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

Optimal. Leaf size=84 \[ -\frac{\left (c+d x^2\right )^{3/2} \left (-5 d (a B+A b)+2 b B c-3 b B d x^2\right )}{15 d^2}+a A \sqrt{c+d x^2}-a A \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right ) \]

[Out]

a*A*Sqrt[c + d*x^2] - ((c + d*x^2)^(3/2)*(2*b*B*c - 5*(A*b + a*B)*d - 3*b*B*d*x^
2))/(15*d^2) - a*A*Sqrt[c]*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]

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Rubi [A]  time = 0.253958, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ -\frac{\left (c+d x^2\right )^{3/2} \left (-5 d (a B+A b)+2 b B c-3 b B d x^2\right )}{15 d^2}+a A \sqrt{c+d x^2}-a A \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

a*A*Sqrt[c + d*x^2] - ((c + d*x^2)^(3/2)*(2*b*B*c - 5*(A*b + a*B)*d - 3*b*B*d*x^
2))/(15*d^2) - a*A*Sqrt[c]*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]

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Rubi in Sympy [A]  time = 20.6677, size = 83, normalized size = 0.99 \[ - A a \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )} + A a \sqrt{c + d x^{2}} + \frac{2 \left (c + d x^{2}\right )^{\frac{3}{2}} \left (- B b c + \frac{3 B b d x^{2}}{2} + \frac{5 d \left (A b + B a\right )}{2}\right )}{15 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a*sqrt(c)*atanh(sqrt(c + d*x**2)/sqrt(c)) + A*a*sqrt(c + d*x**2) + 2*(c + d*x
**2)**(3/2)*(-B*b*c + 3*B*b*d*x**2/2 + 5*d*(A*b + B*a)/2)/(15*d**2)

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Mathematica [A]  time = 0.299313, size = 103, normalized size = 1.23 \[ \frac{\sqrt{c+d x^2} \left (5 a d \left (3 A d+B c+B d x^2\right )-b \left (c+d x^2\right ) \left (-5 A d+2 B c-3 B d x^2\right )\right )}{15 d^2}-a A \sqrt{c} \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+a A \sqrt{c} \log (x) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x^2]*(-(b*(c + d*x^2)*(2*B*c - 5*A*d - 3*B*d*x^2)) + 5*a*d*(B*c + 3*
A*d + B*d*x^2)))/(15*d^2) + a*A*Sqrt[c]*Log[x] - a*A*Sqrt[c]*Log[c + Sqrt[c]*Sqr
t[c + d*x^2]]

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Maple [A]  time = 0.01, size = 112, normalized size = 1.3 \[{\frac{Ab}{3\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-A\sqrt{c}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ) a+aA\sqrt{d{x}^{2}+c}+{\frac{Ba}{3\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{Bb{x}^{2}}{5\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{2\,bBc}{15\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276625, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, A a \sqrt{c} d^{2} \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) + 2 \,{\left (3 \, B b d^{2} x^{4} - 2 \, B b c^{2} + 15 \, A a d^{2} + 5 \,{\left (B a + A b\right )} c d +{\left (B b c d + 5 \,{\left (B a + A b\right )} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{30 \, d^{2}}, -\frac{15 \, A a \sqrt{-c} d^{2} \arctan \left (\frac{c}{\sqrt{d x^{2} + c} \sqrt{-c}}\right ) -{\left (3 \, B b d^{2} x^{4} - 2 \, B b c^{2} + 15 \, A a d^{2} + 5 \,{\left (B a + A b\right )} c d +{\left (B b c d + 5 \,{\left (B a + A b\right )} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{15 \, d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/30*(15*A*a*sqrt(c)*d^2*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) +
2*(3*B*b*d^2*x^4 - 2*B*b*c^2 + 15*A*a*d^2 + 5*(B*a + A*b)*c*d + (B*b*c*d + 5*(B*
a + A*b)*d^2)*x^2)*sqrt(d*x^2 + c))/d^2, -1/15*(15*A*a*sqrt(-c)*d^2*arctan(c/(sq
rt(d*x^2 + c)*sqrt(-c))) - (3*B*b*d^2*x^4 - 2*B*b*c^2 + 15*A*a*d^2 + 5*(B*a + A*
b)*c*d + (B*b*c*d + 5*(B*a + A*b)*d^2)*x^2)*sqrt(d*x^2 + c))/d^2]

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Sympy [A]  time = 20.331, size = 160, normalized size = 1.9 \[ - A a c \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{- c}} \right )}}{\sqrt{- c}} & \text{for}\: - c > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: - c < 0 \wedge c < c + d x^{2} \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: c > c + d x^{2} \wedge - c < 0 \end{cases}\right ) + A a \sqrt{c + d x^{2}} + \frac{B b \left (c + d x^{2}\right )^{\frac{5}{2}}}{5 d^{2}} + \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (2 A b d + 2 B a d - 2 B b c\right )}{6 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a*c*Piecewise((-atan(sqrt(c + d*x**2)/sqrt(-c))/sqrt(-c), -c > 0), (acoth(sqr
t(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*a*sqrt(c + d*x**2) + B
*b*(c + d*x**2)**(5/2)/(5*d**2) + (c + d*x**2)**(3/2)*(2*A*b*d + 2*B*a*d - 2*B*b
*c)/(6*d**2)

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GIAC/XCAS [A]  time = 0.220498, size = 153, normalized size = 1.82 \[ \frac{A a c \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} + \frac{3 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} B b d^{8} - 5 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} B b c d^{8} + 5 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} B a d^{9} + 5 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} A b d^{9} + 15 \, \sqrt{d x^{2} + c} A a d^{10}}{15 \, d^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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