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

Optimal. Leaf size=532 \[ -\frac{a^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{b} (b c-a d)^2}+\frac{a^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{b} (b c-a d)^2}-\frac{a^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{b} (b c-a d)^2}+\frac{a^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} \sqrt [4]{b} (b c-a d)^2}-\frac{\sqrt [4]{c} (b c-5 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} d^{5/4} (b c-a d)^2}+\frac{\sqrt [4]{c} (b c-5 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} d^{5/4} (b c-a d)^2}-\frac{\sqrt [4]{c} (b c-5 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} d^{5/4} (b c-a d)^2}+\frac{\sqrt [4]{c} (b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} d^{5/4} (b c-a d)^2}-\frac{c \sqrt{x}}{2 d \left (c+d x^2\right ) (b c-a d)} \]

[Out]

-(c*Sqrt[x])/(2*d*(b*c - a*d)*(c + d*x^2)) - (a^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4
)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(1/4)*(b*c - a*d)^2) + (a^(5/4)*ArcTan[1 + (Sqrt
[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(1/4)*(b*c - a*d)^2) - (c^(1/4)*(b*c -
5*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*d^(5/4)*(b*c -
a*d)^2) + (c^(1/4)*(b*c - 5*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])
/(4*Sqrt[2]*d^(5/4)*(b*c - a*d)^2) - (a^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1
/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(1/4)*(b*c - a*d)^2) + (a^(5/4)*Log[Sqrt[
a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(1/4)*(b*c - a*d
)^2) - (c^(1/4)*(b*c - 5*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sq
rt[d]*x])/(8*Sqrt[2]*d^(5/4)*(b*c - a*d)^2) + (c^(1/4)*(b*c - 5*a*d)*Log[Sqrt[c]
+ Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*d^(5/4)*(b*c - a*d)^
2)

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Rubi [A]  time = 1.13101, antiderivative size = 532, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{a^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{b} (b c-a d)^2}+\frac{a^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{b} (b c-a d)^2}-\frac{a^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{b} (b c-a d)^2}+\frac{a^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} \sqrt [4]{b} (b c-a d)^2}-\frac{\sqrt [4]{c} (b c-5 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} d^{5/4} (b c-a d)^2}+\frac{\sqrt [4]{c} (b c-5 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} d^{5/4} (b c-a d)^2}-\frac{\sqrt [4]{c} (b c-5 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} d^{5/4} (b c-a d)^2}+\frac{\sqrt [4]{c} (b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} d^{5/4} (b c-a d)^2}-\frac{c \sqrt{x}}{2 d \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-(c*Sqrt[x])/(2*d*(b*c - a*d)*(c + d*x^2)) - (a^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4
)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(1/4)*(b*c - a*d)^2) + (a^(5/4)*ArcTan[1 + (Sqrt
[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(1/4)*(b*c - a*d)^2) - (c^(1/4)*(b*c -
5*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*d^(5/4)*(b*c -
a*d)^2) + (c^(1/4)*(b*c - 5*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])
/(4*Sqrt[2]*d^(5/4)*(b*c - a*d)^2) - (a^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1
/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(1/4)*(b*c - a*d)^2) + (a^(5/4)*Log[Sqrt[
a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(1/4)*(b*c - a*d
)^2) - (c^(1/4)*(b*c - 5*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sq
rt[d]*x])/(8*Sqrt[2]*d^(5/4)*(b*c - a*d)^2) + (c^(1/4)*(b*c - 5*a*d)*Log[Sqrt[c]
+ Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*d^(5/4)*(b*c - a*d)^
2)

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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)/(d*x**2+c)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.515788, size = 523, normalized size = 0.98 \[ \frac{-4 \sqrt{2} a^{5/4} d^{5/4} \left (c+d x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+4 \sqrt{2} a^{5/4} d^{5/4} \left (c+d x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-8 \sqrt{2} a^{5/4} d^{5/4} \left (c+d x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+8 \sqrt{2} a^{5/4} d^{5/4} \left (c+d x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \left (c+d x^2\right ) (b c-5 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \left (c+d x^2\right ) (b c-5 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )-2 \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \left (c+d x^2\right ) (b c-5 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )+2 \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \left (c+d x^2\right ) (b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )-8 \sqrt [4]{b} c \sqrt [4]{d} \sqrt{x} (b c-a d)}{16 \sqrt [4]{b} d^{5/4} \left (c+d x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-8*b^(1/4)*c*d^(1/4)*(b*c - a*d)*Sqrt[x] - 8*Sqrt[2]*a^(5/4)*d^(5/4)*(c + d*x^2
)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 8*Sqrt[2]*a^(5/4)*d^(5/4)*(c +
d*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*Sqrt[2]*b^(1/4)*c^(1/4
)*(b*c - 5*a*d)*(c + d*x^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 2*Sq
rt[2]*b^(1/4)*c^(1/4)*(b*c - 5*a*d)*(c + d*x^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt
[x])/c^(1/4)] - 4*Sqrt[2]*a^(5/4)*d^(5/4)*(c + d*x^2)*Log[Sqrt[a] - Sqrt[2]*a^(1
/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 4*Sqrt[2]*a^(5/4)*d^(5/4)*(c + d*x^2)*Log[Sqr
t[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - Sqrt[2]*b^(1/4)*c^(1/4)*(b
*c - 5*a*d)*(c + d*x^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*
x] + Sqrt[2]*b^(1/4)*c^(1/4)*(b*c - 5*a*d)*(c + d*x^2)*Log[Sqrt[c] + Sqrt[2]*c^(
1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(16*b^(1/4)*d^(5/4)*(b*c - a*d)^2*(c + d*x^2)
)

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Maple [A]  time = 0.024, size = 533, normalized size = 1. \[{\frac{ac}{2\, \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }\sqrt{x}}-{\frac{b{c}^{2}}{2\, \left ( ad-bc \right ) ^{2}d \left ( d{x}^{2}+c \right ) }\sqrt{x}}-{\frac{5\,\sqrt{2}a}{8\, \left ( ad-bc \right ) ^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{c\sqrt{2}b}{8\, \left ( ad-bc \right ) ^{2}d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }-{\frac{5\,\sqrt{2}a}{8\, \left ( ad-bc \right ) ^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{c\sqrt{2}b}{8\, \left ( ad-bc \right ) ^{2}d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }-{\frac{5\,\sqrt{2}a}{16\, \left ( ad-bc \right ) ^{2}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{c\sqrt{2}b}{16\, \left ( ad-bc \right ) ^{2}d}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}a}{4\, \left ( ad-bc \right ) ^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}a}{2\, \left ( ad-bc \right ) ^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{\sqrt{2}a}{2\, \left ( ad-bc \right ) ^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*c/(a*d-b*c)^2*x^(1/2)/(d*x^2+c)*a-1/2*c^2/(a*d-b*c)^2/d*x^(1/2)/(d*x^2+c)*b-
5/8/(a*d-b*c)^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a+1/8*
c/(a*d-b*c)^2/d*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b-5/8/
(a*d-b*c)^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a+1/8*c/(a
*d-b*c)^2/d*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b-5/16/(a*
d-b*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+1/16*c/(a*d-b*c)^2/d*(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+1/4*a/(a*d-b*c)^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)))+1/2*a/(a*d-b
*c)^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+1/2*a/(a*d-b*c)^
2*(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(x^(7/2)/((b*x^2 + a)*(d*x^2 + c)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 7.54025, size = 3505, normalized size = 6.59 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(16*(-a^5/(b^9*c^8 - 8*a*b^8*c^7*d + 28*a^2*b^7*c^6*d^2 - 56*a^3*b^6*c^5*d^
3 + 70*a^4*b^5*c^4*d^4 - 56*a^5*b^4*c^3*d^5 + 28*a^6*b^3*c^2*d^6 - 8*a^7*b^2*c*d
^7 + a^8*b*d^8))^(1/4)*(b*c^2*d - a*c*d^2 + (b*c*d^2 - a*d^3)*x^2)*arctan((-a^5/
(b^9*c^8 - 8*a*b^8*c^7*d + 28*a^2*b^7*c^6*d^2 - 56*a^3*b^6*c^5*d^3 + 70*a^4*b^5*
c^4*d^4 - 56*a^5*b^4*c^3*d^5 + 28*a^6*b^3*c^2*d^6 - 8*a^7*b^2*c*d^7 + a^8*b*d^8)
)^(1/4)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)/(a*sqrt(x) + sqrt(a^2*x + (b^4*c^4 - 4*a
*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(-a^5/(b^9*c^8 - 8
*a*b^8*c^7*d + 28*a^2*b^7*c^6*d^2 - 56*a^3*b^6*c^5*d^3 + 70*a^4*b^5*c^4*d^4 - 56
*a^5*b^4*c^3*d^5 + 28*a^6*b^3*c^2*d^6 - 8*a^7*b^2*c*d^7 + a^8*b*d^8))))) - 4*(b*
c^2*d - a*c*d^2 + (b*c*d^2 - a*d^3)*x^2)*(-(b^4*c^5 - 20*a*b^3*c^4*d + 150*a^2*b
^2*c^3*d^2 - 500*a^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^8*c^8*d^5 - 8*a*b^7*c^7*d^6 +
28*a^2*b^6*c^6*d^7 - 56*a^3*b^5*c^5*d^8 + 70*a^4*b^4*c^4*d^9 - 56*a^5*b^3*c^3*d
^10 + 28*a^6*b^2*c^2*d^11 - 8*a^7*b*c*d^12 + a^8*d^13))^(1/4)*arctan(-(b^2*c^2*d
- 2*a*b*c*d^2 + a^2*d^3)*(-(b^4*c^5 - 20*a*b^3*c^4*d + 150*a^2*b^2*c^3*d^2 - 50
0*a^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^8*c^8*d^5 - 8*a*b^7*c^7*d^6 + 28*a^2*b^6*c^6
*d^7 - 56*a^3*b^5*c^5*d^8 + 70*a^4*b^4*c^4*d^9 - 56*a^5*b^3*c^3*d^10 + 28*a^6*b^
2*c^2*d^11 - 8*a^7*b*c*d^12 + a^8*d^13))^(1/4)/((b*c - 5*a*d)*sqrt(x) - sqrt((b^
2*c^2 - 10*a*b*c*d + 25*a^2*d^2)*x + (b^4*c^4*d^2 - 4*a*b^3*c^3*d^3 + 6*a^2*b^2*
c^2*d^4 - 4*a^3*b*c*d^5 + a^4*d^6)*sqrt(-(b^4*c^5 - 20*a*b^3*c^4*d + 150*a^2*b^2
*c^3*d^2 - 500*a^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^8*c^8*d^5 - 8*a*b^7*c^7*d^6 + 2
8*a^2*b^6*c^6*d^7 - 56*a^3*b^5*c^5*d^8 + 70*a^4*b^4*c^4*d^9 - 56*a^5*b^3*c^3*d^1
0 + 28*a^6*b^2*c^2*d^11 - 8*a^7*b*c*d^12 + a^8*d^13))))) + (b*c^2*d - a*c*d^2 +
(b*c*d^2 - a*d^3)*x^2)*(-(b^4*c^5 - 20*a*b^3*c^4*d + 150*a^2*b^2*c^3*d^2 - 500*a
^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^8*c^8*d^5 - 8*a*b^7*c^7*d^6 + 28*a^2*b^6*c^6*d^
7 - 56*a^3*b^5*c^5*d^8 + 70*a^4*b^4*c^4*d^9 - 56*a^5*b^3*c^3*d^10 + 28*a^6*b^2*c
^2*d^11 - 8*a^7*b*c*d^12 + a^8*d^13))^(1/4)*log(-(b*c - 5*a*d)*sqrt(x) + (b^2*c^
2*d - 2*a*b*c*d^2 + a^2*d^3)*(-(b^4*c^5 - 20*a*b^3*c^4*d + 150*a^2*b^2*c^3*d^2 -
500*a^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^8*c^8*d^5 - 8*a*b^7*c^7*d^6 + 28*a^2*b^6*
c^6*d^7 - 56*a^3*b^5*c^5*d^8 + 70*a^4*b^4*c^4*d^9 - 56*a^5*b^3*c^3*d^10 + 28*a^6
*b^2*c^2*d^11 - 8*a^7*b*c*d^12 + a^8*d^13))^(1/4)) - (b*c^2*d - a*c*d^2 + (b*c*d
^2 - a*d^3)*x^2)*(-(b^4*c^5 - 20*a*b^3*c^4*d + 150*a^2*b^2*c^3*d^2 - 500*a^3*b*c
^2*d^3 + 625*a^4*c*d^4)/(b^8*c^8*d^5 - 8*a*b^7*c^7*d^6 + 28*a^2*b^6*c^6*d^7 - 56
*a^3*b^5*c^5*d^8 + 70*a^4*b^4*c^4*d^9 - 56*a^5*b^3*c^3*d^10 + 28*a^6*b^2*c^2*d^1
1 - 8*a^7*b*c*d^12 + a^8*d^13))^(1/4)*log(-(b*c - 5*a*d)*sqrt(x) - (b^2*c^2*d -
2*a*b*c*d^2 + a^2*d^3)*(-(b^4*c^5 - 20*a*b^3*c^4*d + 150*a^2*b^2*c^3*d^2 - 500*a
^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^8*c^8*d^5 - 8*a*b^7*c^7*d^6 + 28*a^2*b^6*c^6*d^
7 - 56*a^3*b^5*c^5*d^8 + 70*a^4*b^4*c^4*d^9 - 56*a^5*b^3*c^3*d^10 + 28*a^6*b^2*c
^2*d^11 - 8*a^7*b*c*d^12 + a^8*d^13))^(1/4)) - 4*(-a^5/(b^9*c^8 - 8*a*b^8*c^7*d
+ 28*a^2*b^7*c^6*d^2 - 56*a^3*b^6*c^5*d^3 + 70*a^4*b^5*c^4*d^4 - 56*a^5*b^4*c^3*
d^5 + 28*a^6*b^3*c^2*d^6 - 8*a^7*b^2*c*d^7 + a^8*b*d^8))^(1/4)*(b*c^2*d - a*c*d^
2 + (b*c*d^2 - a*d^3)*x^2)*log(a*sqrt(x) + (-a^5/(b^9*c^8 - 8*a*b^8*c^7*d + 28*a
^2*b^7*c^6*d^2 - 56*a^3*b^6*c^5*d^3 + 70*a^4*b^5*c^4*d^4 - 56*a^5*b^4*c^3*d^5 +
28*a^6*b^3*c^2*d^6 - 8*a^7*b^2*c*d^7 + a^8*b*d^8))^(1/4)*(b^2*c^2 - 2*a*b*c*d +
a^2*d^2)) + 4*(-a^5/(b^9*c^8 - 8*a*b^8*c^7*d + 28*a^2*b^7*c^6*d^2 - 56*a^3*b^6*c
^5*d^3 + 70*a^4*b^5*c^4*d^4 - 56*a^5*b^4*c^3*d^5 + 28*a^6*b^3*c^2*d^6 - 8*a^7*b^
2*c*d^7 + a^8*b*d^8))^(1/4)*(b*c^2*d - a*c*d^2 + (b*c*d^2 - a*d^3)*x^2)*log(a*sq
rt(x) - (-a^5/(b^9*c^8 - 8*a*b^8*c^7*d + 28*a^2*b^7*c^6*d^2 - 56*a^3*b^6*c^5*d^3
+ 70*a^4*b^5*c^4*d^4 - 56*a^5*b^4*c^3*d^5 + 28*a^6*b^3*c^2*d^6 - 8*a^7*b^2*c*d^
7 + a^8*b*d^8))^(1/4)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)) + 4*c*sqrt(x))/(b*c^2*d -
a*c*d^2 + (b*c*d^2 - a*d^3)*x^2)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.345843, size = 903, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a*b^3)^(1/4)*a*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4)
)/(sqrt(2)*b^3*c^2 - 2*sqrt(2)*a*b^2*c*d + sqrt(2)*a^2*b*d^2) + (a*b^3)^(1/4)*a*
arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*b^3*
c^2 - 2*sqrt(2)*a*b^2*c*d + sqrt(2)*a^2*b*d^2) + 1/2*(a*b^3)^(1/4)*a*ln(sqrt(2)*
sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*b^3*c^2 - 2*sqrt(2)*a*b^2*c*d + sq
rt(2)*a^2*b*d^2) - 1/2*(a*b^3)^(1/4)*a*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqr
t(a/b))/(sqrt(2)*b^3*c^2 - 2*sqrt(2)*a*b^2*c*d + sqrt(2)*a^2*b*d^2) + 1/4*((c*d^
3)^(1/4)*b*c - 5*(c*d^3)^(1/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*
sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^2*c^2*d^2 - 2*sqrt(2)*a*b*c*d^3 + sqrt(2)*a^2*d
^4) + 1/4*((c*d^3)^(1/4)*b*c - 5*(c*d^3)^(1/4)*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)
*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^2*c^2*d^2 - 2*sqrt(2)*a*b*c*d^
3 + sqrt(2)*a^2*d^4) + 1/8*((c*d^3)^(1/4)*b*c - 5*(c*d^3)^(1/4)*a*d)*ln(sqrt(2)*
sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^2*c^2*d^2 - 2*sqrt(2)*a*b*c*d^3
+ sqrt(2)*a^2*d^4) - 1/8*((c*d^3)^(1/4)*b*c - 5*(c*d^3)^(1/4)*a*d)*ln(-sqrt(2)*s
qrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^2*c^2*d^2 - 2*sqrt(2)*a*b*c*d^3 +
sqrt(2)*a^2*d^4) - 1/2*c*sqrt(x)/((b*c*d - a*d^2)*(d*x^2 + c))