3.618 \(\int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x} \, dx\)

Optimal. Leaf size=111 \[ -a^2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )+\frac{1}{3} a^2 \left (c+d x^2\right )^{3/2}+a^2 c \sqrt{c+d x^2}-\frac{b \left (c+d x^2\right )^{5/2} (b c-2 a d)}{5 d^2}+\frac{b^2 \left (c+d x^2\right )^{7/2}}{7 d^2} \]

[Out]

a^2*c*Sqrt[c + d*x^2] + (a^2*(c + d*x^2)^(3/2))/3 - (b*(b*c - 2*a*d)*(c + d*x^2)
^(5/2))/(5*d^2) + (b^2*(c + d*x^2)^(7/2))/(7*d^2) - a^2*c^(3/2)*ArcTanh[Sqrt[c +
d*x^2]/Sqrt[c]]

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Rubi [A]  time = 0.253041, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -a^2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )+\frac{1}{3} a^2 \left (c+d x^2\right )^{3/2}+a^2 c \sqrt{c+d x^2}-\frac{b \left (c+d x^2\right )^{5/2} (b c-2 a d)}{5 d^2}+\frac{b^2 \left (c+d x^2\right )^{7/2}}{7 d^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x,x]

[Out]

a^2*c*Sqrt[c + d*x^2] + (a^2*(c + d*x^2)^(3/2))/3 - (b*(b*c - 2*a*d)*(c + d*x^2)
^(5/2))/(5*d^2) + (b^2*(c + d*x^2)^(7/2))/(7*d^2) - a^2*c^(3/2)*ArcTanh[Sqrt[c +
d*x^2]/Sqrt[c]]

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Rubi in Sympy [A]  time = 27.487, size = 99, normalized size = 0.89 \[ - a^{2} c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )} + a^{2} c \sqrt{c + d x^{2}} + \frac{a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{3} + \frac{b^{2} \left (c + d x^{2}\right )^{\frac{7}{2}}}{7 d^{2}} + \frac{b \left (c + d x^{2}\right )^{\frac{5}{2}} \left (2 a d - b c\right )}{5 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)**(3/2)/x,x)

[Out]

-a**2*c**(3/2)*atanh(sqrt(c + d*x**2)/sqrt(c)) + a**2*c*sqrt(c + d*x**2) + a**2*
(c + d*x**2)**(3/2)/3 + b**2*(c + d*x**2)**(7/2)/(7*d**2) + b*(c + d*x**2)**(5/2
)*(2*a*d - b*c)/(5*d**2)

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Mathematica [A]  time = 0.21513, size = 116, normalized size = 1.05 \[ \frac{\sqrt{c+d x^2} \left (35 a^2 d^2 \left (4 c+d x^2\right )+42 a b d \left (c+d x^2\right )^2-3 b^2 \left (2 c-5 d x^2\right ) \left (c+d x^2\right )^2\right )}{105 d^2}-a^2 c^{3/2} \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+a^2 c^{3/2} \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x,x]

[Out]

(Sqrt[c + d*x^2]*(42*a*b*d*(c + d*x^2)^2 - 3*b^2*(2*c - 5*d*x^2)*(c + d*x^2)^2 +
35*a^2*d^2*(4*c + d*x^2)))/(105*d^2) + a^2*c^(3/2)*Log[x] - a^2*c^(3/2)*Log[c +
Sqrt[c]*Sqrt[c + d*x^2]]

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Maple [A]  time = 0.015, size = 115, normalized size = 1. \[{\frac{{a}^{2}}{3} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{a}^{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{{\frac{3}{2}}}+{a}^{2}c\sqrt{d{x}^{2}+c}+{\frac{{b}^{2}{x}^{2}}{7\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{b}^{2}c}{35\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{2\,ab}{5\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(3/2)/x,x)

[Out]

1/3*a^2*(d*x^2+c)^(3/2)-a^2*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)*c^(3/2)+a^2*c*
(d*x^2+c)^(1/2)+1/7*b^2*x^2*(d*x^2+c)^(5/2)/d-2/35*b^2*c/d^2*(d*x^2+c)^(5/2)+2/5
*a*b*(d*x^2+c)^(5/2)/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*(d*x^2 + c)^(3/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241282, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, a^{2} c^{\frac{3}{2}} d^{2} \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) + 2 \,{\left (15 \, b^{2} d^{3} x^{6} - 6 \, b^{2} c^{3} + 42 \, a b c^{2} d + 140 \, a^{2} c d^{2} + 6 \,{\left (4 \, b^{2} c d^{2} + 7 \, a b d^{3}\right )} x^{4} +{\left (3 \, b^{2} c^{2} d + 84 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{210 \, d^{2}}, -\frac{105 \, a^{2} \sqrt{-c} c d^{2} \arctan \left (\frac{c}{\sqrt{d x^{2} + c} \sqrt{-c}}\right ) -{\left (15 \, b^{2} d^{3} x^{6} - 6 \, b^{2} c^{3} + 42 \, a b c^{2} d + 140 \, a^{2} c d^{2} + 6 \,{\left (4 \, b^{2} c d^{2} + 7 \, a b d^{3}\right )} x^{4} +{\left (3 \, b^{2} c^{2} d + 84 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{105 \, d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(3/2)/x,x, algorithm="fricas")

[Out]

[1/210*(105*a^2*c^(3/2)*d^2*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2)
+ 2*(15*b^2*d^3*x^6 - 6*b^2*c^3 + 42*a*b*c^2*d + 140*a^2*c*d^2 + 6*(4*b^2*c*d^2
+ 7*a*b*d^3)*x^4 + (3*b^2*c^2*d + 84*a*b*c*d^2 + 35*a^2*d^3)*x^2)*sqrt(d*x^2 + c
))/d^2, -1/105*(105*a^2*sqrt(-c)*c*d^2*arctan(c/(sqrt(d*x^2 + c)*sqrt(-c))) - (1
5*b^2*d^3*x^6 - 6*b^2*c^3 + 42*a*b*c^2*d + 140*a^2*c*d^2 + 6*(4*b^2*c*d^2 + 7*a*
b*d^3)*x^4 + (3*b^2*c^2*d + 84*a*b*c*d^2 + 35*a^2*d^3)*x^2)*sqrt(d*x^2 + c))/d^2
]

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Sympy [A]  time = 26.9036, size = 172, normalized size = 1.55 \[ - a^{2} c^{2} \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{- c}} \right )}}{\sqrt{- c}} & \text{for}\: - c > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: - c < 0 \wedge c < c + d x^{2} \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: c > c + d x^{2} \wedge - c < 0 \end{cases}\right ) + a^{2} c \sqrt{c + d x^{2}} + \frac{a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{3} + \frac{b^{2} \left (c + d x^{2}\right )^{\frac{7}{2}}}{7 d^{2}} + \frac{\left (c + d x^{2}\right )^{\frac{5}{2}} \left (4 a b d - 2 b^{2} c\right )}{10 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**(3/2)/x,x)

[Out]

-a**2*c**2*Piecewise((-atan(sqrt(c + d*x**2)/sqrt(-c))/sqrt(-c), -c > 0), (acoth
(sqrt(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*c*sqrt(c + d*x*
*2) + a**2*(c + d*x**2)**(3/2)/3 + b**2*(c + d*x**2)**(7/2)/(7*d**2) + (c + d*x*
*2)**(5/2)*(4*a*b*d - 2*b**2*c)/(10*d**2)

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GIAC/XCAS [A]  time = 0.240303, size = 163, normalized size = 1.47 \[ \frac{a^{2} c^{2} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} + \frac{15 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} b^{2} d^{12} - 21 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{2} c d^{12} + 42 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a b d^{13} + 35 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} d^{14} + 105 \, \sqrt{d x^{2} + c} a^{2} c d^{14}}{105 \, d^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(3/2)/x,x, algorithm="giac")

[Out]

a^2*c^2*arctan(sqrt(d*x^2 + c)/sqrt(-c))/sqrt(-c) + 1/105*(15*(d*x^2 + c)^(7/2)*
b^2*d^12 - 21*(d*x^2 + c)^(5/2)*b^2*c*d^12 + 42*(d*x^2 + c)^(5/2)*a*b*d^13 + 35*
(d*x^2 + c)^(3/2)*a^2*d^14 + 105*sqrt(d*x^2 + c)*a^2*c*d^14)/d^14