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

Optimal. Leaf size=92 \[ a^2 \sqrt{c+d x^2}-a^2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )-\frac{b \left (c+d x^2\right )^{3/2} (b c-2 a d)}{3 d^2}+\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d^2} \]

[Out]

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

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Rubi [A]  time = 0.219607, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ a^2 \sqrt{c+d x^2}-a^2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )-\frac{b \left (c+d x^2\right )^{3/2} (b c-2 a d)}{3 d^2}+\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.5989, size = 82, normalized size = 0.89 \[ - a^{2} \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )} + a^{2} \sqrt{c + d x^{2}} + \frac{b^{2} \left (c + d x^{2}\right )^{\frac{5}{2}}}{5 d^{2}} + \frac{b \left (c + d x^{2}\right )^{\frac{3}{2}} \left (2 a d - b c\right )}{3 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2*sqrt(c)*atanh(sqrt(c + d*x**2)/sqrt(c)) + a**2*sqrt(c + d*x**2) + b**2*(c
+ d*x**2)**(5/2)/(5*d**2) + b*(c + d*x**2)**(3/2)*(2*a*d - b*c)/(3*d**2)

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Mathematica [A]  time = 0.233078, size = 105, normalized size = 1.14 \[ \frac{\sqrt{c+d x^2} \left (15 a^2 d^2+10 a b d \left (c+d x^2\right )+b^2 \left (-2 c^2+c d x^2+3 d^2 x^4\right )\right )}{15 d^2}-a^2 \sqrt{c} \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+a^2 \sqrt{c} \log (x) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x^2]*(15*a^2*d^2 + 10*a*b*d*(c + d*x^2) + b^2*(-2*c^2 + c*d*x^2 + 3*
d^2*x^4)))/(15*d^2) + a^2*Sqrt[c]*Log[x] - a^2*Sqrt[c]*Log[c + Sqrt[c]*Sqrt[c +
d*x^2]]

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Maple [A]  time = 0.014, size = 100, normalized size = 1.1 \[ -\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ) \sqrt{c}{a}^{2}+{a}^{2}\sqrt{d{x}^{2}+c}+{\frac{{b}^{2}{x}^{2}}{5\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{2\,{b}^{2}c}{15\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{2\,ab}{3\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)*c^(1/2)*a^2+a^2*(d*x^2+c)^(1/2)+1/5*b^2*x
^2*(d*x^2+c)^(3/2)/d-2/15*b^2*c/d^2*(d*x^2+c)^(3/2)+2/3*a*b*(d*x^2+c)^(3/2)/d

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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 = 0.239042, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{2} \sqrt{c} d^{2} \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) + 2 \,{\left (3 \, b^{2} d^{2} x^{4} - 2 \, b^{2} c^{2} + 10 \, a b c d + 15 \, a^{2} d^{2} +{\left (b^{2} c d + 10 \, a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{30 \, d^{2}}, -\frac{15 \, a^{2} \sqrt{-c} d^{2} \arctan \left (\frac{c}{\sqrt{d x^{2} + c} \sqrt{-c}}\right ) -{\left (3 \, b^{2} d^{2} x^{4} - 2 \, b^{2} c^{2} + 10 \, a b c d + 15 \, a^{2} d^{2} +{\left (b^{2} c d + 10 \, a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{15 \, d^{2}}\right ] \]

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]

[1/30*(15*a^2*sqrt(c)*d^2*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) +
2*(3*b^2*d^2*x^4 - 2*b^2*c^2 + 10*a*b*c*d + 15*a^2*d^2 + (b^2*c*d + 10*a*b*d^2)*
x^2)*sqrt(d*x^2 + c))/d^2, -1/15*(15*a^2*sqrt(-c)*d^2*arctan(c/(sqrt(d*x^2 + c)*
sqrt(-c))) - (3*b^2*d^2*x^4 - 2*b^2*c^2 + 10*a*b*c*d + 15*a^2*d^2 + (b^2*c*d + 1
0*a*b*d^2)*x^2)*sqrt(d*x^2 + c))/d^2]

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Sympy [A]  time = 15.1976, size = 153, normalized size = 1.66 \[ - a^{2} c \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{- c}} \right )}}{\sqrt{- c}} & \text{for}\: - c > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: - c < 0 \wedge c < c + d x^{2} \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: c > c + d x^{2} \wedge - c < 0 \end{cases}\right ) + a^{2} \sqrt{c + d x^{2}} + \frac{b^{2} \left (c + d x^{2}\right )^{\frac{5}{2}}}{5 d^{2}} + \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (4 a b d - 2 b^{2} c\right )}{6 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2*c*Piecewise((-atan(sqrt(c + d*x**2)/sqrt(-c))/sqrt(-c), -c > 0), (acoth(sq
rt(c + d*x**2)/sqrt(c))/sqrt(c), (-c < 0) & (c < c + d*x**2)), (atanh(sqrt(c + d
*x**2)/sqrt(c))/sqrt(c), (-c < 0) & (c > c + d*x**2))) + a**2*sqrt(c + d*x**2) +
b**2*(c + d*x**2)**(5/2)/(5*d**2) + (c + d*x**2)**(3/2)*(4*a*b*d - 2*b**2*c)/(6
*d**2)

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GIAC/XCAS [A]  time = 0.235753, size = 136, normalized size = 1.48 \[ \frac{a^{2} c \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} + \frac{3 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{2} d^{8} - 5 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} c d^{8} + 10 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b d^{9} + 15 \, \sqrt{d x^{2} + c} a^{2} d^{10}}{15 \, d^{10}} \]

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*c*arctan(sqrt(d*x^2 + c)/sqrt(-c))/sqrt(-c) + 1/15*(3*(d*x^2 + c)^(5/2)*b^2*
d^8 - 5*(d*x^2 + c)^(3/2)*b^2*c*d^8 + 10*(d*x^2 + c)^(3/2)*a*b*d^9 + 15*sqrt(d*x
^2 + c)*a^2*d^10)/d^10