3.1667 \(\int \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=96 \[ \frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^2 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{3 e^2 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.115019, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^2 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{3 e^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 12.2773, size = 80, normalized size = 0.83 \[ \frac{2 \left (d + e x\right )^{\frac{3}{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 e} + \frac{4 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 e^{2} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0543216, size = 48, normalized size = 0.5 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{3/2} (5 a e-2 b d+3 b e x)}{15 e^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

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

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Maple [A]  time = 0.004, size = 43, normalized size = 0.5 \[{\frac{6\,bex+10\,ae-4\,bd}{15\,{e}^{2} \left ( bx+a \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(e*x+d)^(3/2)*(3*b*e*x+5*a*e-2*b*d)*((b*x+a)^2)^(1/2)/e^2/(b*x+a)

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Maxima [A]  time = 0.759698, size = 62, normalized size = 0.65 \[ \frac{2 \,{\left (3 \, b e^{2} x^{2} - 2 \, b d^{2} + 5 \, a d e +{\left (b d e + 5 \, a e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \, e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.206042, size = 62, normalized size = 0.65 \[ \frac{2 \,{\left (3 \, b e^{2} x^{2} - 2 \, b d^{2} + 5 \, a d e +{\left (b d e + 5 \, a e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \, e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 25.9482, size = 49, normalized size = 0.51 \[ \frac{2 a \left (d + e x\right )^{\frac{3}{2}}}{3 e} - \frac{2 b d \left (d + e x\right )^{\frac{3}{2}}}{3 e^{2}} + \frac{2 b \left (d + e x\right )^{\frac{5}{2}}}{5 e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.21229, size = 73, normalized size = 0.76 \[ \frac{2}{15} \,{\left ({\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} b e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} a{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*((3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*b*e^(-1)*sign(b*x + a) + 5*(x*e
+ d)^(3/2)*a*sign(b*x + a))*e^(-1)