3.2096 \(\int \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=150 \[ \frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^3 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{3 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}{e^3 (a+b x)} \]

[Out]

(2*(b*d - a*e)^2*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^3*(a + b*x)) -
(4*b*(b*d - a*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^2*(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 = 0.212902, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^3 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{3 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}{e^3 (a+b x)} \]

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)^2*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^3*(a + b*x)) -
(4*b*(b*d - a*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^2*(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 = 25.3588, size = 129, normalized size = 0.86 \[ \frac{2 \left (a + b x\right ) \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 e} + \frac{8 \sqrt{d + e x} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 e^{2}} + \frac{16 \sqrt{d + e x} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 e^{3} \left (a + b x\right )} \]

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]

2*(a + b*x)*sqrt(d + e*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(5*e) + 8*sqrt(d + e*
x)*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(15*e**2) + 16*sqrt(d + e*x)*(a*
e - b*d)**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(15*e**3*(a + b*x))

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Mathematica [A]  time = 0.0763755, size = 78, normalized size = 0.52 \[ \frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} \left (15 a^2 e^2+10 a b e (e x-2 d)+b^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3 (a+b x)} \]

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]*(15*a^2*e^2 + 10*a*b*e*(-2*d + e*x) + b^2*(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 = 0.009, size = 79, normalized size = 0.5 \[{\frac{6\,{x}^{2}{b}^{2}{e}^{2}+20\,xab{e}^{2}-8\,x{b}^{2}de+30\,{a}^{2}{e}^{2}-40\,abde+16\,{b}^{2}{d}^{2}}{15\, \left ( bx+a \right ){e}^{3}}\sqrt{ex+d}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

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^2*e^2*x^2+10*a*b*e^2*x-4*b^2*d*e*x+15*a^2*e^2-20*a*b*d*e
+8*b^2*d^2)*((b*x+a)^2)^(1/2)/e^3/(b*x+a)

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Maxima [A]  time = 0.72411, size = 161, normalized size = 1.07 \[ \frac{2 \,{\left (b e^{2} x^{2} - 2 \, b d^{2} + 3 \, a d e -{\left (b d e - 3 \, a e^{2}\right )} x\right )} a}{3 \, \sqrt{e x + d} e^{2}} + \frac{2 \,{\left (3 \, b e^{3} x^{3} + 8 \, b d^{3} - 10 \, a d^{2} e -{\left (b d e^{2} - 5 \, a e^{3}\right )} x^{2} +{\left (4 \, b d^{2} e - 5 \, a d e^{2}\right )} x\right )} b}{15 \, \sqrt{e x + d} e^{3}} \]

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 = 0.274141, size = 86, normalized size = 0.57 \[ \frac{2 \,{\left (3 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} - 20 \, a b d e + 15 \, a^{2} e^{2} - 2 \,{\left (2 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \, e^{3}} \]

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^2*e^2*x^2 + 8*b^2*d^2 - 20*a*b*d*e + 15*a^2*e^2 - 2*(2*b^2*d*e - 5*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 = 0.297058, size = 147, normalized size = 0.98 \[ \frac{2}{15} \,{\left (10 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a b e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) +{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} b^{2} e^{\left (-10\right )}{\rm sign}\left (b x + a\right ) + 15 \, \sqrt{x e + d} a^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )} \]

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*(10*((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^2*e^(-
10)*sign(b*x + a) + 15*sqrt(x*e + d)*a^2*sign(b*x + a))*e^(-1)