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

Optimal. Leaf size=198 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} \left (4 c d-e^2\right ) (2 a d-b e) \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{16 d^{5/2} (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (2 d x+e) (2 a d-b e) \sqrt{c+d x^2+e x}}{8 d^2 (a+b x)}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} \left (c+d x^2+e x\right )^{3/2}}{3 d (a+b x)} \]

[Out]

((2*a*d - b*e)*(e + 2*d*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2])/
(8*d^2*(a + b*x)) + (b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(c + e*x + d*x^2)^(3/2))/(3
*d*(a + b*x)) + ((2*a*d - b*e)*(4*c*d - e^2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTa
nh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + e*x + d*x^2])])/(16*d^(5/2)*(a + b*x))

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Rubi [A]  time = 0.257692, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} \left (4 c d-e^2\right ) (2 a d-b e) \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{16 d^{5/2} (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (2 d x+e) (2 a d-b e) \sqrt{c+d x^2+e x}}{8 d^2 (a+b x)}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} \left (c+d x^2+e x\right )^{3/2}}{3 d (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

((2*a*d - b*e)*(e + 2*d*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2])/
(8*d^2*(a + b*x)) + (b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(c + e*x + d*x^2)^(3/2))/(3
*d*(a + b*x)) + ((2*a*d - b*e)*(4*c*d - e^2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTa
nh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + e*x + d*x^2])])/(16*d^(5/2)*(a + b*x))

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Rubi in Sympy [A]  time = 44.7745, size = 180, normalized size = 0.91 \[ \frac{b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (c + d x^{2} + e x\right )^{\frac{3}{2}}}{3 d \left (a + b x\right )} + \frac{\left (2 a d - b e\right ) \left (2 d x + e\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \sqrt{c + d x^{2} + e x}}{8 d^{2} \left (a + b x\right )} - \frac{\left (2 a d - b e\right ) \left (- 4 c d + e^{2}\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \operatorname{atanh}{\left (\frac{2 d x + e}{2 \sqrt{d} \sqrt{c + d x^{2} + e x}} \right )}}{16 d^{\frac{5}{2}} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(c + d*x**2 + e*x)**(3/2)/(3*d*(a + b*x)) + (
2*a*d - b*e)*(2*d*x + e)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*sqrt(c + d*x**2 + e*x)
/(8*d**2*(a + b*x)) - (2*a*d - b*e)*(-4*c*d + e**2)*sqrt(a**2 + 2*a*b*x + b**2*x
**2)*atanh((2*d*x + e)/(2*sqrt(d)*sqrt(c + d*x**2 + e*x)))/(16*d**(5/2)*(a + b*x
))

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Mathematica [A]  time = 0.247302, size = 132, normalized size = 0.67 \[ \frac{\sqrt{(a+b x)^2} \left (2 \sqrt{d} \sqrt{c+x (d x+e)} \left (6 a d (2 d x+e)+b \left (8 c d+8 d^2 x^2+2 d e x-3 e^2\right )\right )+3 \left (4 c d-e^2\right ) (2 a d-b e) \log \left (2 \sqrt{d} \sqrt{c+x (d x+e)}+2 d x+e\right )\right )}{48 d^{5/2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x)^2]*(2*Sqrt[d]*Sqrt[c + x*(e + d*x)]*(6*a*d*(e + 2*d*x) + b*(8*c*
d - 3*e^2 + 2*d*e*x + 8*d^2*x^2)) + 3*(2*a*d - b*e)*(4*c*d - e^2)*Log[e + 2*d*x
+ 2*Sqrt[d]*Sqrt[c + x*(e + d*x)]]))/(48*d^(5/2)*(a + b*x))

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Maple [C]  time = 0.014, size = 259, normalized size = 1.3 \[{\frac{{\it csgn} \left ( bx+a \right ) }{48} \left ( 16\,b \left ( d{x}^{2}+ex+c \right ) ^{3/2}{d}^{7/2}+24\,a\sqrt{d{x}^{2}+ex+c}x{d}^{9/2}-12\,be\sqrt{d{x}^{2}+ex+c}x{d}^{7/2}+24\,a\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ) c{d}^{4}+12\,a\sqrt{d{x}^{2}+ex+c}e{d}^{7/2}-6\,b{e}^{2}\sqrt{d{x}^{2}+ex+c}{d}^{5/2}-6\,a\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){e}^{2}{d}^{3}-12\,be\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ) c{d}^{3}+3\,b{e}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){d}^{2} \right ){d}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*csgn(b*x+a)*(16*b*(d*x^2+e*x+c)^(3/2)*d^(7/2)+24*a*(d*x^2+e*x+c)^(1/2)*x*d^
(9/2)-12*b*e*(d*x^2+e*x+c)^(1/2)*x*d^(7/2)+24*a*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^
(1/2)+2*d*x+e)/d^(1/2))*c*d^4+12*a*(d*x^2+e*x+c)^(1/2)*e*d^(7/2)-6*b*e^2*(d*x^2+
e*x+c)^(1/2)*d^(5/2)-6*a*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^(1/2)+2*d*x+e)/d^(1/2))
*e^2*d^3-12*b*e*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^(1/2)+2*d*x+e)/d^(1/2))*c*d^3+3*
b*e^3*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^(1/2)+2*d*x+e)/d^(1/2))*d^2)/d^(9/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.329206, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (8 \, b d^{2} x^{2} + 8 \, b c d + 6 \, a d e - 3 \, b e^{2} + 2 \,{\left (6 \, a d^{2} + b d e\right )} x\right )} \sqrt{d x^{2} + e x + c} \sqrt{d} + 3 \,{\left (8 \, a c d^{2} - 4 \, b c d e - 2 \, a d e^{2} + b e^{3}\right )} \log \left (4 \,{\left (2 \, d^{2} x + d e\right )} \sqrt{d x^{2} + e x + c} +{\left (8 \, d^{2} x^{2} + 8 \, d e x + 4 \, c d + e^{2}\right )} \sqrt{d}\right )}{96 \, d^{\frac{5}{2}}}, \frac{2 \,{\left (8 \, b d^{2} x^{2} + 8 \, b c d + 6 \, a d e - 3 \, b e^{2} + 2 \,{\left (6 \, a d^{2} + b d e\right )} x\right )} \sqrt{d x^{2} + e x + c} \sqrt{-d} + 3 \,{\left (8 \, a c d^{2} - 4 \, b c d e - 2 \, a d e^{2} + b e^{3}\right )} \arctan \left (\frac{{\left (2 \, d x + e\right )} \sqrt{-d}}{2 \, \sqrt{d x^{2} + e x + c} d}\right )}{48 \, \sqrt{-d} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(4*(8*b*d^2*x^2 + 8*b*c*d + 6*a*d*e - 3*b*e^2 + 2*(6*a*d^2 + b*d*e)*x)*sqr
t(d*x^2 + e*x + c)*sqrt(d) + 3*(8*a*c*d^2 - 4*b*c*d*e - 2*a*d*e^2 + b*e^3)*log(4
*(2*d^2*x + d*e)*sqrt(d*x^2 + e*x + c) + (8*d^2*x^2 + 8*d*e*x + 4*c*d + e^2)*sqr
t(d)))/d^(5/2), 1/48*(2*(8*b*d^2*x^2 + 8*b*c*d + 6*a*d*e - 3*b*e^2 + 2*(6*a*d^2
+ b*d*e)*x)*sqrt(d*x^2 + e*x + c)*sqrt(-d) + 3*(8*a*c*d^2 - 4*b*c*d*e - 2*a*d*e^
2 + b*e^3)*arctan(1/2*(2*d*x + e)*sqrt(-d)/(sqrt(d*x^2 + e*x + c)*d)))/(sqrt(-d)
*d^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c + d x^{2} + e x} \sqrt{\left (a + b x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(c + d*x**2 + e*x)*sqrt((a + b*x)**2), x)

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GIAC/XCAS [A]  time = 0.282361, size = 250, normalized size = 1.26 \[ \frac{1}{24} \, \sqrt{d x^{2} + x e + c}{\left (2 \,{\left (4 \, b x{\rm sign}\left (b x + a\right ) + \frac{6 \, a d^{2}{\rm sign}\left (b x + a\right ) + b d e{\rm sign}\left (b x + a\right )}{d^{2}}\right )} x + \frac{8 \, b c d{\rm sign}\left (b x + a\right ) + 6 \, a d e{\rm sign}\left (b x + a\right ) - 3 \, b e^{2}{\rm sign}\left (b x + a\right )}{d^{2}}\right )} - \frac{{\left (8 \, a c d^{2}{\rm sign}\left (b x + a\right ) - 4 \, b c d e{\rm sign}\left (b x + a\right ) - 2 \, a d e^{2}{\rm sign}\left (b x + a\right ) + b e^{3}{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + x e + c}\right )} \sqrt{d} - e \right |}\right )}{16 \, d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*sqrt(d*x^2 + x*e + c)*(2*(4*b*x*sign(b*x + a) + (6*a*d^2*sign(b*x + a) + b*
d*e*sign(b*x + a))/d^2)*x + (8*b*c*d*sign(b*x + a) + 6*a*d*e*sign(b*x + a) - 3*b
*e^2*sign(b*x + a))/d^2) - 1/16*(8*a*c*d^2*sign(b*x + a) - 4*b*c*d*e*sign(b*x +
a) - 2*a*d*e^2*sign(b*x + a) + b*e^3*sign(b*x + a))*ln(abs(-2*(sqrt(d)*x - sqrt(
d*x^2 + x*e + c))*sqrt(d) - e))/d^(5/2)