3.424 \(\int \frac{\left (c+d x^2\right )^2}{x^{11/2} \left (a+b x^2\right )} \, dx\)

Optimal. Leaf size=288 \[ -\frac{\sqrt [4]{b} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{13/4}}+\frac{\sqrt [4]{b} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{13/4}}+\frac{\sqrt [4]{b} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{13/4}}-\frac{\sqrt [4]{b} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{13/4}}-\frac{2 (b c-a d)^2}{a^3 \sqrt{x}}+\frac{2 c (b c-2 a d)}{5 a^2 x^{5/2}}-\frac{2 c^2}{9 a x^{9/2}} \]

[Out]

(-2*c^2)/(9*a*x^(9/2)) + (2*c*(b*c - 2*a*d))/(5*a^2*x^(5/2)) - (2*(b*c - a*d)^2)
/(a^3*Sqrt[x]) + (b^(1/4)*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(
1/4)])/(Sqrt[2]*a^(13/4)) - (b^(1/4)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*S
qrt[x])/a^(1/4)])/(Sqrt[2]*a^(13/4)) - (b^(1/4)*(b*c - a*d)^2*Log[Sqrt[a] - Sqrt
[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(13/4)) + (b^(1/4)*(b*c -
a*d)^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a
^(13/4))

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Rubi [A]  time = 0.699275, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{\sqrt [4]{b} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{13/4}}+\frac{\sqrt [4]{b} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{13/4}}+\frac{\sqrt [4]{b} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{13/4}}-\frac{\sqrt [4]{b} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{13/4}}-\frac{2 (b c-a d)^2}{a^3 \sqrt{x}}+\frac{2 c (b c-2 a d)}{5 a^2 x^{5/2}}-\frac{2 c^2}{9 a x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)^2/(x^(11/2)*(a + b*x^2)),x]

[Out]

(-2*c^2)/(9*a*x^(9/2)) + (2*c*(b*c - 2*a*d))/(5*a^2*x^(5/2)) - (2*(b*c - a*d)^2)
/(a^3*Sqrt[x]) + (b^(1/4)*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(
1/4)])/(Sqrt[2]*a^(13/4)) - (b^(1/4)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*S
qrt[x])/a^(1/4)])/(Sqrt[2]*a^(13/4)) - (b^(1/4)*(b*c - a*d)^2*Log[Sqrt[a] - Sqrt
[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(13/4)) + (b^(1/4)*(b*c -
a*d)^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a
^(13/4))

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Rubi in Sympy [A]  time = 110.706, size = 270, normalized size = 0.94 \[ - \frac{2 c^{2}}{9 a x^{\frac{9}{2}}} - \frac{2 c \left (2 a d - b c\right )}{5 a^{2} x^{\frac{5}{2}}} - \frac{2 \left (a d - b c\right )^{2}}{a^{3} \sqrt{x}} - \frac{\sqrt{2} \sqrt [4]{b} \left (a d - b c\right )^{2} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 a^{\frac{13}{4}}} + \frac{\sqrt{2} \sqrt [4]{b} \left (a d - b c\right )^{2} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 a^{\frac{13}{4}}} + \frac{\sqrt{2} \sqrt [4]{b} \left (a d - b c\right )^{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{13}{4}}} - \frac{\sqrt{2} \sqrt [4]{b} \left (a d - b c\right )^{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{13}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**2/x**(11/2)/(b*x**2+a),x)

[Out]

-2*c**2/(9*a*x**(9/2)) - 2*c*(2*a*d - b*c)/(5*a**2*x**(5/2)) - 2*(a*d - b*c)**2/
(a**3*sqrt(x)) - sqrt(2)*b**(1/4)*(a*d - b*c)**2*log(-sqrt(2)*a**(1/4)*b**(1/4)*
sqrt(x) + sqrt(a) + sqrt(b)*x)/(4*a**(13/4)) + sqrt(2)*b**(1/4)*(a*d - b*c)**2*l
og(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(4*a**(13/4)) + sqrt
(2)*b**(1/4)*(a*d - b*c)**2*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(2*a**(1
3/4)) - sqrt(2)*b**(1/4)*(a*d - b*c)**2*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/
4))/(2*a**(13/4))

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Mathematica [A]  time = 0.229892, size = 277, normalized size = 0.96 \[ \frac{-\frac{72 a^{5/4} c (2 a d-b c)}{x^{5/2}}-\frac{40 a^{9/4} c^2}{x^{9/2}}-\frac{360 \sqrt [4]{a} (b c-a d)^2}{\sqrt{x}}-45 \sqrt{2} \sqrt [4]{b} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+45 \sqrt{2} \sqrt [4]{b} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+90 \sqrt{2} \sqrt [4]{b} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-90 \sqrt{2} \sqrt [4]{b} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{180 a^{13/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)^2/(x^(11/2)*(a + b*x^2)),x]

[Out]

((-40*a^(9/4)*c^2)/x^(9/2) - (72*a^(5/4)*c*(-(b*c) + 2*a*d))/x^(5/2) - (360*a^(1
/4)*(b*c - a*d)^2)/Sqrt[x] + 90*Sqrt[2]*b^(1/4)*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2
]*b^(1/4)*Sqrt[x])/a^(1/4)] - 90*Sqrt[2]*b^(1/4)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[
2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 45*Sqrt[2]*b^(1/4)*(b*c - a*d)^2*Log[Sqrt[a] - Sq
rt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 45*Sqrt[2]*b^(1/4)*(b*c - a*d)^2*Lo
g[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(180*a^(13/4))

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Maple [B]  time = 0.021, size = 495, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^2/x^(11/2)/(b*x^2+a),x)

[Out]

-2/9*c^2/a/x^(9/2)-2/a/x^(1/2)*d^2+4/a^2/x^(1/2)*c*b*d-2/a^3/x^(1/2)*b^2*c^2-4/5
*c/a/x^(5/2)*d+2/5*c^2/a^2/x^(5/2)*b-1/2/a/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a
/b)^(1/4)*x^(1/2)+1)*d^2+1/a^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^
(1/2)+1)*b*c*d-1/2/a^3/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)
*b^2*c^2-1/2/a/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*d^2+1/a
^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*b*c*d-1/2/a^3/(a/b)
^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*b^2*c^2-1/4/a/(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^2+1/2/a^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)))*b*c*d-1/4/a^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)))*b^2*c^2

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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((d*x^2 + c)^2/((b*x^2 + a)*x^(11/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.264203, size = 1958, normalized size = 6.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^2/((b*x^2 + a)*x^(11/2)),x, algorithm="fricas")

[Out]

-1/90*(180*a^3*x^(9/2)*(-(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)/a^13)^(1/4)*arctan(a^10*(-(b^9*c^8 - 8*a*b^8*c^7*d + 2
8*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)/a^13)^(3/4)/((b^7*c^6 - 6*a
*b^6*c^5*d + 15*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^3 + 15*a^4*b^3*c^2*d^4 - 6*a^
5*b^2*c*d^5 + a^6*b*d^6)*sqrt(x) + sqrt((b^14*c^12 - 12*a*b^13*c^11*d + 66*a^2*b
^12*c^10*d^2 - 220*a^3*b^11*c^9*d^3 + 495*a^4*b^10*c^8*d^4 - 792*a^5*b^9*c^7*d^5
+ 924*a^6*b^8*c^6*d^6 - 792*a^7*b^7*c^5*d^7 + 495*a^8*b^6*c^4*d^8 - 220*a^9*b^5
*c^3*d^9 + 66*a^10*b^4*c^2*d^10 - 12*a^11*b^3*c*d^11 + a^12*b^2*d^12)*x - (a^7*b
^9*c^8 - 8*a^8*b^8*c^7*d + 28*a^9*b^7*c^6*d^2 - 56*a^10*b^6*c^5*d^3 + 70*a^11*b^
5*c^4*d^4 - 56*a^12*b^4*c^3*d^5 + 28*a^13*b^3*c^2*d^6 - 8*a^14*b^2*c*d^7 + a^15*
b*d^8)*sqrt(-(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)/a^13)))) + 45*a^3*x^(9/2)*(-(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)/a^13)^(1/4)*log(a^10*(-(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)/a^13)^(3/4) +
(b^7*c^6 - 6*a*b^6*c^5*d + 15*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^3 + 15*a^4*b^3
*c^2*d^4 - 6*a^5*b^2*c*d^5 + a^6*b*d^6)*sqrt(x)) - 45*a^3*x^(9/2)*(-(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)/a^13)^(1/4)
*log(-a^10*(-(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)/a^13)^(3/4) + (b^7*c^6 - 6*a*b^6*c^5*d + 15*a^2*b^5*c^4*d^2 - 20*a
^3*b^4*c^3*d^3 + 15*a^4*b^3*c^2*d^4 - 6*a^5*b^2*c*d^5 + a^6*b*d^6)*sqrt(x)) + 18
0*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^4 + 20*a^2*c^2 - 36*(a*b*c^2 - 2*a^2*c*d)*x^
2)/(a^3*x^(9/2))

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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((d*x**2+c)**2/x**(11/2)/(b*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.259147, size = 527, normalized size = 1.83 \[ -\frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 2 \, \left (a b^{3}\right )^{\frac{3}{4}} a b c d + \left (a b^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a^{4} b^{2}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 2 \, \left (a b^{3}\right )^{\frac{3}{4}} a b c d + \left (a b^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a^{4} b^{2}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 2 \, \left (a b^{3}\right )^{\frac{3}{4}} a b c d + \left (a b^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, a^{4} b^{2}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 2 \, \left (a b^{3}\right )^{\frac{3}{4}} a b c d + \left (a b^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, a^{4} b^{2}} - \frac{2 \,{\left (45 \, b^{2} c^{2} x^{4} - 90 \, a b c d x^{4} + 45 \, a^{2} d^{2} x^{4} - 9 \, a b c^{2} x^{2} + 18 \, a^{2} c d x^{2} + 5 \, a^{2} c^{2}\right )}}{45 \, a^{3} x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^2/((b*x^2 + a)*x^(11/2)),x, algorithm="giac")

[Out]

-1/2*sqrt(2)*((a*b^3)^(3/4)*b^2*c^2 - 2*(a*b^3)^(3/4)*a*b*c*d + (a*b^3)^(3/4)*a^
2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^4*b^
2) - 1/2*sqrt(2)*((a*b^3)^(3/4)*b^2*c^2 - 2*(a*b^3)^(3/4)*a*b*c*d + (a*b^3)^(3/4
)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a
^4*b^2) + 1/4*sqrt(2)*((a*b^3)^(3/4)*b^2*c^2 - 2*(a*b^3)^(3/4)*a*b*c*d + (a*b^3)
^(3/4)*a^2*d^2)*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b^2) - 1/4*
sqrt(2)*((a*b^3)^(3/4)*b^2*c^2 - 2*(a*b^3)^(3/4)*a*b*c*d + (a*b^3)^(3/4)*a^2*d^2
)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b^2) - 2/45*(45*b^2*c^2*
x^4 - 90*a*b*c*d*x^4 + 45*a^2*d^2*x^4 - 9*a*b*c^2*x^2 + 18*a^2*c*d*x^2 + 5*a^2*c
^2)/(a^3*x^(9/2))