3.416 \(\int \frac{x^{5/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx\)

Optimal. Leaf size=290 \[ -\frac{c^{3/4} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} d^{15/4}}+\frac{c^{3/4} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} d^{15/4}}+\frac{c^{3/4} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} d^{15/4}}-\frac{c^{3/4} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt{2} d^{15/4}}+\frac{2 x^{3/2} (b c-a d)^2}{3 d^3}-\frac{2 b x^{7/2} (b c-2 a d)}{7 d^2}+\frac{2 b^2 x^{11/2}}{11 d} \]

[Out]

(2*(b*c - a*d)^2*x^(3/2))/(3*d^3) - (2*b*(b*c - 2*a*d)*x^(7/2))/(7*d^2) + (2*b^2
*x^(11/2))/(11*d) + (c^(3/4)*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/
c^(1/4)])/(Sqrt[2]*d^(15/4)) - (c^(3/4)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*d^(1/4
)*Sqrt[x])/c^(1/4)])/(Sqrt[2]*d^(15/4)) - (c^(3/4)*(b*c - a*d)^2*Log[Sqrt[c] - S
qrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*d^(15/4)) + (c^(3/4)*(b*
c - a*d)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2
]*d^(15/4))

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Rubi [A]  time = 0.532838, antiderivative size = 290, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{c^{3/4} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} d^{15/4}}+\frac{c^{3/4} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} d^{15/4}}+\frac{c^{3/4} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} d^{15/4}}-\frac{c^{3/4} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt{2} d^{15/4}}+\frac{2 x^{3/2} (b c-a d)^2}{3 d^3}-\frac{2 b x^{7/2} (b c-2 a d)}{7 d^2}+\frac{2 b^2 x^{11/2}}{11 d} \]

Antiderivative was successfully verified.

[In]  Int[(x^(5/2)*(a + b*x^2)^2)/(c + d*x^2),x]

[Out]

(2*(b*c - a*d)^2*x^(3/2))/(3*d^3) - (2*b*(b*c - 2*a*d)*x^(7/2))/(7*d^2) + (2*b^2
*x^(11/2))/(11*d) + (c^(3/4)*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/
c^(1/4)])/(Sqrt[2]*d^(15/4)) - (c^(3/4)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*d^(1/4
)*Sqrt[x])/c^(1/4)])/(Sqrt[2]*d^(15/4)) - (c^(3/4)*(b*c - a*d)^2*Log[Sqrt[c] - S
qrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*d^(15/4)) + (c^(3/4)*(b*
c - a*d)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2
]*d^(15/4))

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Rubi in Sympy [A]  time = 96.3314, size = 272, normalized size = 0.94 \[ \frac{2 b^{2} x^{\frac{11}{2}}}{11 d} + \frac{2 b x^{\frac{7}{2}} \left (2 a d - b c\right )}{7 d^{2}} - \frac{\sqrt{2} c^{\frac{3}{4}} \left (a d - b c\right )^{2} \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{4 d^{\frac{15}{4}}} + \frac{\sqrt{2} c^{\frac{3}{4}} \left (a d - b c\right )^{2} \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{4 d^{\frac{15}{4}}} + \frac{\sqrt{2} c^{\frac{3}{4}} \left (a d - b c\right )^{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{2 d^{\frac{15}{4}}} - \frac{\sqrt{2} c^{\frac{3}{4}} \left (a d - b c\right )^{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{2 d^{\frac{15}{4}}} + \frac{2 x^{\frac{3}{2}} \left (a d - b c\right )^{2}}{3 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(b*x**2+a)**2/(d*x**2+c),x)

[Out]

2*b**2*x**(11/2)/(11*d) + 2*b*x**(7/2)*(2*a*d - b*c)/(7*d**2) - sqrt(2)*c**(3/4)
*(a*d - b*c)**2*log(-sqrt(2)*c**(1/4)*d**(1/4)*sqrt(x) + sqrt(c) + sqrt(d)*x)/(4
*d**(15/4)) + sqrt(2)*c**(3/4)*(a*d - b*c)**2*log(sqrt(2)*c**(1/4)*d**(1/4)*sqrt
(x) + sqrt(c) + sqrt(d)*x)/(4*d**(15/4)) + sqrt(2)*c**(3/4)*(a*d - b*c)**2*atan(
1 - sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(2*d**(15/4)) - sqrt(2)*c**(3/4)*(a*d - b
*c)**2*atan(1 + sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(2*d**(15/4)) + 2*x**(3/2)*(a
*d - b*c)**2/(3*d**3)

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Mathematica [A]  time = 0.17283, size = 276, normalized size = 0.95 \[ \frac{-231 \sqrt{2} c^{3/4} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )+231 \sqrt{2} c^{3/4} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )+462 \sqrt{2} c^{3/4} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )-462 \sqrt{2} c^{3/4} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )-264 b d^{7/4} x^{7/2} (b c-2 a d)+616 d^{3/4} x^{3/2} (b c-a d)^2+168 b^2 d^{11/4} x^{11/2}}{924 d^{15/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(5/2)*(a + b*x^2)^2)/(c + d*x^2),x]

[Out]

(616*d^(3/4)*(b*c - a*d)^2*x^(3/2) - 264*b*d^(7/4)*(b*c - 2*a*d)*x^(7/2) + 168*b
^2*d^(11/4)*x^(11/2) + 462*Sqrt[2]*c^(3/4)*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*d^(
1/4)*Sqrt[x])/c^(1/4)] - 462*Sqrt[2]*c^(3/4)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*d
^(1/4)*Sqrt[x])/c^(1/4)] - 231*Sqrt[2]*c^(3/4)*(b*c - a*d)^2*Log[Sqrt[c] - Sqrt[
2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] + 231*Sqrt[2]*c^(3/4)*(b*c - a*d)^2*Log[
Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(924*d^(15/4))

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Maple [B]  time = 0.017, size = 504, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(b*x^2+a)^2/(d*x^2+c),x)

[Out]

2/11*b^2*x^(11/2)/d+4/7/d*x^(7/2)*a*b-2/7/d^2*x^(7/2)*b^2*c+2/3/d*x^(3/2)*a^2-4/
3/d^2*x^(3/2)*c*a*b+2/3/d^3*x^(3/2)*b^2*c^2-1/2*c/d^2/(c/d)^(1/4)*2^(1/2)*arctan
(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2+c^2/d^3/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(
c/d)^(1/4)*x^(1/2)+1)*a*b-1/2*c^3/d^4/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(
1/4)*x^(1/2)+1)*b^2-1/2*c/d^2/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(
1/2)-1)*a^2+c^2/d^3/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a*
b-1/2*c^3/d^4/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b^2-1/4*
c/d^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^2+1/2*c^2/d^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*b-1/4*c^3/d^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)))*b^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((b*x^2 + a)^2*x^(5/2)/(d*x^2 + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.262635, size = 1980, normalized size = 6.83 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*x^(5/2)/(d*x^2 + c),x, algorithm="fricas")

[Out]

-1/462*(924*d^3*(-(b^8*c^11 - 8*a*b^7*c^10*d + 28*a^2*b^6*c^9*d^2 - 56*a^3*b^5*c
^8*d^3 + 70*a^4*b^4*c^7*d^4 - 56*a^5*b^3*c^6*d^5 + 28*a^6*b^2*c^5*d^6 - 8*a^7*b*
c^4*d^7 + a^8*c^3*d^8)/d^15)^(1/4)*arctan(d^11*(-(b^8*c^11 - 8*a*b^7*c^10*d + 28
*a^2*b^6*c^9*d^2 - 56*a^3*b^5*c^8*d^3 + 70*a^4*b^4*c^7*d^4 - 56*a^5*b^3*c^6*d^5
+ 28*a^6*b^2*c^5*d^6 - 8*a^7*b*c^4*d^7 + a^8*c^3*d^8)/d^15)^(3/4)/((b^6*c^8 - 6*
a*b^5*c^7*d + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a
^5*b*c^3*d^5 + a^6*c^2*d^6)*sqrt(x) + sqrt((b^12*c^16 - 12*a*b^11*c^15*d + 66*a^
2*b^10*c^14*d^2 - 220*a^3*b^9*c^13*d^3 + 495*a^4*b^8*c^12*d^4 - 792*a^5*b^7*c^11
*d^5 + 924*a^6*b^6*c^10*d^6 - 792*a^7*b^5*c^9*d^7 + 495*a^8*b^4*c^8*d^8 - 220*a^
9*b^3*c^7*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a^11*b*c^5*d^11 + a^12*c^4*d^12)*x - (
b^8*c^11*d^7 - 8*a*b^7*c^10*d^8 + 28*a^2*b^6*c^9*d^9 - 56*a^3*b^5*c^8*d^10 + 70*
a^4*b^4*c^7*d^11 - 56*a^5*b^3*c^6*d^12 + 28*a^6*b^2*c^5*d^13 - 8*a^7*b*c^4*d^14
+ a^8*c^3*d^15)*sqrt(-(b^8*c^11 - 8*a*b^7*c^10*d + 28*a^2*b^6*c^9*d^2 - 56*a^3*b
^5*c^8*d^3 + 70*a^4*b^4*c^7*d^4 - 56*a^5*b^3*c^6*d^5 + 28*a^6*b^2*c^5*d^6 - 8*a^
7*b*c^4*d^7 + a^8*c^3*d^8)/d^15)))) + 231*d^3*(-(b^8*c^11 - 8*a*b^7*c^10*d + 28*
a^2*b^6*c^9*d^2 - 56*a^3*b^5*c^8*d^3 + 70*a^4*b^4*c^7*d^4 - 56*a^5*b^3*c^6*d^5 +
28*a^6*b^2*c^5*d^6 - 8*a^7*b*c^4*d^7 + a^8*c^3*d^8)/d^15)^(1/4)*log(d^11*(-(b^8
*c^11 - 8*a*b^7*c^10*d + 28*a^2*b^6*c^9*d^2 - 56*a^3*b^5*c^8*d^3 + 70*a^4*b^4*c^
7*d^4 - 56*a^5*b^3*c^6*d^5 + 28*a^6*b^2*c^5*d^6 - 8*a^7*b*c^4*d^7 + a^8*c^3*d^8)
/d^15)^(3/4) + (b^6*c^8 - 6*a*b^5*c^7*d + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^
3 + 15*a^4*b^2*c^4*d^4 - 6*a^5*b*c^3*d^5 + a^6*c^2*d^6)*sqrt(x)) - 231*d^3*(-(b^
8*c^11 - 8*a*b^7*c^10*d + 28*a^2*b^6*c^9*d^2 - 56*a^3*b^5*c^8*d^3 + 70*a^4*b^4*c
^7*d^4 - 56*a^5*b^3*c^6*d^5 + 28*a^6*b^2*c^5*d^6 - 8*a^7*b*c^4*d^7 + a^8*c^3*d^8
)/d^15)^(1/4)*log(-d^11*(-(b^8*c^11 - 8*a*b^7*c^10*d + 28*a^2*b^6*c^9*d^2 - 56*a
^3*b^5*c^8*d^3 + 70*a^4*b^4*c^7*d^4 - 56*a^5*b^3*c^6*d^5 + 28*a^6*b^2*c^5*d^6 -
8*a^7*b*c^4*d^7 + a^8*c^3*d^8)/d^15)^(3/4) + (b^6*c^8 - 6*a*b^5*c^7*d + 15*a^2*b
^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a^5*b*c^3*d^5 + a^6*c^2
*d^6)*sqrt(x)) - 4*(21*b^2*d^2*x^5 - 33*(b^2*c*d - 2*a*b*d^2)*x^3 + 77*(b^2*c^2
- 2*a*b*c*d + a^2*d^2)*x)*sqrt(x))/d^3

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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**(5/2)*(b*x**2+a)**2/(d*x**2+c),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.249353, size = 520, normalized size = 1.79 \[ -\frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{2 \, d^{6}} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{2 \, d^{6}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{4 \, d^{6}} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{4 \, d^{6}} + \frac{2 \,{\left (21 \, b^{2} d^{10} x^{\frac{11}{2}} - 33 \, b^{2} c d^{9} x^{\frac{7}{2}} + 66 \, a b d^{10} x^{\frac{7}{2}} + 77 \, b^{2} c^{2} d^{8} x^{\frac{3}{2}} - 154 \, a b c d^{9} x^{\frac{3}{2}} + 77 \, a^{2} d^{10} x^{\frac{3}{2}}\right )}}{231 \, d^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*x^(5/2)/(d*x^2 + c),x, algorithm="giac")

[Out]

-1/2*sqrt(2)*((c*d^3)^(3/4)*b^2*c^2 - 2*(c*d^3)^(3/4)*a*b*c*d + (c*d^3)^(3/4)*a^
2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/d^6 - 1
/2*sqrt(2)*((c*d^3)^(3/4)*b^2*c^2 - 2*(c*d^3)^(3/4)*a*b*c*d + (c*d^3)^(3/4)*a^2*
d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/d^6 + 1/
4*sqrt(2)*((c*d^3)^(3/4)*b^2*c^2 - 2*(c*d^3)^(3/4)*a*b*c*d + (c*d^3)^(3/4)*a^2*d
^2)*ln(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/d^6 - 1/4*sqrt(2)*((c*d^3)^(
3/4)*b^2*c^2 - 2*(c*d^3)^(3/4)*a*b*c*d + (c*d^3)^(3/4)*a^2*d^2)*ln(-sqrt(2)*sqrt
(x)*(c/d)^(1/4) + x + sqrt(c/d))/d^6 + 2/231*(21*b^2*d^10*x^(11/2) - 33*b^2*c*d^
9*x^(7/2) + 66*a*b*d^10*x^(7/2) + 77*b^2*c^2*d^8*x^(3/2) - 154*a*b*c*d^9*x^(3/2)
+ 77*a^2*d^10*x^(3/2))/d^11