3.629 \(\int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x} \, dx\)

Optimal. Leaf size=132 \[ -a^2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )+a^2 c^2 \sqrt{c+d x^2}+\frac{1}{5} a^2 \left (c+d x^2\right )^{5/2}+\frac{1}{3} a^2 c \left (c+d x^2\right )^{3/2}-\frac{b \left (c+d x^2\right )^{7/2} (b c-2 a d)}{7 d^2}+\frac{b^2 \left (c+d x^2\right )^{9/2}}{9 d^2} \]

[Out]

a^2*c^2*Sqrt[c + d*x^2] + (a^2*c*(c + d*x^2)^(3/2))/3 + (a^2*(c + d*x^2)^(5/2))/
5 - (b*(b*c - 2*a*d)*(c + d*x^2)^(7/2))/(7*d^2) + (b^2*(c + d*x^2)^(9/2))/(9*d^2
) - a^2*c^(5/2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]

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Rubi [A]  time = 0.28269, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -a^2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )+a^2 c^2 \sqrt{c+d x^2}+\frac{1}{5} a^2 \left (c+d x^2\right )^{5/2}+\frac{1}{3} a^2 c \left (c+d x^2\right )^{3/2}-\frac{b \left (c+d x^2\right )^{7/2} (b c-2 a d)}{7 d^2}+\frac{b^2 \left (c+d x^2\right )^{9/2}}{9 d^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x,x]

[Out]

a^2*c^2*Sqrt[c + d*x^2] + (a^2*c*(c + d*x^2)^(3/2))/3 + (a^2*(c + d*x^2)^(5/2))/
5 - (b*(b*c - 2*a*d)*(c + d*x^2)^(7/2))/(7*d^2) + (b^2*(c + d*x^2)^(9/2))/(9*d^2
) - a^2*c^(5/2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]

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Rubi in Sympy [A]  time = 31.318, size = 117, normalized size = 0.89 \[ - a^{2} c^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )} + a^{2} c^{2} \sqrt{c + d x^{2}} + \frac{a^{2} c \left (c + d x^{2}\right )^{\frac{3}{2}}}{3} + \frac{a^{2} \left (c + d x^{2}\right )^{\frac{5}{2}}}{5} + \frac{b^{2} \left (c + d x^{2}\right )^{\frac{9}{2}}}{9 d^{2}} + \frac{b \left (c + d x^{2}\right )^{\frac{7}{2}} \left (2 a d - b c\right )}{7 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)**(5/2)/x,x)

[Out]

-a**2*c**(5/2)*atanh(sqrt(c + d*x**2)/sqrt(c)) + a**2*c**2*sqrt(c + d*x**2) + a*
*2*c*(c + d*x**2)**(3/2)/3 + a**2*(c + d*x**2)**(5/2)/5 + b**2*(c + d*x**2)**(9/
2)/(9*d**2) + b*(c + d*x**2)**(7/2)*(2*a*d - b*c)/(7*d**2)

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Mathematica [A]  time = 0.259324, size = 128, normalized size = 0.97 \[ \frac{\sqrt{c+d x^2} \left (21 a^2 d^2 \left (23 c^2+11 c d x^2+3 d^2 x^4\right )+90 a b d \left (c+d x^2\right )^3-5 b^2 \left (2 c-7 d x^2\right ) \left (c+d x^2\right )^3\right )}{315 d^2}-a^2 c^{5/2} \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+a^2 c^{5/2} \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x,x]

[Out]

(Sqrt[c + d*x^2]*(90*a*b*d*(c + d*x^2)^3 - 5*b^2*(2*c - 7*d*x^2)*(c + d*x^2)^3 +
21*a^2*d^2*(23*c^2 + 11*c*d*x^2 + 3*d^2*x^4)))/(315*d^2) + a^2*c^(5/2)*Log[x] -
a^2*c^(5/2)*Log[c + Sqrt[c]*Sqrt[c + d*x^2]]

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Maple [A]  time = 0.014, size = 132, normalized size = 1. \[{\frac{{a}^{2}}{5} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}c}{3} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{a}^{2}{c}^{{\frac{5}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ) +{a}^{2}{c}^{2}\sqrt{d{x}^{2}+c}+{\frac{{b}^{2}{x}^{2}}{9\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{2\,{b}^{2}c}{63\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{2\,ab}{7\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(5/2)/x,x)

[Out]

1/5*a^2*(d*x^2+c)^(5/2)+1/3*a^2*c*(d*x^2+c)^(3/2)-a^2*c^(5/2)*ln((2*c+2*c^(1/2)*
(d*x^2+c)^(1/2))/x)+a^2*c^2*(d*x^2+c)^(1/2)+1/9*b^2*x^2*(d*x^2+c)^(7/2)/d-2/63*b
^2*c/d^2*(d*x^2+c)^(7/2)+2/7*a*b*(d*x^2+c)^(7/2)/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)^(5/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249708, size = 1, normalized size = 0.01 \[ \left [\frac{315 \, a^{2} c^{\frac{5}{2}} d^{2} \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) + 2 \,{\left (35 \, b^{2} d^{4} x^{8} + 5 \,{\left (19 \, b^{2} c d^{3} + 18 \, a b d^{4}\right )} x^{6} - 10 \, b^{2} c^{4} + 90 \, a b c^{3} d + 483 \, a^{2} c^{2} d^{2} + 3 \,{\left (25 \, b^{2} c^{2} d^{2} + 90 \, a b c d^{3} + 21 \, a^{2} d^{4}\right )} x^{4} +{\left (5 \, b^{2} c^{3} d + 270 \, a b c^{2} d^{2} + 231 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{630 \, d^{2}}, -\frac{315 \, a^{2} \sqrt{-c} c^{2} d^{2} \arctan \left (\frac{c}{\sqrt{d x^{2} + c} \sqrt{-c}}\right ) -{\left (35 \, b^{2} d^{4} x^{8} + 5 \,{\left (19 \, b^{2} c d^{3} + 18 \, a b d^{4}\right )} x^{6} - 10 \, b^{2} c^{4} + 90 \, a b c^{3} d + 483 \, a^{2} c^{2} d^{2} + 3 \,{\left (25 \, b^{2} c^{2} d^{2} + 90 \, a b c d^{3} + 21 \, a^{2} d^{4}\right )} x^{4} +{\left (5 \, b^{2} c^{3} d + 270 \, a b c^{2} d^{2} + 231 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{315 \, d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(5/2)/x,x, algorithm="fricas")

[Out]

[1/630*(315*a^2*c^(5/2)*d^2*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2)
+ 2*(35*b^2*d^4*x^8 + 5*(19*b^2*c*d^3 + 18*a*b*d^4)*x^6 - 10*b^2*c^4 + 90*a*b*c^
3*d + 483*a^2*c^2*d^2 + 3*(25*b^2*c^2*d^2 + 90*a*b*c*d^3 + 21*a^2*d^4)*x^4 + (5*
b^2*c^3*d + 270*a*b*c^2*d^2 + 231*a^2*c*d^3)*x^2)*sqrt(d*x^2 + c))/d^2, -1/315*(
315*a^2*sqrt(-c)*c^2*d^2*arctan(c/(sqrt(d*x^2 + c)*sqrt(-c))) - (35*b^2*d^4*x^8
+ 5*(19*b^2*c*d^3 + 18*a*b*d^4)*x^6 - 10*b^2*c^4 + 90*a*b*c^3*d + 483*a^2*c^2*d^
2 + 3*(25*b^2*c^2*d^2 + 90*a*b*c*d^3 + 21*a^2*d^4)*x^4 + (5*b^2*c^3*d + 270*a*b*
c^2*d^2 + 231*a^2*c*d^3)*x^2)*sqrt(d*x^2 + c))/d^2]

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Sympy [A]  time = 48.6221, size = 190, normalized size = 1.44 \[ - a^{2} c^{3} \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{- c}} \right )}}{\sqrt{- c}} & \text{for}\: - c > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: - c < 0 \wedge c < c + d x^{2} \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: c > c + d x^{2} \wedge - c < 0 \end{cases}\right ) + a^{2} c^{2} \sqrt{c + d x^{2}} + \frac{a^{2} c \left (c + d x^{2}\right )^{\frac{3}{2}}}{3} + \frac{a^{2} \left (c + d x^{2}\right )^{\frac{5}{2}}}{5} + \frac{b^{2} \left (c + d x^{2}\right )^{\frac{9}{2}}}{9 d^{2}} + \frac{\left (c + d x^{2}\right )^{\frac{7}{2}} \left (4 a b d - 2 b^{2} c\right )}{14 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**(5/2)/x,x)

[Out]

-a**2*c**3*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**2*sqrt(c + d
*x**2) + a**2*c*(c + d*x**2)**(3/2)/3 + a**2*(c + d*x**2)**(5/2)/5 + b**2*(c + d
*x**2)**(9/2)/(9*d**2) + (c + d*x**2)**(7/2)*(4*a*b*d - 2*b**2*c)/(14*d**2)

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GIAC/XCAS [A]  time = 0.247865, size = 190, normalized size = 1.44 \[ \frac{a^{2} c^{3} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} + \frac{35 \,{\left (d x^{2} + c\right )}^{\frac{9}{2}} b^{2} d^{16} - 45 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} b^{2} c d^{16} + 90 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} a b d^{17} + 63 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a^{2} d^{18} + 105 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} c d^{18} + 315 \, \sqrt{d x^{2} + c} a^{2} c^{2} d^{18}}{315 \, d^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(5/2)/x,x, algorithm="giac")

[Out]

a^2*c^3*arctan(sqrt(d*x^2 + c)/sqrt(-c))/sqrt(-c) + 1/315*(35*(d*x^2 + c)^(9/2)*
b^2*d^16 - 45*(d*x^2 + c)^(7/2)*b^2*c*d^16 + 90*(d*x^2 + c)^(7/2)*a*b*d^17 + 63*
(d*x^2 + c)^(5/2)*a^2*d^18 + 105*(d*x^2 + c)^(3/2)*a^2*c*d^18 + 315*sqrt(d*x^2 +
c)*a^2*c^2*d^18)/d^18