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

Optimal. Leaf size=227 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left (4 c d-e^2\right ) \left (8 a d e+4 b c d-5 b e^2\right ) \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{128 d^{7/2} (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (2 d x+e) \sqrt{c+d x^2+e x} \left (8 a d e+4 b c d-5 b e^2\right )}{64 d^3 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left (c+d x^2+e x\right )^{3/2} (8 a d+6 b d x-5 b e)}{24 d^2 (a+b x)} \]

[Out]

-((4*b*c*d + 8*a*d*e - 5*b*e^2)*(e + 2*d*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c
+ e*x + d*x^2])/(64*d^3*(a + b*x)) + ((8*a*d - 5*b*e + 6*b*d*x)*Sqrt[a^2 + 2*a*
b*x + b^2*x^2]*(c + e*x + d*x^2)^(3/2))/(24*d^2*(a + b*x)) - ((4*c*d - e^2)*(4*b
*c*d + 8*a*d*e - 5*b*e^2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(e + 2*d*x)/(2*S
qrt[d]*Sqrt[c + e*x + d*x^2])])/(128*d^(7/2)*(a + b*x))

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Rubi [A]  time = 0.376417, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.139 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left (4 c d-e^2\right ) \left (8 a d e+4 b c d-5 b e^2\right ) \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{128 d^{7/2} (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (2 d x+e) \sqrt{c+d x^2+e x} \left (8 a d e+4 b c d-5 b e^2\right )}{64 d^3 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left (c+d x^2+e x\right )^{3/2} (8 a d+6 b d x-5 b e)}{24 d^2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

-((4*b*c*d + 8*a*d*e - 5*b*e^2)*(e + 2*d*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c
+ e*x + d*x^2])/(64*d^3*(a + b*x)) + ((8*a*d - 5*b*e + 6*b*d*x)*Sqrt[a^2 + 2*a*
b*x + b^2*x^2]*(c + e*x + d*x^2)^(3/2))/(24*d^2*(a + b*x)) - ((4*c*d - e^2)*(4*b
*c*d + 8*a*d*e - 5*b*e^2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(e + 2*d*x)/(2*S
qrt[d]*Sqrt[c + e*x + d*x^2])])/(128*d^(7/2)*(a + b*x))

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Rubi in Sympy [A]  time = 47.4351, size = 221, normalized size = 0.97 \[ \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (c + d x^{2} + e x\right )^{\frac{3}{2}} \left (8 a d + 6 b d x - 5 b e\right )}{24 d^{2} \left (a + b x\right )} + \frac{\left (2 d x + e\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \sqrt{c + d x^{2} + e x} \left (- 8 a d e - 4 b c d + 5 b e^{2}\right )}{64 d^{3} \left (a + b x\right )} - \frac{\left (- 4 c d + e^{2}\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (- 8 a d e - 4 b c d + 5 b e^{2}\right ) \operatorname{atanh}{\left (\frac{2 d x + e}{2 \sqrt{d} \sqrt{c + d x^{2} + e x}} \right )}}{128 d^{\frac{7}{2}} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a**2 + 2*a*b*x + b**2*x**2)*(c + d*x**2 + e*x)**(3/2)*(8*a*d + 6*b*d*x - 5*
b*e)/(24*d**2*(a + b*x)) + (2*d*x + e)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*sqrt(c +
d*x**2 + e*x)*(-8*a*d*e - 4*b*c*d + 5*b*e**2)/(64*d**3*(a + b*x)) - (-4*c*d + e
**2)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(-8*a*d*e - 4*b*c*d + 5*b*e**2)*atanh((2*d
*x + e)/(2*sqrt(d)*sqrt(c + d*x**2 + e*x)))/(128*d**(7/2)*(a + b*x))

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Mathematica [A]  time = 0.35998, size = 176, normalized size = 0.78 \[ \frac{\sqrt{(a+b x)^2} \left (2 \sqrt{d} \sqrt{c+x (d x+e)} \left (8 a d \left (8 c d+8 d^2 x^2+2 d e x-3 e^2\right )+b \left (4 c d (6 d x-13 e)+48 d^3 x^3+8 d^2 e x^2-10 d e^2 x+15 e^3\right )\right )-3 \left (4 c d-e^2\right ) \left (8 a d e+4 b c d-5 b e^2\right ) \log \left (2 \sqrt{d} \sqrt{c+x (d x+e)}+2 d x+e\right )\right )}{384 d^{7/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[x*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)]*(8*a*d*(8*c*d - 3*e^2 + 2*d*
e*x + 8*d^2*x^2) + b*(15*e^3 - 10*d*e^2*x + 8*d^2*e*x^2 + 48*d^3*x^3 + 4*c*d*(-1
3*e + 6*d*x))) - 3*(4*c*d - e^2)*(4*b*c*d + 8*a*d*e - 5*b*e^2)*Log[e + 2*d*x + 2
*Sqrt[d]*Sqrt[c + x*(e + d*x)]]))/(384*d^(7/2)*(a + b*x))

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Maple [C]  time = 0.016, size = 383, normalized size = 1.7 \[ -{\frac{{\it csgn} \left ( bx+a \right ) }{384} \left ( -96\,bx \left ( d{x}^{2}+ex+c \right ) ^{3/2}{d}^{11/2}-128\,a \left ( d{x}^{2}+ex+c \right ) ^{3/2}{d}^{11/2}+80\,be \left ( d{x}^{2}+ex+c \right ) ^{3/2}{d}^{9/2}+96\,aex\sqrt{d{x}^{2}+ex+c}{d}^{11/2}+48\,bcx\sqrt{d{x}^{2}+ex+c}{d}^{11/2}-60\,b{e}^{2}x\sqrt{d{x}^{2}+ex+c}{d}^{9/2}+48\,a{e}^{2}\sqrt{d{x}^{2}+ex+c}{d}^{9/2}+24\,bc\sqrt{d{x}^{2}+ex+c}e{d}^{9/2}-30\,b{e}^{3}\sqrt{d{x}^{2}+ex+c}{d}^{7/2}+96\,ae\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ) c{d}^{5}+48\,b{c}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){d}^{5}-24\,a{e}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){d}^{4}-72\,b{e}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ) c{d}^{4}+15\,b{e}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){d}^{3} \right ){d}^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/384*csgn(b*x+a)*(-96*b*x*(d*x^2+e*x+c)^(3/2)*d^(11/2)-128*a*(d*x^2+e*x+c)^(3/
2)*d^(11/2)+80*b*e*(d*x^2+e*x+c)^(3/2)*d^(9/2)+96*a*e*x*(d*x^2+e*x+c)^(1/2)*d^(1
1/2)+48*b*c*x*(d*x^2+e*x+c)^(1/2)*d^(11/2)-60*b*e^2*x*(d*x^2+e*x+c)^(1/2)*d^(9/2
)+48*a*e^2*(d*x^2+e*x+c)^(1/2)*d^(9/2)+24*b*c*(d*x^2+e*x+c)^(1/2)*e*d^(9/2)-30*b
*e^3*(d*x^2+e*x+c)^(1/2)*d^(7/2)+96*a*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^5+48*b*c^2*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^(1/2)+2*d*x+e)/d^
(1/2))*d^5-24*a*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^4-
72*b*e^2*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+15*b*e^4*
ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^(1/2)+2*d*x+e)/d^(1/2))*d^3)/d^(13/2)

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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 = 0.336252, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, b d^{3} x^{3} + 64 \, a c d^{2} - 52 \, b c d e - 24 \, a d e^{2} + 15 \, b e^{3} + 8 \,{\left (8 \, a d^{3} + b d^{2} e\right )} x^{2} + 2 \,{\left (12 \, b c d^{2} + 8 \, a d^{2} e - 5 \, b d e^{2}\right )} x\right )} \sqrt{d x^{2} + e x + c} \sqrt{d} + 3 \,{\left (16 \, b c^{2} d^{2} + 32 \, a c d^{2} e - 24 \, b c d e^{2} - 8 \, a d e^{3} + 5 \, b e^{4}\right )} \log \left (-4 \,{\left (2 \, d^{2} x + d e\right )} \sqrt{d x^{2} + e x + c} +{\left (8 \, d^{2} x^{2} + 8 \, d e x + 4 \, c d + e^{2}\right )} \sqrt{d}\right )}{768 \, d^{\frac{7}{2}}}, \frac{2 \,{\left (48 \, b d^{3} x^{3} + 64 \, a c d^{2} - 52 \, b c d e - 24 \, a d e^{2} + 15 \, b e^{3} + 8 \,{\left (8 \, a d^{3} + b d^{2} e\right )} x^{2} + 2 \,{\left (12 \, b c d^{2} + 8 \, a d^{2} e - 5 \, b d e^{2}\right )} x\right )} \sqrt{d x^{2} + e x + c} \sqrt{-d} - 3 \,{\left (16 \, b c^{2} d^{2} + 32 \, a c d^{2} e - 24 \, b c d e^{2} - 8 \, a d e^{3} + 5 \, b e^{4}\right )} \arctan \left (\frac{{\left (2 \, d x + e\right )} \sqrt{-d}}{2 \, \sqrt{d x^{2} + e x + c} d}\right )}{384 \, \sqrt{-d} d^{3}}\right ] \]

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]

[1/768*(4*(48*b*d^3*x^3 + 64*a*c*d^2 - 52*b*c*d*e - 24*a*d*e^2 + 15*b*e^3 + 8*(8
*a*d^3 + b*d^2*e)*x^2 + 2*(12*b*c*d^2 + 8*a*d^2*e - 5*b*d*e^2)*x)*sqrt(d*x^2 + e
*x + c)*sqrt(d) + 3*(16*b*c^2*d^2 + 32*a*c*d^2*e - 24*b*c*d*e^2 - 8*a*d*e^3 + 5*
b*e^4)*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^(7/2), 1/384*(2*(48*b*d^3*x^3 + 64*a*c*d^2 - 52*b*c*d*e -
24*a*d*e^2 + 15*b*e^3 + 8*(8*a*d^3 + b*d^2*e)*x^2 + 2*(12*b*c*d^2 + 8*a*d^2*e -
5*b*d*e^2)*x)*sqrt(d*x^2 + e*x + c)*sqrt(-d) - 3*(16*b*c^2*d^2 + 32*a*c*d^2*e -
24*b*c*d*e^2 - 8*a*d*e^3 + 5*b*e^4)*arctan(1/2*(2*d*x + e)*sqrt(-d)/(sqrt(d*x^2
+ e*x + c)*d)))/(sqrt(-d)*d^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(x*((b*x+a)**2)**(1/2)*(d*x**2+e*x+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.287873, size = 362, normalized size = 1.59 \[ \frac{1}{192} \, \sqrt{d x^{2} + x e + c}{\left (2 \,{\left (4 \,{\left (6 \, b x{\rm sign}\left (b x + a\right ) + \frac{8 \, a d^{3}{\rm sign}\left (b x + a\right ) + b d^{2} e{\rm sign}\left (b x + a\right )}{d^{3}}\right )} x + \frac{12 \, b c d^{2}{\rm sign}\left (b x + a\right ) + 8 \, a d^{2} e{\rm sign}\left (b x + a\right ) - 5 \, b d e^{2}{\rm sign}\left (b x + a\right )}{d^{3}}\right )} x + \frac{64 \, a c d^{2}{\rm sign}\left (b x + a\right ) - 52 \, b c d e{\rm sign}\left (b x + a\right ) - 24 \, a d e^{2}{\rm sign}\left (b x + a\right ) + 15 \, b e^{3}{\rm sign}\left (b x + a\right )}{d^{3}}\right )} + \frac{{\left (16 \, b c^{2} d^{2}{\rm sign}\left (b x + a\right ) + 32 \, a c d^{2} e{\rm sign}\left (b x + a\right ) - 24 \, b c d e^{2}{\rm sign}\left (b x + a\right ) - 8 \, a d e^{3}{\rm sign}\left (b x + a\right ) + 5 \, b e^{4}{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + x e + c}\right )} \sqrt{d} - e \right |}\right )}{128 \, d^{\frac{7}{2}}} \]

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,x, algorithm="giac")

[Out]

1/192*sqrt(d*x^2 + x*e + c)*(2*(4*(6*b*x*sign(b*x + a) + (8*a*d^3*sign(b*x + a)
+ b*d^2*e*sign(b*x + a))/d^3)*x + (12*b*c*d^2*sign(b*x + a) + 8*a*d^2*e*sign(b*x
+ a) - 5*b*d*e^2*sign(b*x + a))/d^3)*x + (64*a*c*d^2*sign(b*x + a) - 52*b*c*d*e
*sign(b*x + a) - 24*a*d*e^2*sign(b*x + a) + 15*b*e^3*sign(b*x + a))/d^3) + 1/128
*(16*b*c^2*d^2*sign(b*x + a) + 32*a*c*d^2*e*sign(b*x + a) - 24*b*c*d*e^2*sign(b*
x + a) - 8*a*d*e^3*sign(b*x + a) + 5*b*e^4*sign(b*x + a))*ln(abs(-2*(sqrt(d)*x -
sqrt(d*x^2 + x*e + c))*sqrt(d) - e))/d^(7/2)