3.500 \(\int \frac{1}{\sqrt{x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=739 \[ -\frac{3 b^{11/4} (b c-5 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^4}+\frac{3 b^{11/4} (b c-5 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^4}-\frac{3 b^{11/4} (b c-5 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} (b c-a d)^4}+\frac{3 b^{11/4} (b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{7/4} (b c-a d)^4}+\frac{d \sqrt{x} \left (-7 a^2 d^2+23 a b c d+8 b^2 c^2\right )}{16 a c^2 \left (c+d x^2\right ) (b c-a d)^3}-\frac{3 d^{7/4} \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{11/4} (b c-a d)^4}+\frac{3 d^{7/4} \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{11/4} (b c-a d)^4}-\frac{3 d^{7/4} \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{11/4} (b c-a d)^4}+\frac{3 d^{7/4} \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{11/4} (b c-a d)^4}+\frac{b \sqrt{x}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}+\frac{d \sqrt{x} (a d+2 b c)}{4 a c \left (c+d x^2\right )^2 (b c-a d)^2} \]

[Out]

(d*(2*b*c + a*d)*Sqrt[x])/(4*a*c*(b*c - a*d)^2*(c + d*x^2)^2) + (b*Sqrt[x])/(2*a
*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) + (d*(8*b^2*c^2 + 23*a*b*c*d - 7*a^2*d^2
)*Sqrt[x])/(16*a*c^2*(b*c - a*d)^3*(c + d*x^2)) - (3*b^(11/4)*(b*c - 5*a*d)*ArcT
an[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) + (
3*b^(11/4)*(b*c - 5*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[
2]*a^(7/4)*(b*c - a*d)^4) - (3*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*Arc
Tan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(11/4)*(b*c - a*d)^4)
+ (3*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*S
qrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(11/4)*(b*c - a*d)^4) - (3*b^(11/4)*(b*c - 5*a*d
)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)
*(b*c - a*d)^4) + (3*b^(11/4)*(b*c - 5*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) - (3*d^(7/4)*(55*b^2*c
^2 - 30*a*b*c*d + 7*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqr
t[d]*x])/(64*Sqrt[2]*c^(11/4)*(b*c - a*d)^4) + (3*d^(7/4)*(55*b^2*c^2 - 30*a*b*c
*d + 7*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*
Sqrt[2]*c^(11/4)*(b*c - a*d)^4)

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Rubi [A]  time = 2.05292, antiderivative size = 739, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{3 b^{11/4} (b c-5 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^4}+\frac{3 b^{11/4} (b c-5 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^4}-\frac{3 b^{11/4} (b c-5 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} (b c-a d)^4}+\frac{3 b^{11/4} (b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{7/4} (b c-a d)^4}+\frac{d \sqrt{x} \left (-7 a^2 d^2+23 a b c d+8 b^2 c^2\right )}{16 a c^2 \left (c+d x^2\right ) (b c-a d)^3}-\frac{3 d^{7/4} \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{11/4} (b c-a d)^4}+\frac{3 d^{7/4} \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{11/4} (b c-a d)^4}-\frac{3 d^{7/4} \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{11/4} (b c-a d)^4}+\frac{3 d^{7/4} \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{11/4} (b c-a d)^4}+\frac{b \sqrt{x}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}+\frac{d \sqrt{x} (a d+2 b c)}{4 a c \left (c+d x^2\right )^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

(d*(2*b*c + a*d)*Sqrt[x])/(4*a*c*(b*c - a*d)^2*(c + d*x^2)^2) + (b*Sqrt[x])/(2*a
*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) + (d*(8*b^2*c^2 + 23*a*b*c*d - 7*a^2*d^2
)*Sqrt[x])/(16*a*c^2*(b*c - a*d)^3*(c + d*x^2)) - (3*b^(11/4)*(b*c - 5*a*d)*ArcT
an[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) + (
3*b^(11/4)*(b*c - 5*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[
2]*a^(7/4)*(b*c - a*d)^4) - (3*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*Arc
Tan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(11/4)*(b*c - a*d)^4)
+ (3*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*S
qrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(11/4)*(b*c - a*d)^4) - (3*b^(11/4)*(b*c - 5*a*d
)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)
*(b*c - a*d)^4) + (3*b^(11/4)*(b*c - 5*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) - (3*d^(7/4)*(55*b^2*c
^2 - 30*a*b*c*d + 7*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqr
t[d]*x])/(64*Sqrt[2]*c^(11/4)*(b*c - a*d)^4) + (3*d^(7/4)*(55*b^2*c^2 - 30*a*b*c
*d + 7*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*
Sqrt[2]*c^(11/4)*(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(1/(b*x**2+a)**2/(d*x**2+c)**3/x**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 5.70825, size = 692, normalized size = 0.94 \[ \frac{1}{128} \left (\frac{24 \sqrt{2} b^{11/4} (5 a d-b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{7/4} (b c-a d)^4}+\frac{24 \sqrt{2} b^{11/4} (b c-5 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{7/4} (b c-a d)^4}+\frac{48 \sqrt{2} b^{11/4} (5 a d-b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{7/4} (b c-a d)^4}+\frac{48 \sqrt{2} b^{11/4} (b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{7/4} (b c-a d)^4}-\frac{3 \sqrt{2} d^{7/4} \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{11/4} (b c-a d)^4}+\frac{3 \sqrt{2} d^{7/4} \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{11/4} (b c-a d)^4}-\frac{6 \sqrt{2} d^{7/4} \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{11/4} (b c-a d)^4}+\frac{6 \sqrt{2} d^{7/4} \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{11/4} (b c-a d)^4}-\frac{64 b^3 \sqrt{x}}{a \left (a+b x^2\right ) (a d-b c)^3}+\frac{8 d^2 \sqrt{x} (23 b c-7 a d)}{c^2 \left (c+d x^2\right ) (b c-a d)^3}+\frac{32 d^2 \sqrt{x}}{c \left (c+d x^2\right )^2 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-64*b^3*Sqrt[x])/(a*(-(b*c) + a*d)^3*(a + b*x^2)) + (32*d^2*Sqrt[x])/(c*(b*c -
a*d)^2*(c + d*x^2)^2) + (8*d^2*(23*b*c - 7*a*d)*Sqrt[x])/(c^2*(b*c - a*d)^3*(c
+ d*x^2)) + (48*Sqrt[2]*b^(11/4)*(-(b*c) + 5*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sq
rt[x])/a^(1/4)])/(a^(7/4)*(b*c - a*d)^4) + (48*Sqrt[2]*b^(11/4)*(b*c - 5*a*d)*Ar
cTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(7/4)*(b*c - a*d)^4) - (6*Sqrt[2
]*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt
[x])/c^(1/4)])/(c^(11/4)*(b*c - a*d)^4) + (6*Sqrt[2]*d^(7/4)*(55*b^2*c^2 - 30*a*
b*c*d + 7*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(11/4)*(b*c
- a*d)^4) + (24*Sqrt[2]*b^(11/4)*(-(b*c) + 5*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)
*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(7/4)*(b*c - a*d)^4) + (24*Sqrt[2]*b^(11/4)*(b
*c - 5*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(7/4)
*(b*c - a*d)^4) - (3*Sqrt[2]*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*Log[S
qrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(11/4)*(b*c - a*d)^4)
+ (3*Sqrt[2]*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]
*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(11/4)*(b*c - a*d)^4))/128

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Maple [A]  time = 0.033, size = 1124, normalized size = 1.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/16*d^5/(a*d-b*c)^4/(d*x^2+c)^2/c^2*x^(5/2)*a^2-15/8*d^4/(a*d-b*c)^4/(d*x^2+c)^
2/c*x^(5/2)*a*b+23/16*d^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(5/2)*b^2+11/16*d^4/(a*d-b*c
)^4/(d*x^2+c)^2/c*x^(1/2)*a^2-19/8*d^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(1/2)*a*b+27/16
*d^2/(a*d-b*c)^4/(d*x^2+c)^2*c*x^(1/2)*b^2+21/64*d^4/(a*d-b*c)^4/c^3*(c/d)^(1/4)
*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2-45/32*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*b+165/64*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)*b^2+21/64*d^4/(a*d
-b*c)^4/c^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2-45/32*
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*
b+165/64*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)*b^2+21/128*d^4/(a*d-b*c)^4/c^3*(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^2-45/64*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*b+165/128*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)))*b^2-1/2*b^3/(a*d-b*c)^4*x^(1/2)/(b*x^2+a)*d+1/2*
b^4/(a*d-b*c)^4/a*x^(1/2)/(b*x^2+a)*c-15/8*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)*d+3/8*b^4/(a*d-b*c)^4/a^2*(a/b)^(1/4)*2^(
1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*c-15/8*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)*d+3/8*b^4/(a*d-b*c)^4/a^2*(a/b)^(1
/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c-15/16*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)))*d+3/16*b^4/(a*d-b*c)^4/a^2*(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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.498943, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^3*sqrt(x)),x, algorithm="giac")

[Out]

Done