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

Optimal. Leaf size=286 \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2+e x} \left (8 a d^2-2 b d e x-b e^2\right )}{8 d^2 \left (a+b x^2\right )}+\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4} \left (8 a d^2-b \left (4 c d-e^2\right )\right ) \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{16 d^{5/2} \left (a+b x^2\right )}+\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{3 d \left (a+b x^2\right )}-\frac{a \sqrt{c} \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right )}{a+b x^2} \]

[Out]

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

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Rubi [A]  time = 1.53886, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175 \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2+e x} \left (8 a d^2-2 b d e x-b e^2\right )}{8 d^2 \left (a+b x^2\right )}+\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4} \left (8 a d^2-b \left (4 c d-e^2\right )\right ) \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{16 d^{5/2} \left (a+b x^2\right )}+\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{3 d \left (a+b x^2\right )}-\frac{a \sqrt{c} \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right )}{a+b x^2} \]

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,x]

[Out]

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

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

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,x)

[Out]

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

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Mathematica [A]  time = 0.355841, size = 302, normalized size = 1.06 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (48 a d^{5/2} \sqrt{c+x (d x+e)}-48 a \sqrt{c} d^{5/2} \log \left (2 \sqrt{c} \sqrt{c+x (d x+e)}+2 c+e x\right )+48 a \sqrt{c} d^{5/2} \log (x)+24 a d^2 e \log \left (2 \sqrt{d} \sqrt{c+x (d x+e)}+2 d x+e\right )+16 b d^{5/2} x^2 \sqrt{c+x (d x+e)}+4 b d^{3/2} e x \sqrt{c+x (d x+e)}+16 b c d^{3/2} \sqrt{c+x (d x+e)}+3 b e^3 \log \left (2 \sqrt{d} \sqrt{c+x (d x+e)}+2 d x+e\right )-6 b \sqrt{d} e^2 \sqrt{c+x (d x+e)}-12 b c d e \log \left (2 \sqrt{d} \sqrt{c+x (d x+e)}+2 d x+e\right )\right )}{48 d^{5/2} \left (a+b x^2\right )} \]

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,x]

[Out]

(Sqrt[(a + b*x^2)^2]*(16*b*c*d^(3/2)*Sqrt[c + x*(e + d*x)] + 48*a*d^(5/2)*Sqrt[c
+ x*(e + d*x)] - 6*b*Sqrt[d]*e^2*Sqrt[c + x*(e + d*x)] + 4*b*d^(3/2)*e*x*Sqrt[c
+ x*(e + d*x)] + 16*b*d^(5/2)*x^2*Sqrt[c + x*(e + d*x)] + 48*a*Sqrt[c]*d^(5/2)*
Log[x] - 48*a*Sqrt[c]*d^(5/2)*Log[2*c + e*x + 2*Sqrt[c]*Sqrt[c + x*(e + d*x)]] -
12*b*c*d*e*Log[e + 2*d*x + 2*Sqrt[d]*Sqrt[c + x*(e + d*x)]] + 24*a*d^2*e*Log[e
+ 2*d*x + 2*Sqrt[d]*Sqrt[c + x*(e + d*x)]] + 3*b*e^3*Log[e + 2*d*x + 2*Sqrt[d]*S
qrt[c + x*(e + d*x)]]))/(48*d^(5/2)*(a + b*x^2))

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Maple [A]  time = 0.014, size = 253, normalized size = 0.9 \[{\frac{1}{48\,b{x}^{2}+48\,a}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( -48\,a\sqrt{c}\ln \left ({\frac{2\,c+ex+2\,\sqrt{c}\sqrt{d{x}^{2}+ex+c}}{x}} \right ){d}^{9/2}+16\,b \left ( d{x}^{2}+ex+c \right ) ^{3/2}{d}^{7/2}-12\,be\sqrt{d{x}^{2}+ex+c}x{d}^{7/2}+24\,ae\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){d}^{4}+48\,a\sqrt{d{x}^{2}+ex+c}{d}^{9/2}-6\,b{e}^{2}\sqrt{d{x}^{2}+ex+c}{d}^{5/2}-12\,be\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ) c{d}^{3}+3\,b{e}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){d}^{2} \right ){d}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.905756, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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,x, algorithm="fricas")

[Out]

[1/96*(48*a*sqrt(c)*d^(5/2)*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*(8*b*d^2*x^2 + 2*b*d*e*x + 8*b*c*d
+ 24*a*d^2 - 3*b*e^2)*sqrt(d*x^2 + e*x + c)*sqrt(d) + 3*(b*e^3 - 4*(b*c*d - 2*a*
d^2)*e)*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^(5/2), 1/48*(24*a*sqrt(c)*sqrt(-d)*d^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) + 2*(8
*b*d^2*x^2 + 2*b*d*e*x + 8*b*c*d + 24*a*d^2 - 3*b*e^2)*sqrt(d*x^2 + e*x + c)*sqr
t(-d) + 3*(b*e^3 - 4*(b*c*d - 2*a*d^2)*e)*arctan(1/2*(2*d*x + e)*sqrt(-d)/(sqrt(
d*x^2 + e*x + c)*d)))/(sqrt(-d)*d^2), -1/96*(96*a*sqrt(-c)*d^(5/2)*arctan(1/2*(e
*x + 2*c)/(sqrt(d*x^2 + e*x + c)*sqrt(-c))) - 4*(8*b*d^2*x^2 + 2*b*d*e*x + 8*b*c
*d + 24*a*d^2 - 3*b*e^2)*sqrt(d*x^2 + e*x + c)*sqrt(d) - 3*(b*e^3 - 4*(b*c*d - 2
*a*d^2)*e)*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^(5/2), -1/48*(48*a*sqrt(-c)*sqrt(-d)*d^2*arctan(1/2*(e*
x + 2*c)/(sqrt(d*x^2 + e*x + c)*sqrt(-c))) - 2*(8*b*d^2*x^2 + 2*b*d*e*x + 8*b*c*
d + 24*a*d^2 - 3*b*e^2)*sqrt(d*x^2 + e*x + c)*sqrt(-d) - 3*(b*e^3 - 4*(b*c*d - 2
*a*d^2)*e)*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(-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,x)

[Out]

Timed out

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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^2 + a)^2)/x,x, algorithm="giac")

[Out]

Exception raised: TypeError