3.497 \(\int \frac{x^{5/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=703 \[ -\frac{3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} (b c-a d)^4}+\frac{3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} (b c-a d)^4}+\frac{3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{5/4} (b c-a d)^4}-\frac{3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{5/4} (b c-a d)^4}+\frac{3 b^{5/4} (3 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}-\frac{3 b^{5/4} (3 a d+b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}-\frac{3 b^{5/4} (3 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}+\frac{3 b^{5/4} (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}-\frac{3 d x^{3/2} (a d+7 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^3}-\frac{x^{3/2}}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac{3 d x^{3/2}}{4 \left (c+d x^2\right )^2 (b c-a d)^2} \]

[Out]

(-3*d*x^(3/2))/(4*(b*c - a*d)^2*(c + d*x^2)^2) - x^(3/2)/(2*(b*c - a*d)*(a + b*x
^2)*(c + d*x^2)^2) - (3*d*(7*b*c + a*d)*x^(3/2))/(16*c*(b*c - a*d)^3*(c + d*x^2)
) - (3*b^(5/4)*(b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*S
qrt[2]*a^(1/4)*(b*c - a*d)^4) + (3*b^(5/4)*(b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*b^(
1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(1/4)*(b*c - a*d)^4) + (3*d^(1/4)*(15*b^2*c
^2 + 18*a*b*c*d - a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sq
rt[2]*c^(5/4)*(b*c - a*d)^4) - (3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*Ar
cTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*(b*c - a*d)^4)
+ (3*b^(5/4)*(b*c + 3*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[
b]*x])/(8*Sqrt[2]*a^(1/4)*(b*c - a*d)^4) - (3*b^(5/4)*(b*c + 3*a*d)*Log[Sqrt[a]
+ Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*(b*c - a*d)^4
) - (3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)
*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(5/4)*(b*c - a*d)^4) + (3*d^(1/4)*(
15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x]
+ Sqrt[d]*x])/(64*Sqrt[2]*c^(5/4)*(b*c - a*d)^4)

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Rubi [A]  time = 2.19898, antiderivative size = 703, 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 \[ -\frac{3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} (b c-a d)^4}+\frac{3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} (b c-a d)^4}+\frac{3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{5/4} (b c-a d)^4}-\frac{3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{5/4} (b c-a d)^4}+\frac{3 b^{5/4} (3 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}-\frac{3 b^{5/4} (3 a d+b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}-\frac{3 b^{5/4} (3 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}+\frac{3 b^{5/4} (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}-\frac{3 d x^{3/2} (a d+7 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^3}-\frac{x^{3/2}}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac{3 d x^{3/2}}{4 \left (c+d x^2\right )^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/((a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

(-3*d*x^(3/2))/(4*(b*c - a*d)^2*(c + d*x^2)^2) - x^(3/2)/(2*(b*c - a*d)*(a + b*x
^2)*(c + d*x^2)^2) - (3*d*(7*b*c + a*d)*x^(3/2))/(16*c*(b*c - a*d)^3*(c + d*x^2)
) - (3*b^(5/4)*(b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*S
qrt[2]*a^(1/4)*(b*c - a*d)^4) + (3*b^(5/4)*(b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*b^(
1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(1/4)*(b*c - a*d)^4) + (3*d^(1/4)*(15*b^2*c
^2 + 18*a*b*c*d - a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sq
rt[2]*c^(5/4)*(b*c - a*d)^4) - (3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*Ar
cTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*(b*c - a*d)^4)
+ (3*b^(5/4)*(b*c + 3*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[
b]*x])/(8*Sqrt[2]*a^(1/4)*(b*c - a*d)^4) - (3*b^(5/4)*(b*c + 3*a*d)*Log[Sqrt[a]
+ Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*(b*c - a*d)^4
) - (3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)
*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(5/4)*(b*c - a*d)^4) + (3*d^(1/4)*(
15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x]
+ Sqrt[d]*x])/(64*Sqrt[2]*c^(5/4)*(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**(5/2)/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 3.22736, size = 604, normalized size = 0.86 \[ \frac{\frac{3 \sqrt{2} \sqrt [4]{d} \left (a^2 d^2-18 a b c d-15 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4}}+\frac{3 \sqrt{2} \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4}}+\frac{6 \sqrt{2} \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{5/4}}+\frac{6 \sqrt{2} \sqrt [4]{d} \left (a^2 d^2-18 a b c d-15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{5/4}}+\frac{24 \sqrt{2} b^{5/4} (3 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{a}}-\frac{24 \sqrt{2} b^{5/4} (3 a d+b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{a}}-\frac{48 \sqrt{2} b^{5/4} (3 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}+\frac{48 \sqrt{2} b^{5/4} (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a}}-\frac{64 b^2 x^{3/2} (b c-a d)}{a+b x^2}+\frac{8 d x^{3/2} (a d-b c) (3 a d+13 b c)}{c \left (c+d x^2\right )}-\frac{32 d x^{3/2} (b c-a d)^2}{\left (c+d x^2\right )^2}}{128 (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)/((a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

((-64*b^2*(b*c - a*d)*x^(3/2))/(a + b*x^2) - (32*d*(b*c - a*d)^2*x^(3/2))/(c + d
*x^2)^2 + (8*d*(-(b*c) + a*d)*(13*b*c + 3*a*d)*x^(3/2))/(c*(c + d*x^2)) - (48*Sq
rt[2]*b^(5/4)*(b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(1/
4) + (48*Sqrt[2]*b^(5/4)*(b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1
/4)])/a^(1/4) + (6*Sqrt[2]*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*ArcTan[1
- (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(5/4) + (6*Sqrt[2]*d^(1/4)*(-15*b^2*c^2
- 18*a*b*c*d + a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(5/4) +
(24*Sqrt[2]*b^(5/4)*(b*c + 3*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x]
+ Sqrt[b]*x])/a^(1/4) - (24*Sqrt[2]*b^(5/4)*(b*c + 3*a*d)*Log[Sqrt[a] + Sqrt[2]
*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(1/4) + (3*Sqrt[2]*d^(1/4)*(-15*b^2*c^2
- 18*a*b*c*d + a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]
*x])/c^(5/4) + (3*Sqrt[2]*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*Log[Sqrt[c
] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(5/4))/(128*(b*c - a*d)^4)

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Maple [A]  time = 0.035, size = 1067, normalized size = 1.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

3/16*d^4/(a*d-b*c)^4/(d*x^2+c)^2/c*x^(7/2)*a^2+5/8*d^3/(a*d-b*c)^4/(d*x^2+c)^2*x
^(7/2)*a*b-13/16*d^2/(a*d-b*c)^4/(d*x^2+c)^2*c*x^(7/2)*b^2-1/16*d^3/(a*d-b*c)^4/
(d*x^2+c)^2*x^(3/2)*a^2+9/8*d^2/(a*d-b*c)^4/(d*x^2+c)^2*x^(3/2)*c*a*b-17/16*d/(a
*d-b*c)^4/(d*x^2+c)^2*x^(3/2)*b^2*c^2+3/64*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)*a^2-27/32*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)*a*b-45/64/(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+3/64*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)*a^2-27/32*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)*a*b-45/64/(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+3/128*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)))*a^2-27/64*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)))*a*b-45/128/(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^2/(a*
d-b*c)^4*x^(3/2)/(b*x^2+a)*a*d-1/2*b^3/(a*d-b*c)^4*x^(3/2)/(b*x^2+a)*c+9/16*b/(a
*d-b*c)^4/(a/b)^(1/4)*2^(1/2)*a*d*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)))+9/8*b/(a*d-b*c)^4/(a/b)^(1/4)*2^(1
/2)*a*d*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+9/8*b/(a*d-b*c)^4/(a/b)^(1/4)*2^(1
/2)*a*d*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+3/16*b^2/(a*d-b*c)^4/(a/b)^(1/4)*2
^(1/2)*c*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)))+3/8*b^2/(a*d-b*c)^4/(a/b)^(1/4)*2^(1/2)*c*arctan(2^(1/2)/(a
/b)^(1/4)*x^(1/2)+1)+3/8*b^2/(a*d-b*c)^4/(a/b)^(1/4)*2^(1/2)*c*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(x^(5/2)/((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(x^(5/2)/((b*x^2 + a)^2*(d*x^2 + c)^3),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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**(5/2)/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.529946, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/((b*x^2 + a)^2*(d*x^2 + c)^3),x, algorithm="giac")

[Out]

Done