3.466 \(\int \frac{1}{\sqrt{x} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx\)

Optimal. Leaf size=463 \[ -\frac{b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} (b c-a d)}+\frac{b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} (b c-a d)}-\frac{b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{3/4} (b c-a d)}+\frac{b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{3/4} (b c-a d)}+\frac{d^{3/4} \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} c^{3/4} (b c-a d)}-\frac{d^{3/4} \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} c^{3/4} (b c-a d)}+\frac{d^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} c^{3/4} (b c-a d)}-\frac{d^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt{2} c^{3/4} (b c-a d)} \]

[Out]

-((b^(3/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(3/4)*(b*c
- a*d))) + (b^(3/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(3
/4)*(b*c - a*d)) + (d^(3/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt
[2]*c^(3/4)*(b*c - a*d)) - (d^(3/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)
])/(Sqrt[2]*c^(3/4)*(b*c - a*d)) - (b^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(3/4)*(b*c - a*d)) + (b^(3/4)*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(3/4)*(b*c - a*d)) +
(d^(3/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]
*c^(3/4)*(b*c - a*d)) - (d^(3/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] +
Sqrt[d]*x])/(2*Sqrt[2]*c^(3/4)*(b*c - a*d))

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Rubi [A]  time = 0.716622, antiderivative size = 463, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} (b c-a d)}+\frac{b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} (b c-a d)}-\frac{b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{3/4} (b c-a d)}+\frac{b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{3/4} (b c-a d)}+\frac{d^{3/4} \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} c^{3/4} (b c-a d)}-\frac{d^{3/4} \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} c^{3/4} (b c-a d)}+\frac{d^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} c^{3/4} (b c-a d)}-\frac{d^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt{2} c^{3/4} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-((b^(3/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(3/4)*(b*c
- a*d))) + (b^(3/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(3
/4)*(b*c - a*d)) + (d^(3/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt
[2]*c^(3/4)*(b*c - a*d)) - (d^(3/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)
])/(Sqrt[2]*c^(3/4)*(b*c - a*d)) - (b^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(3/4)*(b*c - a*d)) + (b^(3/4)*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(3/4)*(b*c - a*d)) +
(d^(3/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]
*c^(3/4)*(b*c - a*d)) - (d^(3/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] +
Sqrt[d]*x])/(2*Sqrt[2]*c^(3/4)*(b*c - a*d))

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Rubi in Sympy [A]  time = 145.862, size = 420, normalized size = 0.91 \[ - \frac{\sqrt{2} d^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{4 c^{\frac{3}{4}} \left (a d - b c\right )} + \frac{\sqrt{2} d^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{4 c^{\frac{3}{4}} \left (a d - b c\right )} - \frac{\sqrt{2} d^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{2 c^{\frac{3}{4}} \left (a d - b c\right )} + \frac{\sqrt{2} d^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{2 c^{\frac{3}{4}} \left (a d - b c\right )} + \frac{\sqrt{2} b^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 a^{\frac{3}{4}} \left (a d - b c\right )} - \frac{\sqrt{2} b^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 a^{\frac{3}{4}} \left (a d - b c\right )} + \frac{\sqrt{2} b^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} \left (a d - b c\right )} - \frac{\sqrt{2} b^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*d**(3/4)*log(-sqrt(2)*c**(1/4)*d**(1/4)*sqrt(x) + sqrt(c) + sqrt(d)*x)/
(4*c**(3/4)*(a*d - b*c)) + sqrt(2)*d**(3/4)*log(sqrt(2)*c**(1/4)*d**(1/4)*sqrt(x
) + sqrt(c) + sqrt(d)*x)/(4*c**(3/4)*(a*d - b*c)) - sqrt(2)*d**(3/4)*atan(1 - sq
rt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(2*c**(3/4)*(a*d - b*c)) + sqrt(2)*d**(3/4)*ata
n(1 + sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(2*c**(3/4)*(a*d - b*c)) + sqrt(2)*b**(
3/4)*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(4*a**(3/4)*(
a*d - b*c)) - sqrt(2)*b**(3/4)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) +
sqrt(b)*x)/(4*a**(3/4)*(a*d - b*c)) + sqrt(2)*b**(3/4)*atan(1 - sqrt(2)*b**(1/4
)*sqrt(x)/a**(1/4))/(2*a**(3/4)*(a*d - b*c)) - sqrt(2)*b**(3/4)*atan(1 + sqrt(2)
*b**(1/4)*sqrt(x)/a**(1/4))/(2*a**(3/4)*(a*d - b*c))

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Mathematica [A]  time = 0.254887, size = 364, normalized size = 0.79 \[ \frac{a^{3/4} d^{3/4} \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )-a^{3/4} d^{3/4} \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )+2 a^{3/4} d^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )-2 a^{3/4} d^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )-b^{3/4} c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+b^{3/4} c^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-2 b^{3/4} c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+2 b^{3/4} c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} c^{3/4} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*b^(3/4)*c^(3/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 2*b^(3/4)*c^
(3/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 2*a^(3/4)*d^(3/4)*ArcTan[1
- (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 2*a^(3/4)*d^(3/4)*ArcTan[1 + (Sqrt[2]*d^
(1/4)*Sqrt[x])/c^(1/4)] - b^(3/4)*c^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*
Sqrt[x] + Sqrt[b]*x] + b^(3/4)*c^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqr
t[x] + Sqrt[b]*x] + a^(3/4)*d^(3/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x
] + Sqrt[d]*x] - a^(3/4)*d^(3/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] +
Sqrt[d]*x])/(2*Sqrt[2]*a^(3/4)*c^(3/4)*(b*c - a*d))

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Maple [A]  time = 0.016, size = 328, normalized size = 0.7 \[{\frac{d\sqrt{2}}{ \left ( 4\,ad-4\,bc \right ) c}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{d\sqrt{2}}{ \left ( 2\,ad-2\,bc \right ) c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{d\sqrt{2}}{ \left ( 2\,ad-2\,bc \right ) c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }-{\frac{b\sqrt{2}}{ \left ( 4\,ad-4\,bc \right ) a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{b\sqrt{2}}{ \left ( 2\,ad-2\,bc \right ) a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{b\sqrt{2}}{ \left ( 2\,ad-2\,bc \right ) a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*d/(a*d-b*c)*(c/d)^(1/4)/c*2^(1/2)*ln((x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1
/2))/(x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+1/2*d/(a*d-b*c)*(c/d)^(1/4)/c*
2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+1/2*d/(a*d-b*c)*(c/d)^(1/4)/c*2^(1
/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-1/4*b/(a*d-b*c)*(a/b)^(1/4)/a*2^(1/2)*
ln((x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a
/b)^(1/2)))-1/2*b/(a*d-b*c)*(a/b)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(
1/2)+1)-1/2*b/(a*d-b*c)*(a/b)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)
-1)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.417589, size = 1467, normalized size = 3.17 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)*sqrt(x)),x, algorithm="fricas")

[Out]

2*(-b^3/(a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7
*d^4))^(1/4)*arctan(-(a*b*c - a^2*d)*(-b^3/(a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^
5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4))^(1/4)/(b*sqrt(x) + sqrt(b^2*x + (a^2*b
^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*sqrt(-b^3/(a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5
*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4))))) - 2*(-d^3/(b^4*c^7 - 4*a*b^3*c^6*d +
6*a^2*b^2*c^5*d^2 - 4*a^3*b*c^4*d^3 + a^4*c^3*d^4))^(1/4)*arctan(-(b*c^2 - a*c*
d)*(-d^3/(b^4*c^7 - 4*a*b^3*c^6*d + 6*a^2*b^2*c^5*d^2 - 4*a^3*b*c^4*d^3 + a^4*c^
3*d^4))^(1/4)/(d*sqrt(x) + sqrt(d^2*x + (b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*sq
rt(-d^3/(b^4*c^7 - 4*a*b^3*c^6*d + 6*a^2*b^2*c^5*d^2 - 4*a^3*b*c^4*d^3 + a^4*c^3
*d^4))))) + 1/2*(-b^3/(a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6
*b*c*d^3 + a^7*d^4))^(1/4)*log(b*sqrt(x) + (a*b*c - a^2*d)*(-b^3/(a^3*b^4*c^4 -
4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4))^(1/4)) - 1/2*(-b
^3/(a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4)
)^(1/4)*log(b*sqrt(x) - (a*b*c - a^2*d)*(-b^3/(a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6
*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4))^(1/4)) - 1/2*(-d^3/(b^4*c^7 - 4*a*b
^3*c^6*d + 6*a^2*b^2*c^5*d^2 - 4*a^3*b*c^4*d^3 + a^4*c^3*d^4))^(1/4)*log(d*sqrt(
x) + (b*c^2 - a*c*d)*(-d^3/(b^4*c^7 - 4*a*b^3*c^6*d + 6*a^2*b^2*c^5*d^2 - 4*a^3*
b*c^4*d^3 + a^4*c^3*d^4))^(1/4)) + 1/2*(-d^3/(b^4*c^7 - 4*a*b^3*c^6*d + 6*a^2*b^
2*c^5*d^2 - 4*a^3*b*c^4*d^3 + a^4*c^3*d^4))^(1/4)*log(d*sqrt(x) - (b*c^2 - a*c*d
)*(-d^3/(b^4*c^7 - 4*a*b^3*c^6*d + 6*a^2*b^2*c^5*d^2 - 4*a^3*b*c^4*d^3 + a^4*c^3
*d^4))^(1/4))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )} \sqrt{x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)*sqrt(x)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)*(d*x^2 + c)*sqrt(x)), x)