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

Optimal. Leaf size=739 \[ \frac{b^{9/4} (b c-13 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} (b c-a d)^4}-\frac{b^{9/4} (b c-13 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} (b c-a d)^4}-\frac{b^{9/4} (b c-13 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} (b c-a d)^4}+\frac{b^{9/4} (b c-13 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4} (b c-a d)^4}+\frac{d x^{3/2} \left (-5 a^2 d^2+21 a b c d+8 b^2 c^2\right )}{16 a c^2 \left (c+d x^2\right ) (b c-a d)^3}+\frac{d^{5/4} \left (5 a^2 d^2-26 a b c d+117 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{9/4} (b c-a d)^4}-\frac{d^{5/4} \left (5 a^2 d^2-26 a b c d+117 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{9/4} (b c-a d)^4}-\frac{d^{5/4} \left (5 a^2 d^2-26 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{9/4} (b c-a d)^4}+\frac{d^{5/4} \left (5 a^2 d^2-26 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{9/4} (b c-a d)^4}+\frac{b x^{3/2}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}+\frac{d x^{3/2} (a d+2 b c)}{4 a c \left (c+d x^2\right )^2 (b c-a d)^2} \]

[Out]

(d*(2*b*c + a*d)*x^(3/2))/(4*a*c*(b*c - a*d)^2*(c + d*x^2)^2) + (b*x^(3/2))/(2*a
*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) + (d*(8*b^2*c^2 + 21*a*b*c*d - 5*a^2*d^2
)*x^(3/2))/(16*a*c^2*(b*c - a*d)^3*(c + d*x^2)) - (b^(9/4)*(b*c - 13*a*d)*ArcTan
[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(5/4)*(b*c - a*d)^4) + (b^
(9/4)*(b*c - 13*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a
^(5/4)*(b*c - a*d)^4) - (d^(5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)*ArcTan[1
- (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(9/4)*(b*c - a*d)^4) + (d^(
5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])
/c^(1/4)])/(32*Sqrt[2]*c^(9/4)*(b*c - a*d)^4) + (b^(9/4)*(b*c - 13*a*d)*Log[Sqrt
[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(5/4)*(b*c - a*
d)^4) - (b^(9/4)*(b*c - 13*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/(8*Sqrt[2]*a^(5/4)*(b*c - a*d)^4) + (d^(5/4)*(117*b^2*c^2 - 26*a*b*c
*d + 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*
Sqrt[2]*c^(9/4)*(b*c - a*d)^4) - (d^(5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)
*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(9/4)
*(b*c - a*d)^4)

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Rubi [A]  time = 2.45956, antiderivative size = 739, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ \frac{b^{9/4} (b c-13 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} (b c-a d)^4}-\frac{b^{9/4} (b c-13 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} (b c-a d)^4}-\frac{b^{9/4} (b c-13 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} (b c-a d)^4}+\frac{b^{9/4} (b c-13 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4} (b c-a d)^4}+\frac{d x^{3/2} \left (-5 a^2 d^2+21 a b c d+8 b^2 c^2\right )}{16 a c^2 \left (c+d x^2\right ) (b c-a d)^3}+\frac{d^{5/4} \left (5 a^2 d^2-26 a b c d+117 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{9/4} (b c-a d)^4}-\frac{d^{5/4} \left (5 a^2 d^2-26 a b c d+117 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{9/4} (b c-a d)^4}-\frac{d^{5/4} \left (5 a^2 d^2-26 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{9/4} (b c-a d)^4}+\frac{d^{5/4} \left (5 a^2 d^2-26 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{9/4} (b c-a d)^4}+\frac{b x^{3/2}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}+\frac{d x^{3/2} (a d+2 b c)}{4 a c \left (c+d x^2\right )^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

(d*(2*b*c + a*d)*x^(3/2))/(4*a*c*(b*c - a*d)^2*(c + d*x^2)^2) + (b*x^(3/2))/(2*a
*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) + (d*(8*b^2*c^2 + 21*a*b*c*d - 5*a^2*d^2
)*x^(3/2))/(16*a*c^2*(b*c - a*d)^3*(c + d*x^2)) - (b^(9/4)*(b*c - 13*a*d)*ArcTan
[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(5/4)*(b*c - a*d)^4) + (b^
(9/4)*(b*c - 13*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a
^(5/4)*(b*c - a*d)^4) - (d^(5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)*ArcTan[1
- (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(9/4)*(b*c - a*d)^4) + (d^(
5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])
/c^(1/4)])/(32*Sqrt[2]*c^(9/4)*(b*c - a*d)^4) + (b^(9/4)*(b*c - 13*a*d)*Log[Sqrt
[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(5/4)*(b*c - a*
d)^4) - (b^(9/4)*(b*c - 13*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/(8*Sqrt[2]*a^(5/4)*(b*c - a*d)^4) + (d^(5/4)*(117*b^2*c^2 - 26*a*b*c
*d + 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*
Sqrt[2]*c^(9/4)*(b*c - a*d)^4) - (d^(5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)
*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(9/4)
*(b*c - a*d)^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 5.42775, size = 691, normalized size = 0.94 \[ \frac{1}{128} \left (\frac{8 \sqrt{2} b^{9/4} (b c-13 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{5/4} (b c-a d)^4}+\frac{8 \sqrt{2} b^{9/4} (13 a d-b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{5/4} (b c-a d)^4}+\frac{16 \sqrt{2} b^{9/4} (13 a d-b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{5/4} (b c-a d)^4}+\frac{16 \sqrt{2} b^{9/4} (b c-13 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{5/4} (b c-a d)^4}+\frac{\sqrt{2} d^{5/4} \left (5 a^2 d^2-26 a b c d+117 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{9/4} (b c-a d)^4}-\frac{\sqrt{2} d^{5/4} \left (5 a^2 d^2-26 a b c d+117 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{9/4} (b c-a d)^4}-\frac{2 \sqrt{2} d^{5/4} \left (5 a^2 d^2-26 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{9/4} (b c-a d)^4}+\frac{2 \sqrt{2} d^{5/4} \left (5 a^2 d^2-26 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{9/4} (b c-a d)^4}-\frac{64 b^3 x^{3/2}}{a \left (a+b x^2\right ) (a d-b c)^3}+\frac{8 d^2 x^{3/2} (21 b c-5 a d)}{c^2 \left (c+d x^2\right ) (b c-a d)^3}+\frac{32 d^2 x^{3/2}}{c \left (c+d x^2\right )^2 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-64*b^3*x^(3/2))/(a*(-(b*c) + a*d)^3*(a + b*x^2)) + (32*d^2*x^(3/2))/(c*(b*c -
a*d)^2*(c + d*x^2)^2) + (8*d^2*(21*b*c - 5*a*d)*x^(3/2))/(c^2*(b*c - a*d)^3*(c
+ d*x^2)) + (16*Sqrt[2]*b^(9/4)*(-(b*c) + 13*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sq
rt[x])/a^(1/4)])/(a^(5/4)*(b*c - a*d)^4) + (16*Sqrt[2]*b^(9/4)*(b*c - 13*a*d)*Ar
cTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(5/4)*(b*c - a*d)^4) - (2*Sqrt[2
]*d^(5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqr
t[x])/c^(1/4)])/(c^(9/4)*(b*c - a*d)^4) + (2*Sqrt[2]*d^(5/4)*(117*b^2*c^2 - 26*a
*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(9/4)*(b*c
- a*d)^4) + (8*Sqrt[2]*b^(9/4)*(b*c - 13*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(
1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(5/4)*(b*c - a*d)^4) + (8*Sqrt[2]*b^(9/4)*(-(b*c)
+ 13*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(5/4)*(
b*c - a*d)^4) + (Sqrt[2]*d^(5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)*Log[Sqrt
[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(9/4)*(b*c - a*d)^4) - (S
qrt[2]*d^(5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1
/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(9/4)*(b*c - a*d)^4))/128

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Maple [A]  time = 0.034, size = 1100, normalized size = 1.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

5/16*d^5/(a*d-b*c)^4/(d*x^2+c)^2/c^2*x^(7/2)*a^2-13/8*d^4/(a*d-b*c)^4/(d*x^2+c)^
2/c*x^(7/2)*a*b+21/16*d^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(7/2)*b^2+9/16*d^4/(a*d-b*c)
^4/(d*x^2+c)^2/c*x^(3/2)*a^2-17/8*d^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(3/2)*a*b+25/16*
d^2/(a*d-b*c)^4/(d*x^2+c)^2*c*x^(3/2)*b^2+5/64*d^3/(a*d-b*c)^4/c^2/(c/d)^(1/4)*2
^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2-13/32*d^2/(a*d-b*c)^4/c/(c/d)^(
1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a*b+117/64*d/(a*d-b*c)^4/(c/d
)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b^2+5/128*d^3/(a*d-b*c)^4/
c^2/(c/d)^(1/4)*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)))*a^2-13/64*d^2/(a*d-b*c)^4/c/(c/d)^(1/4)*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)))*a*b+117/128*d/(a*d-b*c)^4/(c/d)^(1/4)*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)))*b^2+5/
64*d^3/(a*d-b*c)^4/c^2/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)
*a^2-13/32*d^2/(a*d-b*c)^4/c/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1
/2)+1)*a*b+117/64*d/(a*d-b*c)^4/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x
^(1/2)+1)*b^2-1/2*b^3/(a*d-b*c)^4*x^(3/2)/(b*x^2+a)*d+1/2*b^4/(a*d-b*c)^4/a*x^(3
/2)/(b*x^2+a)*c-13/8*b^2/(a*d-b*c)^4/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1
/4)*x^(1/2)-1)*d+1/8*b^3/(a*d-b*c)^4/a/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^
(1/4)*x^(1/2)-1)*c-13/16*b^2/(a*d-b*c)^4/(a/b)^(1/4)*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)))*d+1/16*
b^3/(a*d-b*c)^4/a/(a/b)^(1/4)*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)))*c-13/8*b^2/(a*d-b*c)^4/(a/b)^(
1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*d+1/8*b^3/(a*d-b*c)^4/a/(a/b)
^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*c

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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(sqrt(x)/((b*x^2 + a)^2*(d*x^2 + c)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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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(x**(1/2)/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.566899, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done