - b d\right )} + \frac{2 \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (3 A b e + B a e - 4 B b d\right )}{3 e^{2} \left (a e - b d\right )} + \frac{4 \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (3 A b e + B a e - 4 B b d\right )}{3 e^{3} \left (a + b x\right )} \]

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)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0995576, size = 85, normalized size = 0.53 \[ \frac{2 \sqrt{(a+b x)^2} \left (3 a e (-A e+2 B d+B e x)+3 A b e (2 d+e x)+b B \left (-8 d^2-4 d e x+e^2 x^2\right )\right )}{3 e^3 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.008, size = 89, normalized size = 0.6 \[ -{\frac{-2\,B{x}^{2}b{e}^{2}-6\,Ab{e}^{2}x-6\,aB{e}^{2}x+8\,Bbdex+6\,A{e}^{2}a-12\,Abde-12\,aBde+16\,Bb{d}^{2}}{3\, \left ( bx+a \right ){e}^{3}}\sqrt{ \left ( bx+a \right ) ^{2}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3/(e*x+d)^(1/2)*(-B*b*e^2*x^2-3*A*b*e^2*x-3*B*a*e^2*x+4*B*b*d*e*x+3*A*a*e^2-6
*A*b*d*e-6*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 = 0.731031, size = 101, normalized size = 0.63 \[ \frac{2 \,{\left (b e x + 2 \, b d - a e\right )} A}{\sqrt{e x + d} e^{2}} + \frac{2 \,{\left (b e^{2} x^{2} - 8 \, b d^{2} + 6 \, a d e -{\left (4 \, b d e - 3 \, a e^{2}\right )} x\right )} B}{3 \, \sqrt{e x + d} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)/(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.30228, size = 93, normalized size = 0.58 \[ \frac{2 \,{\left (B b e^{2} x^{2} - 8 \, B b d^{2} - 3 \, A a e^{2} + 6 \,{\left (B a + A b\right )} d e -{\left (4 \, B b d e - 3 \,{\left (B a + A b\right )} e^{2}\right )} x\right )}}{3 \, \sqrt{e x + d} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)/(e*x + d)^(3/2),x, algorithm="fricas")

[Out]

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

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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)**2)**(1/2)/(e*x+d)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.286872, size = 200, normalized size = 1.25 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B b e^{6}{\rm sign}\left (b x + a\right ) - 6 \, \sqrt{x e + d} B b d e^{6}{\rm sign}\left (b x + a\right ) + 3 \, \sqrt{x e + d} B a e^{7}{\rm sign}\left (b x + a\right ) + 3 \, \sqrt{x e + d} A b e^{7}{\rm sign}\left (b x + a\right )\right )} e^{\left (-9\right )} - \frac{2 \,{\left (B b d^{2}{\rm sign}\left (b x + a\right ) - B a d e{\rm sign}\left (b x + a\right ) - A b d e{\rm sign}\left (b x + a\right ) + A a e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-3\right )}}{\sqrt{x e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)/(e*x + d)^(3/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*B*b*e^6*sign(b*x + a) - 6*sqrt(x*e + d)*B*b*d*e^6*sign(b*x
+ a) + 3*sqrt(x*e + d)*B*a*e^7*sign(b*x + a) + 3*sqrt(x*e + d)*A*b*e^7*sign(b*x
+ a))*e^(-9) - 2*(B*b*d^2*sign(b*x + a) - B*a*d*e*sign(b*x + a) - A*b*d*e*sign(b
*x + a) + A*a*e^2*sign(b*x + a))*e^(-3)/sqrt(x*e + d)