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

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

[Out]

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

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Rubi [A]  time = 0.249962, 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 \[ -\frac{\left (a+b x^2\right )^{3/2} \left (2 a B d-5 b (A d+B c)-3 b B d x^2\right )}{15 b^2}+A c \sqrt{a+b x^2}-\sqrt{a} A c \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.3805, size = 83, normalized size = 0.99 \[ - A \sqrt{a} c \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )} + A c \sqrt{a + b x^{2}} + \frac{2 \left (a + b x^{2}\right )^{\frac{3}{2}} \left (- B a d + \frac{3 B b d x^{2}}{2} + \frac{5 b \left (A d + B c\right )}{2}\right )}{15 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.271426, size = 103, normalized size = 1.23 \[ \frac{\sqrt{a+b x^2} \left (5 A b \left (a d+3 b c+b d x^2\right )-B \left (a+b x^2\right ) \left (2 a d-5 b c-3 b d x^2\right )\right )}{15 b^2}-\sqrt{a} A c \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+\sqrt{a} A c \log (x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 112, normalized size = 1.3 \[{\frac{Ad}{3\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-A\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) c+Ac\sqrt{b{x}^{2}+a}+{\frac{Bc}{3\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Bd{x}^{2}}{5\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{2\,aBd}{15\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*A*d/b*(b*x^2+a)^(3/2)-A*a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)*c+A*c*
(b*x^2+a)^(1/2)+1/3*B*c/b*(b*x^2+a)^(3/2)+1/5*B*d*x^2*(b*x^2+a)^(3/2)/b-2/15*B*d
*a/b^2*(b*x^2+a)^(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)*sqrt(b*x^2 + a)*(d*x^2 + c)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.269431, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, A \sqrt{a} b^{2} c \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (3 \, B b^{2} d x^{4} +{\left (5 \, B b^{2} c +{\left (B a b + 5 \, A b^{2}\right )} d\right )} x^{2} + 5 \,{\left (B a b + 3 \, A b^{2}\right )} c -{\left (2 \, B a^{2} - 5 \, A a b\right )} d\right )} \sqrt{b x^{2} + a}}{30 \, b^{2}}, -\frac{15 \, A \sqrt{-a} b^{2} c \arctan \left (\frac{a}{\sqrt{b x^{2} + a} \sqrt{-a}}\right ) -{\left (3 \, B b^{2} d x^{4} +{\left (5 \, B b^{2} c +{\left (B a b + 5 \, A b^{2}\right )} d\right )} x^{2} + 5 \,{\left (B a b + 3 \, A b^{2}\right )} c -{\left (2 \, B a^{2} - 5 \, A a b\right )} d\right )} \sqrt{b x^{2} + a}}{15 \, b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 19.4673, size = 160, normalized size = 1.9 \[ - A a c \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x^{2} \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x^{2} \wedge - a < 0 \end{cases}\right ) + A c \sqrt{a + b x^{2}} + \frac{B d \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 b^{2}} + \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (2 A b d - 2 B a d + 2 B b c\right )}{6 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.220893, size = 153, normalized size = 1.82 \[ \frac{A a c \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B b^{9} c + 15 \, \sqrt{b x^{2} + a} A b^{10} c + 3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B b^{8} d - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a b^{8} d + 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A b^{9} d}{15 \, b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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