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

Optimal. Leaf size=718 \[ -\frac{\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac{\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}-\frac{\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac{\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac{\sqrt [4]{a} b^{3/4} (7 a d+5 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} (b c-a d)^4}-\frac{\sqrt [4]{a} b^{3/4} (7 a d+5 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} (b c-a d)^4}+\frac{\sqrt [4]{a} b^{3/4} (7 a d+5 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} (b c-a d)^4}-\frac{\sqrt [4]{a} b^{3/4} (7 a d+5 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} (b c-a d)^4}+\frac{a \sqrt{x}}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}+\frac{\sqrt{x} (17 a d+7 b c)}{16 \left (c+d x^2\right ) (b c-a d)^3}+\frac{\sqrt{x} (2 a d+b c)}{4 b \left (c+d x^2\right )^2 (b c-a d)^2} \]

[Out]

((b*c + 2*a*d)*Sqrt[x])/(4*b*(b*c - a*d)^2*(c + d*x^2)^2) + (a*Sqrt[x])/(2*b*(b*
c - a*d)*(a + b*x^2)*(c + d*x^2)^2) + ((7*b*c + 17*a*d)*Sqrt[x])/(16*(b*c - a*d)
^3*(c + d*x^2)) + (a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*S
qrt[x])/a^(1/4)])/(4*Sqrt[2]*(b*c - a*d)^4) - (a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*A
rcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*(b*c - a*d)^4) - ((21*b
^2*c^2 + 70*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^(3/4)*d^(1/4)*(b*c - a*d)^4) + ((21*b^2*c^2 + 70*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^(3/4)*d^(1/4)*(
b*c - a*d)^4) + (a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b
^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*(b*c - a*d)^4) - (a^(1/4)*b^(3/4)*(5*b*c
+ 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]
*(b*c - a*d)^4) - ((21*b^2*c^2 + 70*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^(3/4)*d^(1/4)*(b*c - a*d)^4)
+ ((21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*S
qrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(3/4)*d^(1/4)*(b*c - a*d)^4)

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Rubi [A]  time = 2.1069, antiderivative size = 718, 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 \[ -\frac{\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac{\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}-\frac{\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac{\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac{\sqrt [4]{a} b^{3/4} (7 a d+5 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} (b c-a d)^4}-\frac{\sqrt [4]{a} b^{3/4} (7 a d+5 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} (b c-a d)^4}+\frac{\sqrt [4]{a} b^{3/4} (7 a d+5 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} (b c-a d)^4}-\frac{\sqrt [4]{a} b^{3/4} (7 a d+5 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} (b c-a d)^4}+\frac{a \sqrt{x}}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}+\frac{\sqrt{x} (17 a d+7 b c)}{16 \left (c+d x^2\right ) (b c-a d)^3}+\frac{\sqrt{x} (2 a d+b c)}{4 b \left (c+d x^2\right )^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

((b*c + 2*a*d)*Sqrt[x])/(4*b*(b*c - a*d)^2*(c + d*x^2)^2) + (a*Sqrt[x])/(2*b*(b*
c - a*d)*(a + b*x^2)*(c + d*x^2)^2) + ((7*b*c + 17*a*d)*Sqrt[x])/(16*(b*c - a*d)
^3*(c + d*x^2)) + (a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*S
qrt[x])/a^(1/4)])/(4*Sqrt[2]*(b*c - a*d)^4) - (a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*A
rcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*(b*c - a*d)^4) - ((21*b
^2*c^2 + 70*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^(3/4)*d^(1/4)*(b*c - a*d)^4) + ((21*b^2*c^2 + 70*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^(3/4)*d^(1/4)*(
b*c - a*d)^4) + (a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b
^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*(b*c - a*d)^4) - (a^(1/4)*b^(3/4)*(5*b*c
+ 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]
*(b*c - a*d)^4) - ((21*b^2*c^2 + 70*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^(3/4)*d^(1/4)*(b*c - a*d)^4)
+ ((21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*S
qrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(3/4)*d^(1/4)*(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**(7/2)/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 2.50188, size = 604, normalized size = 0.84 \[ \frac{-\frac{\sqrt{2} \left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{3/4} \sqrt [4]{d}}+\frac{\sqrt{2} \left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{3/4} \sqrt [4]{d}}-\frac{2 \sqrt{2} \left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{3/4} \sqrt [4]{d}}+\frac{2 \sqrt{2} \left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{3/4} \sqrt [4]{d}}+8 \sqrt{2} \sqrt [4]{a} b^{3/4} (7 a d+5 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-8 \sqrt{2} \sqrt [4]{a} b^{3/4} (7 a d+5 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+16 \sqrt{2} \sqrt [4]{a} b^{3/4} (7 a d+5 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-16 \sqrt{2} \sqrt [4]{a} b^{3/4} (7 a d+5 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+\frac{32 c \sqrt{x} (b c-a d)^2}{\left (c+d x^2\right )^2}+\frac{64 a b \sqrt{x} (b c-a d)}{a+b x^2}+\frac{8 \sqrt{x} (9 a d+7 b c) (b c-a d)}{c+d x^2}}{128 (b c-a d)^4} \]

Antiderivative was successfully verified.

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

[Out]

((64*a*b*(b*c - a*d)*Sqrt[x])/(a + b*x^2) + (32*c*(b*c - a*d)^2*Sqrt[x])/(c + d*
x^2)^2 + (8*(b*c - a*d)*(7*b*c + 9*a*d)*Sqrt[x])/(c + d*x^2) + 16*Sqrt[2]*a^(1/4
)*b^(3/4)*(5*b*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 16*Sqr
t[2]*a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4
)] - (2*Sqrt[2]*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4
)*Sqrt[x])/c^(1/4)])/(c^(3/4)*d^(1/4)) + (2*Sqrt[2]*(21*b^2*c^2 + 70*a*b*c*d + 5
*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(3/4)*d^(1/4)) + 8*S
qrt[2]*a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqr
t[x] + Sqrt[b]*x] - 8*Sqrt[2]*a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*Log[Sqrt[a] + Sqrt
[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - (Sqrt[2]*(21*b^2*c^2 + 70*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^(3/4)*d
^(1/4)) + (Sqrt[2]*(21*b^2*c^2 + 70*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^(3/4)*d^(1/4)))/(128*(b*c - a*d)^4)

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Maple [A]  time = 0.033, size = 1066, normalized size = 1.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/16/(a*d-b*c)^4/(d*x^2+c)^2*x^(5/2)*a^2*d^3+1/8/(a*d-b*c)^4/(d*x^2+c)^2*x^(5/2
)*a*b*c*d^2+7/16/(a*d-b*c)^4/(d*x^2+c)^2*x^(5/2)*b^2*c^2*d-5/16/(a*d-b*c)^4/(d*x
^2+c)^2*x^(1/2)*a^2*c*d^2-3/8/(a*d-b*c)^4/(d*x^2+c)^2*x^(1/2)*a*b*c^2*d+11/16/(a
*d-b*c)^4/(d*x^2+c)^2*x^(1/2)*b^2*c^3+5/64/(a*d-b*c)^4*(c/d)^(1/4)/c*2^(1/2)*arc
tan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2*d^2+35/32/(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*d+21/64/(a*d-b*c)^4*(c/d)^(1/4)*c*2^(
1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b^2+5/64/(a*d-b*c)^4*(c/d)^(1/4)/c*2^
(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2*d^2+35/32/(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*d+21/64/(a*d-b*c)^4*(c/d)^(1
/4)*c*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b^2+5/128/(a*d-b*c)^4*(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)))*a^2*d^2+35/64/(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*d+21/128/(a*d-b*c)^4*(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)))*b^2-1/2*a^2*b/(
a*d-b*c)^4*x^(1/2)/(b*x^2+a)*d+1/2*a*b^2/(a*d-b*c)^4*x^(1/2)/(b*x^2+a)*c-7/8*a*b
/(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-5/8*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)*c-7/8*a*b
/(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-5/8*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)*c-7/16*a*
b/(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-5/16*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)))*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(x^(7/2)/((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^(7/2)/((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**(7/2)/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.508318, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done