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

Optimal. Leaf size=211 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} \left (4 a d e+4 b c d-b e^2\right ) \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{8 d^{3/2} (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2+e x} (4 a d+2 b d x+b e)}{4 d (a+b x)}-\frac{a \sqrt{c} \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right )}{a+b x} \]

[Out]

((4*a*d + b*e + 2*b*d*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2])/(4
*d*(a + b*x)) + ((4*b*c*d + 4*a*d*e - b*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])])/(8*d^(3/2)*(a + b*x)) - (a*Sq
rt[c]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + e*x
+ d*x^2])])/(a + b*x)

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Rubi [A]  time = 0.537125, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} \left (4 a d e+4 b c d-b e^2\right ) \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{8 d^{3/2} (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2+e x} (4 a d+2 b d x+b e)}{4 d (a+b x)}-\frac{a \sqrt{c} \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right )}{a+b x} \]

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,x]

[Out]

((4*a*d + b*e + 2*b*d*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2])/(4
*d*(a + b*x)) + ((4*b*c*d + 4*a*d*e - b*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])])/(8*d^(3/2)*(a + b*x)) - (a*Sq
rt[c]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + e*x
+ d*x^2])])/(a + b*x)

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Rubi in Sympy [A]  time = 68.4531, size = 197, normalized size = 0.93 \[ - \frac{a \sqrt{c} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \operatorname{atanh}{\left (\frac{2 c + e x}{2 \sqrt{c} \sqrt{c + d x^{2} + e x}} \right )}}{a + b x} + \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \sqrt{c + d x^{2} + e x} \left (4 a d + 2 b d x + b e\right )}{4 d \left (a + b x\right )} - \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (- 4 a d e - 4 b c d + b e^{2}\right ) \operatorname{atanh}{\left (\frac{2 d x + e}{2 \sqrt{d} \sqrt{c + d x^{2} + e x}} \right )}}{8 d^{\frac{3}{2}} \left (a + b x\right )} \]

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,x)

[Out]

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

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Mathematica [A]  time = 0.273794, size = 246, normalized size = 1.17 \[ \frac{\sqrt{(a+b x)^2} \left (8 a d^{3/2} \sqrt{c+x (d x+e)}-8 a \sqrt{c} d^{3/2} \log \left (2 \sqrt{c} \sqrt{c+x (d x+e)}+2 c+e x\right )+8 a \sqrt{c} d^{3/2} \log (x)+4 a d e \log \left (2 \sqrt{d} \sqrt{c+x (d x+e)}+2 d x+e\right )+4 b d^{3/2} x \sqrt{c+x (d x+e)}-b e^2 \log \left (2 \sqrt{d} \sqrt{c+x (d x+e)}+2 d x+e\right )+2 b \sqrt{d} e \sqrt{c+x (d x+e)}+4 b c d \log \left (2 \sqrt{d} \sqrt{c+x (d x+e)}+2 d x+e\right )\right )}{8 d^{3/2} (a+b x)} \]

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,x]

[Out]

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

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Maple [C]  time = 0.015, size = 214, normalized size = 1. \[ -{\frac{{\it csgn} \left ( bx+a \right ) }{8} \left ( 8\,a\sqrt{c}\ln \left ({\frac{2\,c+ex+2\,\sqrt{c}\sqrt{d{x}^{2}+ex+c}}{x}} \right ){d}^{5/2}-4\,b\sqrt{d{x}^{2}+ex+c}x{d}^{5/2}-4\,ae\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){d}^{2}-4\,b\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ) c{d}^{2}-8\,a\sqrt{d{x}^{2}+ex+c}{d}^{5/2}-2\,b\sqrt{d{x}^{2}+ex+c}e{d}^{3/2}+b\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e \right ){\frac{1}{\sqrt{d}}}} \right ){e}^{2}d \right ){d}^{-{\frac{5}{2}}}} \]

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,x)

[Out]

-1/8*csgn(b*x+a)*(8*a*c^(1/2)*ln((2*c+e*x+2*c^(1/2)*(d*x^2+e*x+c)^(1/2))/x)*d^(5
/2)-4*b*(d*x^2+e*x+c)^(1/2)*x*d^(5/2)-4*a*e*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^(1/2
)+2*d*x+e)/d^(1/2))*d^2-4*b*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^(1/2)+2*d*x+e)/d^(1/
2))*c*d^2-8*a*(d*x^2+e*x+c)^(1/2)*d^(5/2)-2*b*(d*x^2+e*x+c)^(1/2)*e*d^(3/2)+b*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)/d^(5/2)

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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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.995471, size = 1, normalized size = 0. \[ \left [\frac{8 \, a \sqrt{c} d^{\frac{3}{2}} \log \left (\frac{8 \, c e x +{\left (4 \, c d + e^{2}\right )} x^{2} - 4 \, \sqrt{d x^{2} + e x + c}{\left (e x + 2 \, c\right )} \sqrt{c} + 8 \, c^{2}}{x^{2}}\right ) + 4 \,{\left (2 \, b d x + 4 \, a d + b e\right )} \sqrt{d x^{2} + e x + c} \sqrt{d} -{\left (4 \, b c d + 4 \, a d e - b e^{2}\right )} \log \left (-4 \,{\left (2 \, d^{2} x + d e\right )} \sqrt{d x^{2} + e x + c} +{\left (8 \, d^{2} x^{2} + 8 \, d e x + 4 \, c d + e^{2}\right )} \sqrt{d}\right )}{16 \, d^{\frac{3}{2}}}, \frac{4 \, a \sqrt{c} \sqrt{-d} d \log \left (\frac{8 \, c e x +{\left (4 \, c d + e^{2}\right )} x^{2} - 4 \, \sqrt{d x^{2} + e x + c}{\left (e x + 2 \, c\right )} \sqrt{c} + 8 \, c^{2}}{x^{2}}\right ) + 2 \,{\left (2 \, b d x + 4 \, a d + b e\right )} \sqrt{d x^{2} + e x + c} \sqrt{-d} +{\left (4 \, b c d + 4 \, a d e - b e^{2}\right )} \arctan \left (\frac{{\left (2 \, d x + e\right )} \sqrt{-d}}{2 \, \sqrt{d x^{2} + e x + c} d}\right )}{8 \, \sqrt{-d} d}, -\frac{16 \, a \sqrt{-c} d^{\frac{3}{2}} \arctan \left (\frac{e x + 2 \, c}{2 \, \sqrt{d x^{2} + e x + c} \sqrt{-c}}\right ) - 4 \,{\left (2 \, b d x + 4 \, a d + b e\right )} \sqrt{d x^{2} + e x + c} \sqrt{d} +{\left (4 \, b c d + 4 \, a d e - b e^{2}\right )} \log \left (-4 \,{\left (2 \, d^{2} x + d e\right )} \sqrt{d x^{2} + e x + c} +{\left (8 \, d^{2} x^{2} + 8 \, d e x + 4 \, c d + e^{2}\right )} \sqrt{d}\right )}{16 \, d^{\frac{3}{2}}}, -\frac{8 \, a \sqrt{-c} \sqrt{-d} d \arctan \left (\frac{e x + 2 \, c}{2 \, \sqrt{d x^{2} + e x + c} \sqrt{-c}}\right ) - 2 \,{\left (2 \, b d x + 4 \, a d + b e\right )} \sqrt{d x^{2} + e x + c} \sqrt{-d} -{\left (4 \, b c d + 4 \, a d e - b e^{2}\right )} \arctan \left (\frac{{\left (2 \, d x + e\right )} \sqrt{-d}}{2 \, \sqrt{d x^{2} + e x + c} d}\right )}{8 \, \sqrt{-d} d}\right ] \]

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,x, algorithm="fricas")

[Out]

[1/16*(8*a*sqrt(c)*d^(3/2)*log((8*c*e*x + (4*c*d + e^2)*x^2 - 4*sqrt(d*x^2 + e*x
+ c)*(e*x + 2*c)*sqrt(c) + 8*c^2)/x^2) + 4*(2*b*d*x + 4*a*d + b*e)*sqrt(d*x^2 +
e*x + c)*sqrt(d) - (4*b*c*d + 4*a*d*e - b*e^2)*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)*sqrt(d)))/d^(3/2), 1/8*(4*a*s
qrt(c)*sqrt(-d)*d*log((8*c*e*x + (4*c*d + e^2)*x^2 - 4*sqrt(d*x^2 + e*x + c)*(e*
x + 2*c)*sqrt(c) + 8*c^2)/x^2) + 2*(2*b*d*x + 4*a*d + b*e)*sqrt(d*x^2 + e*x + c)
*sqrt(-d) + (4*b*c*d + 4*a*d*e - b*e^2)*arctan(1/2*(2*d*x + e)*sqrt(-d)/(sqrt(d*
x^2 + e*x + c)*d)))/(sqrt(-d)*d), -1/16*(16*a*sqrt(-c)*d^(3/2)*arctan(1/2*(e*x +
2*c)/(sqrt(d*x^2 + e*x + c)*sqrt(-c))) - 4*(2*b*d*x + 4*a*d + b*e)*sqrt(d*x^2 +
e*x + c)*sqrt(d) + (4*b*c*d + 4*a*d*e - b*e^2)*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)*sqrt(d)))/d^(3/2), -1/8*(8*a*
sqrt(-c)*sqrt(-d)*d*arctan(1/2*(e*x + 2*c)/(sqrt(d*x^2 + e*x + c)*sqrt(-c))) - 2
*(2*b*d*x + 4*a*d + b*e)*sqrt(d*x^2 + e*x + c)*sqrt(-d) - (4*b*c*d + 4*a*d*e - b
*e^2)*arctan(1/2*(2*d*x + e)*sqrt(-d)/(sqrt(d*x^2 + e*x + c)*d)))/(sqrt(-d)*d)]

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

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,x)

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

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,x, algorithm="giac")

[Out]

Exception raised: TypeError