3.487 \(\int \frac{1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=743 \[ \frac{b^{17/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{b^{17/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{b^{17/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4} (b c-a d)^3}+\frac{b^{17/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{9/4} (b c-a d)^3}-\frac{d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^3}+\frac{d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^3}+\frac{d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^3}-\frac{d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^3}-\frac{117 a^2 d^2-189 a b c d+32 b^2 c^2}{80 a c^3 x^{5/2} (b c-a d)^2}+\frac{117 a^3 d^3-189 a^2 b c d^2+32 a b^2 c^2 d+32 b^3 c^3}{16 a^2 c^4 \sqrt{x} (b c-a d)^2}-\frac{d (21 b c-13 a d)}{16 c^2 x^{5/2} \left (c+d x^2\right ) (b c-a d)^2}-\frac{d}{4 c x^{5/2} \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

-(32*b^2*c^2 - 189*a*b*c*d + 117*a^2*d^2)/(80*a*c^3*(b*c - a*d)^2*x^(5/2)) + (32
*b^3*c^3 + 32*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 117*a^3*d^3)/(16*a^2*c^4*(b*c - a*
d)^2*Sqrt[x]) - d/(4*c*(b*c - a*d)*x^(5/2)*(c + d*x^2)^2) - (d*(21*b*c - 13*a*d)
)/(16*c^2*(b*c - a*d)^2*x^(5/2)*(c + d*x^2)) - (b^(17/4)*ArcTan[1 - (Sqrt[2]*b^(
1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(9/4)*(b*c - a*d)^3) + (b^(17/4)*ArcTan[1 + (
Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(9/4)*(b*c - a*d)^3) + (d^(9/4)*(2
21*b^2*c^2 - 306*a*b*c*d + 117*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(
1/4)])/(32*Sqrt[2]*c^(17/4)*(b*c - a*d)^3) - (d^(9/4)*(221*b^2*c^2 - 306*a*b*c*d
+ 117*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(17
/4)*(b*c - a*d)^3) + (b^(17/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + S
qrt[b]*x])/(2*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) - (b^(17/4)*Log[Sqrt[a] + Sqrt[2]*a
^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) - (d^(9/4
)*(221*b^2*c^2 - 306*a*b*c*d + 117*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4
)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(17/4)*(b*c - a*d)^3) + (d^(9/4)*(221*b^2*
c^2 - 306*a*b*c*d + 117*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] +
Sqrt[d]*x])/(64*Sqrt[2]*c^(17/4)*(b*c - a*d)^3)

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Rubi [A]  time = 2.75816, antiderivative size = 743, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458 \[ \frac{b^{17/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{b^{17/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{b^{17/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4} (b c-a d)^3}+\frac{b^{17/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{9/4} (b c-a d)^3}-\frac{d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^3}+\frac{d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^3}+\frac{d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^3}-\frac{d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^3}-\frac{117 a^2 d^2-189 a b c d+32 b^2 c^2}{80 a c^3 x^{5/2} (b c-a d)^2}+\frac{117 a^3 d^3-189 a^2 b c d^2+32 a b^2 c^2 d+32 b^3 c^3}{16 a^2 c^4 \sqrt{x} (b c-a d)^2}-\frac{d (21 b c-13 a d)}{16 c^2 x^{5/2} \left (c+d x^2\right ) (b c-a d)^2}-\frac{d}{4 c x^{5/2} \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)^3),x]

[Out]

-(32*b^2*c^2 - 189*a*b*c*d + 117*a^2*d^2)/(80*a*c^3*(b*c - a*d)^2*x^(5/2)) + (32
*b^3*c^3 + 32*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 117*a^3*d^3)/(16*a^2*c^4*(b*c - a*
d)^2*Sqrt[x]) - d/(4*c*(b*c - a*d)*x^(5/2)*(c + d*x^2)^2) - (d*(21*b*c - 13*a*d)
)/(16*c^2*(b*c - a*d)^2*x^(5/2)*(c + d*x^2)) - (b^(17/4)*ArcTan[1 - (Sqrt[2]*b^(
1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(9/4)*(b*c - a*d)^3) + (b^(17/4)*ArcTan[1 + (
Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(9/4)*(b*c - a*d)^3) + (d^(9/4)*(2
21*b^2*c^2 - 306*a*b*c*d + 117*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(
1/4)])/(32*Sqrt[2]*c^(17/4)*(b*c - a*d)^3) - (d^(9/4)*(221*b^2*c^2 - 306*a*b*c*d
+ 117*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(17
/4)*(b*c - a*d)^3) + (b^(17/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + S
qrt[b]*x])/(2*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) - (b^(17/4)*Log[Sqrt[a] + Sqrt[2]*a
^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) - (d^(9/4
)*(221*b^2*c^2 - 306*a*b*c*d + 117*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4
)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(17/4)*(b*c - a*d)^3) + (d^(9/4)*(221*b^2*
c^2 - 306*a*b*c*d + 117*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] +
Sqrt[d]*x])/(64*Sqrt[2]*c^(17/4)*(b*c - a*d)^3)

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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/x**(7/2)/(b*x**2+a)/(d*x**2+c)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 3.25835, size = 660, normalized size = 0.89 \[ \frac{1}{640} \left (\frac{160 \sqrt{2} b^{17/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{9/4} (b c-a d)^3}+\frac{160 \sqrt{2} b^{17/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{9/4} (a d-b c)^3}+\frac{320 \sqrt{2} b^{17/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{9/4} (a d-b c)^3}-\frac{320 \sqrt{2} b^{17/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{9/4} (a d-b c)^3}+\frac{5 \sqrt{2} d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{17/4} (a d-b c)^3}+\frac{5 \sqrt{2} d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{17/4} (b c-a d)^3}+\frac{10 \sqrt{2} d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{17/4} (b c-a d)^3}-\frac{10 \sqrt{2} d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{17/4} (b c-a d)^3}+\frac{1280 (3 a d+b c)}{a^2 c^4 \sqrt{x}}+\frac{40 d^3 x^{3/2} (21 a d-29 b c)}{c^4 \left (c+d x^2\right ) (b c-a d)^2}-\frac{160 d^3 x^{3/2}}{c^3 \left (c+d x^2\right )^2 (b c-a d)}-\frac{256}{a c^3 x^{5/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)^3),x]

[Out]

(-256/(a*c^3*x^(5/2)) + (1280*(b*c + 3*a*d))/(a^2*c^4*Sqrt[x]) - (160*d^3*x^(3/2
))/(c^3*(b*c - a*d)*(c + d*x^2)^2) + (40*d^3*(-29*b*c + 21*a*d)*x^(3/2))/(c^4*(b
*c - a*d)^2*(c + d*x^2)) + (320*Sqrt[2]*b^(17/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqr
t[x])/a^(1/4)])/(a^(9/4)*(-(b*c) + a*d)^3) - (320*Sqrt[2]*b^(17/4)*ArcTan[1 + (S
qrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(9/4)*(-(b*c) + a*d)^3) + (10*Sqrt[2]*d^(9/
4)*(221*b^2*c^2 - 306*a*b*c*d + 117*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x]
)/c^(1/4)])/(c^(17/4)*(b*c - a*d)^3) - (10*Sqrt[2]*d^(9/4)*(221*b^2*c^2 - 306*a*
b*c*d + 117*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(17/4)*(b
*c - a*d)^3) + (160*Sqrt[2]*b^(17/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[
x] + Sqrt[b]*x])/(a^(9/4)*(b*c - a*d)^3) + (160*Sqrt[2]*b^(17/4)*Log[Sqrt[a] + S
qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(9/4)*(-(b*c) + a*d)^3) + (5*Sqr
t[2]*d^(9/4)*(221*b^2*c^2 - 306*a*b*c*d + 117*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(
1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(17/4)*(-(b*c) + a*d)^3) + (5*Sqrt[2]*d^(9
/4)*(221*b^2*c^2 - 306*a*b*c*d + 117*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1
/4)*Sqrt[x] + Sqrt[d]*x])/(c^(17/4)*(b*c - a*d)^3))/640

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Maple [A]  time = 0.038, size = 933, normalized size = 1.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(7/2)/(b*x^2+a)/(d*x^2+c)^3,x)

[Out]

21/16*d^6/c^4/(a*d-b*c)^3/(d*x^2+c)^2*x^(7/2)*a^2-25/8*d^5/c^3/(a*d-b*c)^3/(d*x^
2+c)^2*x^(7/2)*a*b+29/16*d^4/c^2/(a*d-b*c)^3/(d*x^2+c)^2*x^(7/2)*b^2+25/16*d^5/c
^3/(a*d-b*c)^3/(d*x^2+c)^2*x^(3/2)*a^2-29/8*d^4/c^2/(a*d-b*c)^3/(d*x^2+c)^2*x^(3
/2)*a*b+33/16*d^3/c/(a*d-b*c)^3/(d*x^2+c)^2*x^(3/2)*b^2+117/128*d^4/c^4/(a*d-b*c
)^3/(c/d)^(1/4)*2^(1/2)*a^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)))+117/64*d^4/c^4/(a*d-b*c)^3/(c/d)^(1/4)*2
^(1/2)*a^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+117/64*d^4/c^4/(a*d-b*c)^3/(c/d
)^(1/4)*2^(1/2)*a^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-153/64*d^3/c^3/(a*d-b*
c)^3/(c/d)^(1/4)*2^(1/2)*a*b*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)))-153/32*d^3/c^3/(a*d-b*c)^3/(c/d)^(1/4)*
2^(1/2)*a*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)-153/32*d^3/c^3/(a*d-b*c)^3/(c/
d)^(1/4)*2^(1/2)*a*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)+221/128*d^2/c^2/(a*d-
b*c)^3/(c/d)^(1/4)*2^(1/2)*b^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)))+221/64*d^2/c^2/(a*d-b*c)^3/(c/d)^(1/4
)*2^(1/2)*b^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+221/64*d^2/c^2/(a*d-b*c)^3/(
c/d)^(1/4)*2^(1/2)*b^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-2/5/a/c^3/x^(5/2)+6
/x^(1/2)/a/c^4*d+2/x^(1/2)/a^2/c^3*b-1/4*b^4/a^2/(a*d-b*c)^3/(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)))-1/2*b^4/a^2/(a*d-b*c)^3/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1
/4)*x^(1/2)+1)-1/2*b^4/a^2/(a*d-b*c)^3/(a/b)^(1/4)*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)^3*x^(7/2)),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)*(d*x^2 + c)^3*x^(7/2)),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/x**(7/2)/(b*x**2+a)/(d*x**2+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.456854, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)^3*x^(7/2)),x, algorithm="giac")

[Out]

Done