3.423 \(\int \frac{\left (a+b x^2\right )^2}{x^{9/2} \left (c+d x^2\right )} \, dx\)

Optimal. Leaf size=269 \[ -\frac{2 a^2}{7 c x^{7/2}}-\frac{(b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} c^{11/4} \sqrt [4]{d}}+\frac{(b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} c^{11/4} \sqrt [4]{d}}-\frac{(b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} c^{11/4} \sqrt [4]{d}}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt{2} c^{11/4} \sqrt [4]{d}}-\frac{2 a (2 b c-a d)}{3 c^2 x^{3/2}} \]

[Out]

(-2*a^2)/(7*c*x^(7/2)) - (2*a*(2*b*c - a*d))/(3*c^2*x^(3/2)) - ((b*c - a*d)^2*Ar
cTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt[2]*c^(11/4)*d^(1/4)) + ((b*c
- a*d)^2*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt[2]*c^(11/4)*d^(1/4
)) - ((b*c - a*d)^2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/
(2*Sqrt[2]*c^(11/4)*d^(1/4)) + ((b*c - a*d)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1
/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*c^(11/4)*d^(1/4))

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Rubi [A]  time = 0.587205, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{2 a^2}{7 c x^{7/2}}-\frac{(b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} c^{11/4} \sqrt [4]{d}}+\frac{(b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} c^{11/4} \sqrt [4]{d}}-\frac{(b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} c^{11/4} \sqrt [4]{d}}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt{2} c^{11/4} \sqrt [4]{d}}-\frac{2 a (2 b c-a d)}{3 c^2 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/(x^(9/2)*(c + d*x^2)),x]

[Out]

(-2*a^2)/(7*c*x^(7/2)) - (2*a*(2*b*c - a*d))/(3*c^2*x^(3/2)) - ((b*c - a*d)^2*Ar
cTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt[2]*c^(11/4)*d^(1/4)) + ((b*c
- a*d)^2*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt[2]*c^(11/4)*d^(1/4
)) - ((b*c - a*d)^2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/
(2*Sqrt[2]*c^(11/4)*d^(1/4)) + ((b*c - a*d)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1
/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*c^(11/4)*d^(1/4))

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Rubi in Sympy [A]  time = 97.1718, size = 252, normalized size = 0.94 \[ - \frac{2 a^{2}}{7 c x^{\frac{7}{2}}} + \frac{2 a \left (a d - 2 b c\right )}{3 c^{2} x^{\frac{3}{2}}} - \frac{\sqrt{2} \left (a d - b c\right )^{2} \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{4 c^{\frac{11}{4}} \sqrt [4]{d}} + \frac{\sqrt{2} \left (a d - b c\right )^{2} \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{4 c^{\frac{11}{4}} \sqrt [4]{d}} - \frac{\sqrt{2} \left (a d - b c\right )^{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{2 c^{\frac{11}{4}} \sqrt [4]{d}} + \frac{\sqrt{2} \left (a d - b c\right )^{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{2 c^{\frac{11}{4}} \sqrt [4]{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2/x**(9/2)/(d*x**2+c),x)

[Out]

-2*a**2/(7*c*x**(7/2)) + 2*a*(a*d - 2*b*c)/(3*c**2*x**(3/2)) - sqrt(2)*(a*d - b*
c)**2*log(-sqrt(2)*c**(1/4)*d**(1/4)*sqrt(x) + sqrt(c) + sqrt(d)*x)/(4*c**(11/4)
*d**(1/4)) + sqrt(2)*(a*d - b*c)**2*log(sqrt(2)*c**(1/4)*d**(1/4)*sqrt(x) + sqrt
(c) + sqrt(d)*x)/(4*c**(11/4)*d**(1/4)) - sqrt(2)*(a*d - b*c)**2*atan(1 - sqrt(2
)*d**(1/4)*sqrt(x)/c**(1/4))/(2*c**(11/4)*d**(1/4)) + sqrt(2)*(a*d - b*c)**2*ata
n(1 + sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(2*c**(11/4)*d**(1/4))

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Mathematica [A]  time = 0.192612, size = 254, normalized size = 0.94 \[ \frac{-\frac{24 a^2 c^{7/4}}{x^{7/2}}+\frac{56 a c^{3/4} (a d-2 b c)}{x^{3/2}}-\frac{21 \sqrt{2} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{\sqrt [4]{d}}+\frac{21 \sqrt{2} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{\sqrt [4]{d}}-\frac{42 \sqrt{2} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt [4]{d}}+\frac{42 \sqrt{2} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt [4]{d}}}{84 c^{11/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2/(x^(9/2)*(c + d*x^2)),x]

[Out]

((-24*a^2*c^(7/4))/x^(7/2) + (56*a*c^(3/4)*(-2*b*c + a*d))/x^(3/2) - (42*Sqrt[2]
*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(1/4) + (42*Sqrt
[2]*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(1/4) - (21*S
qrt[2]*(b*c - a*d)^2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])
/d^(1/4) + (21*Sqrt[2]*(b*c - a*d)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[
x] + Sqrt[d]*x])/d^(1/4))/(84*c^(11/4))

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Maple [B]  time = 0.02, size = 461, normalized size = 1.7 \[{\frac{\sqrt{2}{a}^{2}{d}^{2}}{2\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }-{\frac{\sqrt{2}abd}{{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{\sqrt{2}{b}^{2}}{2\,c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{\sqrt{2}{a}^{2}{d}^{2}}{2\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }-{\frac{\sqrt{2}abd}{{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{\sqrt{2}{b}^{2}}{2\,c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{\sqrt{2}{a}^{2}{d}^{2}}{4\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}abd}{2\,{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}{b}^{2}}{4\,c}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }-{\frac{2\,{a}^{2}}{7\,c}{x}^{-{\frac{7}{2}}}}+{\frac{2\,{a}^{2}d}{3\,{c}^{2}}{x}^{-{\frac{3}{2}}}}-{\frac{4\,ab}{3\,c}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/x^(9/2)/(d*x^2+c),x)

[Out]

1/2/c^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2*d^2-1/c^2*
(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a*b*d+1/2/c*(c/d)^(1/4
)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b^2+1/2/c^3*(c/d)^(1/4)*2^(1/2)*
arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2*d^2-1/c^2*(c/d)^(1/4)*2^(1/2)*arctan(2
^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a*b*d+1/2/c*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/
d)^(1/4)*x^(1/2)-1)*b^2+1/4/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*d^2-1/2/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*d+1/4/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-
2/7*a^2/c/x^(7/2)+2/3*a^2/c^2/x^(3/2)*d-4/3*a/c/x^(3/2)*b

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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((b*x^2 + a)^2/((d*x^2 + c)*x^(9/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.257235, size = 1370, normalized size = 5.09 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)*x^(9/2)),x, algorithm="fricas")

[Out]

-1/42*(84*c^2*x^(7/2)*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b
^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^
7*b*c*d^7 + a^8*d^8)/(c^11*d))^(1/4)*arctan(c^3*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*
a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 +
28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^11*d))^(1/4)/((b^2*c^2 - 2*a*b
*c*d + a^2*d^2)*sqrt(x) + sqrt(c^6*sqrt(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c
^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b
^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^11*d)) + (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)*x))) - 21*c^2*x^(7/2)*(-(b^8*c^8 - 8*a
*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a
^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^11*d))^(1/4)*l
og(c^3*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70
*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8
*d^8)/(c^11*d))^(1/4) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(x)) + 21*c^2*x^(7/2
)*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*
b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)
/(c^11*d))^(1/4)*log(-c^3*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a
^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 -
8*a^7*b*c*d^7 + a^8*d^8)/(c^11*d))^(1/4) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(
x)) + 12*a^2*c + 28*(2*a*b*c - a^2*d)*x^2)/(c^2*x^(7/2))

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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((b*x**2+a)**2/x**(9/2)/(d*x**2+c),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.239125, size = 478, normalized size = 1.78 \[ \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{2 \, c^{3} d} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{2 \, c^{3} d} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{4 \, c^{3} d} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{4 \, c^{3} d} - \frac{2 \,{\left (14 \, a b c x^{2} - 7 \, a^{2} d x^{2} + 3 \, a^{2} c\right )}}{21 \, c^{2} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)*x^(9/2)),x, algorithm="giac")

[Out]

1/2*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4)*a^2
*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(c^3*d)
+ 1/2*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4)*a
^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(c^3*
d) + 1/4*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4
)*a^2*d^2)*ln(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^3*d) - 1/4*sqrt(2)
*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4)*a^2*d^2)*ln(-s
qrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^3*d) - 2/21*(14*a*b*c*x^2 - 7*a^2
*d*x^2 + 3*a^2*c)/(c^2*x^(7/2))