3.641 \(\int \frac{\left (a+b x^2\right )^2}{x \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=75 \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{\sqrt{c}}-\frac{b \sqrt{c+d x^2} (b c-2 a d)}{d^2}+\frac{b^2 \left (c+d x^2\right )^{3/2}}{3 d^2} \]

[Out]

-((b*(b*c - 2*a*d)*Sqrt[c + d*x^2])/d^2) + (b^2*(c + d*x^2)^(3/2))/(3*d^2) - (a^
2*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/Sqrt[c]

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Rubi [A]  time = 0.198181, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{\sqrt{c}}-\frac{b \sqrt{c+d x^2} (b c-2 a d)}{d^2}+\frac{b^2 \left (c+d x^2\right )^{3/2}}{3 d^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/(x*Sqrt[c + d*x^2]),x]

[Out]

-((b*(b*c - 2*a*d)*Sqrt[c + d*x^2])/d^2) + (b^2*(c + d*x^2)^(3/2))/(3*d^2) - (a^
2*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/Sqrt[c]

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Rubi in Sympy [A]  time = 22.4573, size = 66, normalized size = 0.88 \[ - \frac{a^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{\sqrt{c}} + \frac{b^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{3 d^{2}} + \frac{b \sqrt{c + d x^{2}} \left (2 a d - b c\right )}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.130006, size = 76, normalized size = 1.01 \[ -\frac{a^2 \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{\sqrt{c}}+\frac{a^2 \log (x)}{\sqrt{c}}+\frac{b \sqrt{c+d x^2} \left (6 a d-2 b c+b d x^2\right )}{3 d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2/(x*Sqrt[c + d*x^2]),x]

[Out]

(b*Sqrt[c + d*x^2]*(-2*b*c + 6*a*d + b*d*x^2))/(3*d^2) + (a^2*Log[x])/Sqrt[c] -
(a^2*Log[c + Sqrt[c]*Sqrt[c + d*x^2]])/Sqrt[c]

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Maple [A]  time = 0.015, size = 87, normalized size = 1.2 \[ -{{a}^{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){\frac{1}{\sqrt{c}}}}+{\frac{{b}^{2}{x}^{2}}{3\,d}\sqrt{d{x}^{2}+c}}-{\frac{2\,{b}^{2}c}{3\,{d}^{2}}\sqrt{d{x}^{2}+c}}+2\,{\frac{\sqrt{d{x}^{2}+c}ab}{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a^2/c^(1/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)+1/3*b^2*x^2/d*(d*x^2+c)^(1/2)
-2/3*b^2*c/d^2*(d*x^2+c)^(1/2)+2*a*b/d*(d*x^2+c)^(1/2)

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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((b*x^2 + a)^2/(sqrt(d*x^2 + c)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258776, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} d^{2} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right ) + 2 \,{\left (b^{2} d x^{2} - 2 \, b^{2} c + 6 \, a b d\right )} \sqrt{d x^{2} + c} \sqrt{c}}{6 \, \sqrt{c} d^{2}}, -\frac{3 \, a^{2} d^{2} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) -{\left (b^{2} d x^{2} - 2 \, b^{2} c + 6 \, a b d\right )} \sqrt{d x^{2} + c} \sqrt{-c}}{3 \, \sqrt{-c} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(3*a^2*d^2*log(-((d*x^2 + 2*c)*sqrt(c) - 2*sqrt(d*x^2 + c)*c)/x^2) + 2*(b^2
*d*x^2 - 2*b^2*c + 6*a*b*d)*sqrt(d*x^2 + c)*sqrt(c))/(sqrt(c)*d^2), -1/3*(3*a^2*
d^2*arctan(sqrt(-c)/sqrt(d*x^2 + c)) - (b^2*d*x^2 - 2*b^2*c + 6*a*b*d)*sqrt(d*x^
2 + c)*sqrt(-c))/(sqrt(-c)*d^2)]

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Sympy [A]  time = 21.7103, size = 167, normalized size = 2.23 \[ a^{2} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{c}} \sqrt{c + d x^{2}}} \right )}}{c \sqrt{- \frac{1}{c}}} & \text{for}\: - \frac{1}{c} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{c + d x^{2}} \sqrt{\frac{1}{c}}} \right )}}{c \sqrt{\frac{1}{c}}} & \text{for}\: - \frac{1}{c} < 0 \wedge \frac{1}{c} < \frac{1}{c + d x^{2}} \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{c + d x^{2}} \sqrt{\frac{1}{c}}} \right )}}{c \sqrt{\frac{1}{c}}} & \text{for}\: \frac{1}{c} > \frac{1}{c + d x^{2}} \wedge - \frac{1}{c} < 0 \end{cases}\right ) + \frac{b^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{3 d^{2}} + \frac{b \sqrt{c + d x^{2}} \left (2 a d - b c\right )}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*Piecewise((atan(1/(sqrt(-1/c)*sqrt(c + d*x**2)))/(c*sqrt(-1/c)), -1/c > 0),
(-acoth(1/(sqrt(c + d*x**2)*sqrt(1/c)))/(c*sqrt(1/c)), (-1/c < 0) & (1/c < 1/(c
+ d*x**2))), (-atanh(1/(sqrt(c + d*x**2)*sqrt(1/c)))/(c*sqrt(1/c)), (-1/c < 0)
& (1/c > 1/(c + d*x**2)))) + b**2*(c + d*x**2)**(3/2)/(3*d**2) + b*sqrt(c + d*x*
*2)*(2*a*d - b*c)/d**2

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GIAC/XCAS [A]  time = 0.236444, size = 111, normalized size = 1.48 \[ \frac{a^{2} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} + \frac{{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} d^{4} - 3 \, \sqrt{d x^{2} + c} b^{2} c d^{4} + 6 \, \sqrt{d x^{2} + c} a b d^{5}}{3 \, d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^2*arctan(sqrt(d*x^2 + c)/sqrt(-c))/sqrt(-c) + 1/3*((d*x^2 + c)^(3/2)*b^2*d^4 -
3*sqrt(d*x^2 + c)*b^2*c*d^4 + 6*sqrt(d*x^2 + c)*a*b*d^5)/d^6