3.14 \(\int \sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx\)

Optimal. Leaf size=283 \[ -\frac{5 b e \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{24 d^2 \left (a+b x^2\right )}-\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (4 c d-e^2\right ) \left (-16 a d^2+4 b c d-5 b e^2\right ) \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{128 d^{7/2} \left (a+b x^2\right )}-\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} (2 d x+e) \sqrt{c+d x^2+e x} \left (-16 a d^2+4 b c d-5 b e^2\right )}{64 d^3 \left (a+b x^2\right )}+\frac{b x \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{4 d \left (a+b x^2\right )} \]

[Out]

-((4*b*c*d - 16*a*d^2 - 5*b*e^2)*(e + 2*d*x)*Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*
a*b*x^2 + b^2*x^4])/(64*d^3*(a + b*x^2)) - (5*b*e*(c + e*x + d*x^2)^(3/2)*Sqrt[a
^2 + 2*a*b*x^2 + b^2*x^4])/(24*d^2*(a + b*x^2)) + (b*x*(c + e*x + d*x^2)^(3/2)*S
qrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(4*d*(a + b*x^2)) - ((4*c*d - e^2)*(4*b*c*d - 16
*a*d^2 - 5*b*e^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]
*Sqrt[c + e*x + d*x^2])])/(128*d^(7/2)*(a + b*x^2))

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Rubi [A]  time = 0.705794, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162 \[ -\frac{5 b e \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{24 d^2 \left (a+b x^2\right )}-\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (4 c d-e^2\right ) \left (-16 a d^2+4 b c d-5 b e^2\right ) \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{128 d^{7/2} \left (a+b x^2\right )}-\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} (2 d x+e) \sqrt{c+d x^2+e x} \left (-16 a d^2+4 b c d-5 b e^2\right )}{64 d^3 \left (a+b x^2\right )}+\frac{b x \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{4 d \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

-((4*b*c*d - 16*a*d^2 - 5*b*e^2)*(e + 2*d*x)*Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*
a*b*x^2 + b^2*x^4])/(64*d^3*(a + b*x^2)) - (5*b*e*(c + e*x + d*x^2)^(3/2)*Sqrt[a
^2 + 2*a*b*x^2 + b^2*x^4])/(24*d^2*(a + b*x^2)) + (b*x*(c + e*x + d*x^2)^(3/2)*S
qrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(4*d*(a + b*x^2)) - ((4*c*d - e^2)*(4*b*c*d - 16
*a*d^2 - 5*b*e^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]
*Sqrt[c + e*x + d*x^2])])/(128*d^(7/2)*(a + b*x^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.321261, size = 166, normalized size = 0.59 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (2 \sqrt{d} \sqrt{c+x (d x+e)} \left (48 a d^2 (2 d x+e)+b \left (4 c d (6 d x-13 e)+48 d^3 x^3+8 d^2 e x^2-10 d e^2 x+15 e^3\right )\right )-3 \left (4 c d-e^2\right ) \left (-16 a d^2+4 b c d-5 b e^2\right ) \log \left (2 \sqrt{d} \sqrt{c+x (d x+e)}+2 d x+e\right )\right )}{384 d^{7/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

(Sqrt[(a + b*x^2)^2]*(2*Sqrt[d]*Sqrt[c + x*(e + d*x)]*(48*a*d^2*(e + 2*d*x) + b*
(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))) - 3*(4
*c*d - e^2)*(4*b*c*d - 16*a*d^2 - 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^2))

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Maple [A]  time = 0.011, size = 375, normalized size = 1.3 \[{\frac{1}{384\,b{x}^{2}+384\,a}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( 96\,bx \left ( d{x}^{2}+ex+c \right ) ^{3/2}{d}^{11/2}-80\,be \left ( d{x}^{2}+ex+c \right ) ^{3/2}{d}^{9/2}+192\,a\sqrt{d{x}^{2}+ex+c}x{d}^{13/2}-48\,bcx\sqrt{d{x}^{2}+ex+c}{d}^{11/2}+60\,b{e}^{2}x\sqrt{d{x}^{2}+ex+c}{d}^{9/2}+192\,a\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ) c{d}^{6}+96\,a\sqrt{d{x}^{2}+ex+c}e{d}^{11/2}-24\,bc\sqrt{d{x}^{2}+ex+c}e{d}^{9/2}+30\,b{e}^{3}\sqrt{d{x}^{2}+ex+c}{d}^{7/2}-48\,a\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){e}^{2}{d}^{5}-48\,b{c}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){d}^{5}+72\,b{e}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ) c{d}^{4}-15\,b{e}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){d}^{3} \right ){d}^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/384*((b*x^2+a)^2)^(1/2)*(96*b*x*(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)+192*a*(d*x^2+e*x+c)^(1/2)*x*d^(13/2)-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)+192*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^6+96*a*(d*x^2+e*x+c)^(1/2)*e*d^(11/2)-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)-48*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^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+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)/(b*x^2+a)/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^2 + a)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.297719, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, b d^{3} x^{3} + 8 \, b d^{2} e x^{2} + 15 \, b e^{3} - 4 \,{\left (13 \, b c d - 12 \, a d^{2}\right )} e + 2 \,{\left (12 \, b c d^{2} + 48 \, a d^{3} - 5 \, b d e^{2}\right )} x\right )} \sqrt{d x^{2} + e x + c} \sqrt{d} + 3 \,{\left (16 \, b c^{2} d^{2} - 64 \, a c d^{3} + 5 \, b e^{4} - 8 \,{\left (3 \, b c d - 2 \, a d^{2}\right )} e^{2}\right )} \log \left (-4 \,{\left (2 \, d^{2} x + d e\right )} \sqrt{d x^{2} + e x + c} +{\left (8 \, d^{2} x^{2} + 8 \, d e x + 4 \, c d + e^{2}\right )} \sqrt{d}\right )}{768 \, d^{\frac{7}{2}}}, \frac{2 \,{\left (48 \, b d^{3} x^{3} + 8 \, b d^{2} e x^{2} + 15 \, b e^{3} - 4 \,{\left (13 \, b c d - 12 \, a d^{2}\right )} e + 2 \,{\left (12 \, b c d^{2} + 48 \, a d^{3} - 5 \, b d e^{2}\right )} x\right )} \sqrt{d x^{2} + e x + c} \sqrt{-d} - 3 \,{\left (16 \, b c^{2} d^{2} - 64 \, a c d^{3} + 5 \, b e^{4} - 8 \,{\left (3 \, b c d - 2 \, a d^{2}\right )} e^{2}\right )} \arctan \left (\frac{{\left (2 \, d x + e\right )} \sqrt{-d}}{2 \, \sqrt{d x^{2} + e x + c} d}\right )}{384 \, \sqrt{-d} d^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(4*(48*b*d^3*x^3 + 8*b*d^2*e*x^2 + 15*b*e^3 - 4*(13*b*c*d - 12*a*d^2)*e +
2*(12*b*c*d^2 + 48*a*d^3 - 5*b*d*e^2)*x)*sqrt(d*x^2 + e*x + c)*sqrt(d) + 3*(16*
b*c^2*d^2 - 64*a*c*d^3 + 5*b*e^4 - 8*(3*b*c*d - 2*a*d^2)*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^(7/
2), 1/384*(2*(48*b*d^3*x^3 + 8*b*d^2*e*x^2 + 15*b*e^3 - 4*(13*b*c*d - 12*a*d^2)*
e + 2*(12*b*c*d^2 + 48*a*d^3 - 5*b*d*e^2)*x)*sqrt(d*x^2 + e*x + c)*sqrt(-d) - 3*
(16*b*c^2*d^2 - 64*a*c*d^3 + 5*b*e^4 - 8*(3*b*c*d - 2*a*d^2)*e^2)*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((d*x**2+e*x+c)**(1/2)*((b*x**2+a)**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.28478, size = 358, normalized size = 1.27 \[ \frac{1}{192} \, \sqrt{d x^{2} + x e + c}{\left (2 \,{\left (4 \,{\left (6 \, b x{\rm sign}\left (b x^{2} + a\right ) + \frac{b e{\rm sign}\left (b x^{2} + a\right )}{d}\right )} x + \frac{12 \, b c d^{2}{\rm sign}\left (b x^{2} + a\right ) + 48 \, a d^{3}{\rm sign}\left (b x^{2} + a\right ) - 5 \, b d e^{2}{\rm sign}\left (b x^{2} + a\right )}{d^{3}}\right )} x - \frac{52 \, b c d e{\rm sign}\left (b x^{2} + a\right ) - 48 \, a d^{2} e{\rm sign}\left (b x^{2} + a\right ) - 15 \, b e^{3}{\rm sign}\left (b x^{2} + a\right )}{d^{3}}\right )} + \frac{{\left (16 \, b c^{2} d^{2}{\rm sign}\left (b x^{2} + a\right ) - 64 \, a c d^{3}{\rm sign}\left (b x^{2} + a\right ) - 24 \, b c d e^{2}{\rm sign}\left (b x^{2} + a\right ) + 16 \, a d^{2} e^{2}{\rm sign}\left (b x^{2} + a\right ) + 5 \, b e^{4}{\rm sign}\left (b x^{2} + a\right )\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + x e + c}\right )} \sqrt{d} - e \right |}\right )}{128 \, d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*sqrt(d*x^2 + x*e + c)*(2*(4*(6*b*x*sign(b*x^2 + a) + b*e*sign(b*x^2 + a)/d
)*x + (12*b*c*d^2*sign(b*x^2 + a) + 48*a*d^3*sign(b*x^2 + a) - 5*b*d*e^2*sign(b*
x^2 + a))/d^3)*x - (52*b*c*d*e*sign(b*x^2 + a) - 48*a*d^2*e*sign(b*x^2 + a) - 15
*b*e^3*sign(b*x^2 + a))/d^3) + 1/128*(16*b*c^2*d^2*sign(b*x^2 + a) - 64*a*c*d^3*
sign(b*x^2 + a) - 24*b*c*d*e^2*sign(b*x^2 + a) + 16*a*d^2*e^2*sign(b*x^2 + a) +
5*b*e^4*sign(b*x^2 + a))*ln(abs(-2*(sqrt(d)*x - sqrt(d*x^2 + x*e + c))*sqrt(d) -
e))/d^(7/2)