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

Optimal. Leaf size=317 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (2 d x+e) \sqrt{c+d x^2+e x} \left (2 a d \left (4 c d-5 e^2\right )-b \left (12 c d e-7 e^3\right )\right )}{128 d^4 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left (c+d x^2+e x\right )^{3/2} \left (-6 d x (10 a d-7 b e)+50 a d e+32 b c d-35 b e^2\right )}{240 d^3 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left (4 c d-e^2\right ) \left (8 a c d^2-10 a d e^2-12 b c d e+7 b e^3\right ) \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{256 d^{9/2} (a+b x)}+\frac{b x^2 \sqrt{a^2+2 a b x+b^2 x^2} \left (c+d x^2+e x\right )^{3/2}}{5 d (a+b x)} \]

[Out]

-((2*a*d*(4*c*d - 5*e^2) - b*(12*c*d*e - 7*e^3))*(e + 2*d*x)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]*Sqrt[c + e*x + d*x^2])/(128*d^4*(a + b*x)) + (b*x^2*Sqrt[a^2 + 2*a*b*
x + b^2*x^2]*(c + e*x + d*x^2)^(3/2))/(5*d*(a + b*x)) - ((32*b*c*d + 50*a*d*e -
35*b*e^2 - 6*d*(10*a*d - 7*b*e)*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(c + e*x + d*x^
2)^(3/2))/(240*d^3*(a + b*x)) - ((4*c*d - e^2)*(8*a*c*d^2 - 12*b*c*d*e - 10*a*d*
e^2 + 7*b*e^3)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt
[c + e*x + d*x^2])])/(256*d^(9/2)*(a + b*x))

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Rubi [A]  time = 0.833034, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left (c+d x^2+e x\right )^{3/2} \left (-6 d x (10 a d-7 b e)+50 a d e+32 b c d-35 b e^2\right )}{240 d^3 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left (4 c d-e^2\right ) \left (8 a c d^2-10 a d e^2-12 b c d e+7 b e^3\right ) \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{256 d^{9/2} (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (2 d x+e) \sqrt{c+d x^2+e x} \left (8 a c d^2-10 a d e^2-12 b c d e+7 b e^3\right )}{128 d^4 (a+b x)}+\frac{b x^2 \sqrt{a^2+2 a b x+b^2 x^2} \left (c+d x^2+e x\right )^{3/2}}{5 d (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

-((8*a*c*d^2 - 12*b*c*d*e - 10*a*d*e^2 + 7*b*e^3)*(e + 2*d*x)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]*Sqrt[c + e*x + d*x^2])/(128*d^4*(a + b*x)) + (b*x^2*Sqrt[a^2 + 2*a*b
*x + b^2*x^2]*(c + e*x + d*x^2)^(3/2))/(5*d*(a + b*x)) - ((32*b*c*d + 50*a*d*e -
35*b*e^2 - 6*d*(10*a*d - 7*b*e)*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(c + e*x + d*x
^2)^(3/2))/(240*d^3*(a + b*x)) - ((4*c*d - e^2)*(8*a*c*d^2 - 12*b*c*d*e - 10*a*d
*e^2 + 7*b*e^3)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqr
t[c + e*x + d*x^2])])/(256*d^(9/2)*(a + b*x))

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Rubi in Sympy [A]  time = 71.2277, size = 313, normalized size = 0.99 \[ \frac{b x^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (c + d x^{2} + e x\right )^{\frac{3}{2}}}{5 d \left (a + b x\right )} + \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (c + d x^{2} + e x\right )^{\frac{3}{2}} \left (- 25 a d e - 16 b c d + \frac{35 b e^{2}}{2} + 3 d x \left (10 a d - 7 b e\right )\right )}{120 d^{3} \left (a + b x\right )} - \frac{\left (2 d x + e\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \sqrt{c + d x^{2} + e x} \left (8 a c d^{2} - 10 a d e^{2} - 12 b c d e + 7 b e^{3}\right )}{128 d^{4} \left (a + b x\right )} + \frac{\left (- 4 c d + e^{2}\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (8 a c d^{2} - 10 a d e^{2} - 12 b c d e + 7 b e^{3}\right ) \operatorname{atanh}{\left (\frac{2 d x + e}{2 \sqrt{d} \sqrt{c + d x^{2} + e x}} \right )}}{256 d^{\frac{9}{2}} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(c + d*x**2 + e*x)**(3/2)/(5*d*(a + b*x)
) + sqrt(a**2 + 2*a*b*x + b**2*x**2)*(c + d*x**2 + e*x)**(3/2)*(-25*a*d*e - 16*b
*c*d + 35*b*e**2/2 + 3*d*x*(10*a*d - 7*b*e))/(120*d**3*(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*c*d**2 - 10*a*d*e**
2 - 12*b*c*d*e + 7*b*e**3)/(128*d**4*(a + b*x)) + (-4*c*d + e**2)*sqrt(a**2 + 2*
a*b*x + b**2*x**2)*(8*a*c*d**2 - 10*a*d*e**2 - 12*b*c*d*e + 7*b*e**3)*atanh((2*d
*x + e)/(2*sqrt(d)*sqrt(c + d*x**2 + e*x)))/(256*d**(9/2)*(a + b*x))

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Mathematica [A]  time = 0.431116, size = 236, normalized size = 0.74 \[ \frac{\sqrt{(a+b x)^2} \left (\frac{2 \sqrt{c+x (d x+e)} \left (10 a d \left (4 c d (6 d x-13 e)+48 d^3 x^3+8 d^2 e x^2-10 d e^2 x+15 e^3\right )+b \left (-256 c^2 d^2+4 c d \left (32 d^2 x^2-58 d e x+115 e^2\right )+384 d^4 x^4+48 d^3 e x^3-56 d^2 e^2 x^2+70 d e^3 x-105 e^4\right )\right )}{15 d^4}-\frac{\left (4 c d-e^2\right ) \left (2 a d \left (4 c d-5 e^2\right )+b \left (7 e^3-12 c d e\right )\right ) \log \left (2 \sqrt{d} \sqrt{c+x (d x+e)}+2 d x+e\right )}{d^{9/2}}\right )}{256 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*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[c + x*(e + d*x)]*(10*a*d*(15*e^3 - 10*d*e^2*x + 8*d^
2*e*x^2 + 48*d^3*x^3 + 4*c*d*(-13*e + 6*d*x)) + b*(-256*c^2*d^2 - 105*e^4 + 70*d
*e^3*x - 56*d^2*e^2*x^2 + 48*d^3*e*x^3 + 384*d^4*x^4 + 4*c*d*(115*e^2 - 58*d*e*x
+ 32*d^2*x^2))))/(15*d^4) - ((4*c*d - e^2)*(2*a*d*(4*c*d - 5*e^2) + b*(-12*c*d*
e + 7*e^3))*Log[e + 2*d*x + 2*Sqrt[d]*Sqrt[c + x*(e + d*x)]])/d^(9/2)))/(256*(a
+ b*x))

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Maple [C]  time = 0.022, size = 532, normalized size = 1.7 \[ -{\frac{{\it csgn} \left ( bx+a \right ) }{3840} \left ( -768\,b{x}^{2} \left ( d{x}^{2}+ex+c \right ) ^{3/2}{d}^{15/2}-960\,ax \left ( d{x}^{2}+ex+c \right ) ^{3/2}{d}^{15/2}+672\,bex \left ( d{x}^{2}+ex+c \right ) ^{3/2}{d}^{13/2}+800\,ae \left ( d{x}^{2}+ex+c \right ) ^{3/2}{d}^{13/2}+512\,bc \left ( d{x}^{2}+ex+c \right ) ^{3/2}{d}^{13/2}-560\,b{e}^{2} \left ( d{x}^{2}+ex+c \right ) ^{3/2}{d}^{11/2}+480\,ac\sqrt{d{x}^{2}+ex+c}x{d}^{15/2}-600\,a{e}^{2}\sqrt{d{x}^{2}+ex+c}x{d}^{13/2}-720\,bec\sqrt{d{x}^{2}+ex+c}x{d}^{13/2}+420\,b{e}^{3}\sqrt{d{x}^{2}+ex+c}x{d}^{11/2}+240\,ac\sqrt{d{x}^{2}+ex+c}e{d}^{13/2}-300\,a{e}^{3}\sqrt{d{x}^{2}+ex+c}{d}^{11/2}-360\,b{e}^{2}c\sqrt{d{x}^{2}+ex+c}{d}^{11/2}+210\,b{e}^{4}\sqrt{d{x}^{2}+ex+c}{d}^{9/2}+480\,a{c}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){d}^{7}-720\,a{e}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ) c{d}^{6}-720\,be{c}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){d}^{6}+150\,a{e}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){d}^{5}+600\,b{e}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ) c{d}^{5}-105\,b{e}^{5}\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){d}^{4} \right ){d}^{-{\frac{17}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3840*csgn(b*x+a)*(-768*b*x^2*(d*x^2+e*x+c)^(3/2)*d^(15/2)-960*a*x*(d*x^2+e*x+
c)^(3/2)*d^(15/2)+672*b*e*x*(d*x^2+e*x+c)^(3/2)*d^(13/2)+800*a*e*(d*x^2+e*x+c)^(
3/2)*d^(13/2)+512*b*c*(d*x^2+e*x+c)^(3/2)*d^(13/2)-560*b*e^2*(d*x^2+e*x+c)^(3/2)
*d^(11/2)+480*a*c*(d*x^2+e*x+c)^(1/2)*x*d^(15/2)-600*a*e^2*(d*x^2+e*x+c)^(1/2)*x
*d^(13/2)-720*b*e*c*(d*x^2+e*x+c)^(1/2)*x*d^(13/2)+420*b*e^3*(d*x^2+e*x+c)^(1/2)
*x*d^(11/2)+240*a*c*(d*x^2+e*x+c)^(1/2)*e*d^(13/2)-300*a*e^3*(d*x^2+e*x+c)^(1/2)
*d^(11/2)-360*b*e^2*c*(d*x^2+e*x+c)^(1/2)*d^(11/2)+210*b*e^4*(d*x^2+e*x+c)^(1/2)
*d^(9/2)+480*a*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^7-7
20*a*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^6-720*b*e*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^6+150*a*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^5+600*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))*c*d^5-105*b*e^5*ln(1/2*(2*(d*x^2+e*x+c)^(
1/2)*d^(1/2)+2*d*x+e)/d^(1/2))*d^4)/d^(17/2)

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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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.353961, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (384 \, b d^{4} x^{4} - 256 \, b c^{2} d^{2} - 520 \, a c d^{2} e + 460 \, b c d e^{2} + 150 \, a d e^{3} - 105 \, b e^{4} + 48 \,{\left (10 \, a d^{4} + b d^{3} e\right )} x^{3} + 8 \,{\left (16 \, b c d^{3} + 10 \, a d^{3} e - 7 \, b d^{2} e^{2}\right )} x^{2} + 2 \,{\left (120 \, a c d^{3} - 116 \, b c d^{2} e - 50 \, a d^{2} e^{2} + 35 \, b d e^{3}\right )} x\right )} \sqrt{d x^{2} + e x + c} \sqrt{d} - 15 \,{\left (32 \, a c^{2} d^{3} - 48 \, b c^{2} d^{2} e - 48 \, a c d^{2} e^{2} + 40 \, b c d e^{3} + 10 \, a d e^{4} - 7 \, b e^{5}\right )} \log \left (4 \,{\left (2 \, d^{2} x + d e\right )} \sqrt{d x^{2} + e x + c} +{\left (8 \, d^{2} x^{2} + 8 \, d e x + 4 \, c d + e^{2}\right )} \sqrt{d}\right )}{7680 \, d^{\frac{9}{2}}}, \frac{2 \,{\left (384 \, b d^{4} x^{4} - 256 \, b c^{2} d^{2} - 520 \, a c d^{2} e + 460 \, b c d e^{2} + 150 \, a d e^{3} - 105 \, b e^{4} + 48 \,{\left (10 \, a d^{4} + b d^{3} e\right )} x^{3} + 8 \,{\left (16 \, b c d^{3} + 10 \, a d^{3} e - 7 \, b d^{2} e^{2}\right )} x^{2} + 2 \,{\left (120 \, a c d^{3} - 116 \, b c d^{2} e - 50 \, a d^{2} e^{2} + 35 \, b d e^{3}\right )} x\right )} \sqrt{d x^{2} + e x + c} \sqrt{-d} - 15 \,{\left (32 \, a c^{2} d^{3} - 48 \, b c^{2} d^{2} e - 48 \, a c d^{2} e^{2} + 40 \, b c d e^{3} + 10 \, a d e^{4} - 7 \, b e^{5}\right )} \arctan \left (\frac{{\left (2 \, d x + e\right )} \sqrt{-d}}{2 \, \sqrt{d x^{2} + e x + c} d}\right )}{3840 \, \sqrt{-d} d^{4}}\right ] \]

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^2,x, algorithm="fricas")

[Out]

[1/7680*(4*(384*b*d^4*x^4 - 256*b*c^2*d^2 - 520*a*c*d^2*e + 460*b*c*d*e^2 + 150*
a*d*e^3 - 105*b*e^4 + 48*(10*a*d^4 + b*d^3*e)*x^3 + 8*(16*b*c*d^3 + 10*a*d^3*e -
7*b*d^2*e^2)*x^2 + 2*(120*a*c*d^3 - 116*b*c*d^2*e - 50*a*d^2*e^2 + 35*b*d*e^3)*
x)*sqrt(d*x^2 + e*x + c)*sqrt(d) - 15*(32*a*c^2*d^3 - 48*b*c^2*d^2*e - 48*a*c*d^
2*e^2 + 40*b*c*d*e^3 + 10*a*d*e^4 - 7*b*e^5)*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^(9/2), 1/3840*(2*(384
*b*d^4*x^4 - 256*b*c^2*d^2 - 520*a*c*d^2*e + 460*b*c*d*e^2 + 150*a*d*e^3 - 105*b
*e^4 + 48*(10*a*d^4 + b*d^3*e)*x^3 + 8*(16*b*c*d^3 + 10*a*d^3*e - 7*b*d^2*e^2)*x
^2 + 2*(120*a*c*d^3 - 116*b*c*d^2*e - 50*a*d^2*e^2 + 35*b*d*e^3)*x)*sqrt(d*x^2 +
e*x + c)*sqrt(-d) - 15*(32*a*c^2*d^3 - 48*b*c^2*d^2*e - 48*a*c*d^2*e^2 + 40*b*c
*d*e^3 + 10*a*d*e^4 - 7*b*e^5)*arctan(1/2*(2*d*x + e)*sqrt(-d)/(sqrt(d*x^2 + e*x
+ c)*d)))/(sqrt(-d)*d^4)]

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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**2*((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 = 0.289018, size = 497, normalized size = 1.57 \[ \frac{1}{1920} \, \sqrt{d x^{2} + x e + c}{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, b x{\rm sign}\left (b x + a\right ) + \frac{10 \, a d^{4}{\rm sign}\left (b x + a\right ) + b d^{3} e{\rm sign}\left (b x + a\right )}{d^{4}}\right )} x + \frac{16 \, b c d^{3}{\rm sign}\left (b x + a\right ) + 10 \, a d^{3} e{\rm sign}\left (b x + a\right ) - 7 \, b d^{2} e^{2}{\rm sign}\left (b x + a\right )}{d^{4}}\right )} x + \frac{120 \, a c d^{3}{\rm sign}\left (b x + a\right ) - 116 \, b c d^{2} e{\rm sign}\left (b x + a\right ) - 50 \, a d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 35 \, b d e^{3}{\rm sign}\left (b x + a\right )}{d^{4}}\right )} x - \frac{256 \, b c^{2} d^{2}{\rm sign}\left (b x + a\right ) + 520 \, a c d^{2} e{\rm sign}\left (b x + a\right ) - 460 \, b c d e^{2}{\rm sign}\left (b x + a\right ) - 150 \, a d e^{3}{\rm sign}\left (b x + a\right ) + 105 \, b e^{4}{\rm sign}\left (b x + a\right )}{d^{4}}\right )} + \frac{{\left (32 \, a c^{2} d^{3}{\rm sign}\left (b x + a\right ) - 48 \, b c^{2} d^{2} e{\rm sign}\left (b x + a\right ) - 48 \, a c d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 40 \, b c d e^{3}{\rm sign}\left (b x + a\right ) + 10 \, a d e^{4}{\rm sign}\left (b x + a\right ) - 7 \, b e^{5}{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + x e + c}\right )} \sqrt{d} - e \right |}\right )}{256 \, d^{\frac{9}{2}}} \]

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^2,x, algorithm="giac")

[Out]

1/1920*sqrt(d*x^2 + x*e + c)*(2*(4*(6*(8*b*x*sign(b*x + a) + (10*a*d^4*sign(b*x
+ a) + b*d^3*e*sign(b*x + a))/d^4)*x + (16*b*c*d^3*sign(b*x + a) + 10*a*d^3*e*si
gn(b*x + a) - 7*b*d^2*e^2*sign(b*x + a))/d^4)*x + (120*a*c*d^3*sign(b*x + a) - 1
16*b*c*d^2*e*sign(b*x + a) - 50*a*d^2*e^2*sign(b*x + a) + 35*b*d*e^3*sign(b*x +
a))/d^4)*x - (256*b*c^2*d^2*sign(b*x + a) + 520*a*c*d^2*e*sign(b*x + a) - 460*b*
c*d*e^2*sign(b*x + a) - 150*a*d*e^3*sign(b*x + a) + 105*b*e^4*sign(b*x + a))/d^4
) + 1/256*(32*a*c^2*d^3*sign(b*x + a) - 48*b*c^2*d^2*e*sign(b*x + a) - 48*a*c*d^
2*e^2*sign(b*x + a) + 40*b*c*d*e^3*sign(b*x + a) + 10*a*d*e^4*sign(b*x + a) - 7*
b*e^5*sign(b*x + a))*ln(abs(-2*(sqrt(d)*x - sqrt(d*x^2 + x*e + c))*sqrt(d) - e))
/d^(9/2)