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

Optimal. Leaf size=311 \[ -\frac{c^{5/4} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} d^{17/4}}+\frac{c^{5/4} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} d^{17/4}}-\frac{c^{5/4} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} d^{17/4}}+\frac{c^{5/4} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt{2} d^{17/4}}-\frac{2 c \sqrt{x} (b c-a d)^2}{d^4}+\frac{2 x^{5/2} (b c-a d)^2}{5 d^3}-\frac{2 b x^{9/2} (b c-2 a d)}{9 d^2}+\frac{2 b^2 x^{13/2}}{13 d} \]

[Out]

(-2*c*(b*c - a*d)^2*Sqrt[x])/d^4 + (2*(b*c - a*d)^2*x^(5/2))/(5*d^3) - (2*b*(b*c
- 2*a*d)*x^(9/2))/(9*d^2) + (2*b^2*x^(13/2))/(13*d) - (c^(5/4)*(b*c - a*d)^2*Ar
cTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt[2]*d^(17/4)) + (c^(5/4)*(b*c
- a*d)^2*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt[2]*d^(17/4)) - (c^
(5/4)*(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^(17/4)) + (c^(5/4)*(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^(17/4))

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Rubi [A]  time = 0.630695, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{c^{5/4} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} d^{17/4}}+\frac{c^{5/4} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} d^{17/4}}-\frac{c^{5/4} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} d^{17/4}}+\frac{c^{5/4} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt{2} d^{17/4}}-\frac{2 c \sqrt{x} (b c-a d)^2}{d^4}+\frac{2 x^{5/2} (b c-a d)^2}{5 d^3}-\frac{2 b x^{9/2} (b c-2 a d)}{9 d^2}+\frac{2 b^2 x^{13/2}}{13 d} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c*(b*c - a*d)^2*Sqrt[x])/d^4 + (2*(b*c - a*d)^2*x^(5/2))/(5*d^3) - (2*b*(b*c
- 2*a*d)*x^(9/2))/(9*d^2) + (2*b^2*x^(13/2))/(13*d) - (c^(5/4)*(b*c - a*d)^2*Ar
cTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt[2]*d^(17/4)) + (c^(5/4)*(b*c
- a*d)^2*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt[2]*d^(17/4)) - (c^
(5/4)*(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^(17/4)) + (c^(5/4)*(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^(17/4))

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Rubi in Sympy [A]  time = 102.279, size = 292, normalized size = 0.94 \[ \frac{2 b^{2} x^{\frac{13}{2}}}{13 d} + \frac{2 b x^{\frac{9}{2}} \left (2 a d - b c\right )}{9 d^{2}} - \frac{\sqrt{2} c^{\frac{5}{4}} \left (a d - b c\right )^{2} \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{4 d^{\frac{17}{4}}} + \frac{\sqrt{2} c^{\frac{5}{4}} \left (a d - b c\right )^{2} \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{4 d^{\frac{17}{4}}} - \frac{\sqrt{2} c^{\frac{5}{4}} \left (a d - b c\right )^{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{2 d^{\frac{17}{4}}} + \frac{\sqrt{2} c^{\frac{5}{4}} \left (a d - b c\right )^{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{2 d^{\frac{17}{4}}} - \frac{2 c \sqrt{x} \left (a d - b c\right )^{2}}{d^{4}} + \frac{2 x^{\frac{5}{2}} \left (a d - b c\right )^{2}}{5 d^{3}} \]

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),x)

[Out]

2*b**2*x**(13/2)/(13*d) + 2*b*x**(9/2)*(2*a*d - b*c)/(9*d**2) - sqrt(2)*c**(5/4)
*(a*d - b*c)**2*log(-sqrt(2)*c**(1/4)*d**(1/4)*sqrt(x) + sqrt(c) + sqrt(d)*x)/(4
*d**(17/4)) + sqrt(2)*c**(5/4)*(a*d - b*c)**2*log(sqrt(2)*c**(1/4)*d**(1/4)*sqrt
(x) + sqrt(c) + sqrt(d)*x)/(4*d**(17/4)) - sqrt(2)*c**(5/4)*(a*d - b*c)**2*atan(
1 - sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(2*d**(17/4)) + sqrt(2)*c**(5/4)*(a*d - b
*c)**2*atan(1 + sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(2*d**(17/4)) - 2*c*sqrt(x)*(
a*d - b*c)**2/d**4 + 2*x**(5/2)*(a*d - b*c)**2/(5*d**3)

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Mathematica [A]  time = 0.224267, size = 299, normalized size = 0.96 \[ \frac{-585 \sqrt{2} c^{5/4} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )+585 \sqrt{2} c^{5/4} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )-1170 \sqrt{2} c^{5/4} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )+1170 \sqrt{2} c^{5/4} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )-520 b d^{9/4} x^{9/2} (b c-2 a d)+936 d^{5/4} x^{5/2} (b c-a d)^2-4680 c \sqrt [4]{d} \sqrt{x} (b c-a d)^2+360 b^2 d^{13/4} x^{13/2}}{2340 d^{17/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-4680*c*d^(1/4)*(b*c - a*d)^2*Sqrt[x] + 936*d^(5/4)*(b*c - a*d)^2*x^(5/2) - 520
*b*d^(9/4)*(b*c - 2*a*d)*x^(9/2) + 360*b^2*d^(13/4)*x^(13/2) - 1170*Sqrt[2]*c^(5
/4)*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 1170*Sqrt[2]*c
^(5/4)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 585*Sqrt[2]
*c^(5/4)*(b*c - a*d)^2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x
] + 585*Sqrt[2]*c^(5/4)*(b*c - a*d)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt
[x] + Sqrt[d]*x])/(2340*d^(17/4))

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Maple [B]  time = 0.023, size = 545, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/13*b^2*x^(13/2)/d+4/9/d*x^(9/2)*a*b-2/9/d^2*x^(9/2)*b^2*c+2/5/d*x^(5/2)*a^2-4/
5/d^2*x^(5/2)*a*b*c+2/5/d^3*x^(5/2)*b^2*c^2-2/d^2*x^(1/2)*a^2*c+4/d^3*x^(1/2)*a*
b*c^2-2/d^4*x^(1/2)*b^2*c^3+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+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

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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^(7/2)/(d*x^2 + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246855, size = 1485, normalized size = 4.77 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1170*(2340*d^4*(-(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^
5*c^10*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^
7*b*c^6*d^7 + a^8*c^5*d^8)/d^17)^(1/4)*arctan(d^4*(-(b^8*c^13 - 8*a*b^7*c^12*d +
28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8
*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8)/d^17)^(1/4)/((b^2*c^3
- 2*a*b*c^2*d + a^2*c*d^2)*sqrt(x) + sqrt(d^8*sqrt(-(b^8*c^13 - 8*a*b^7*c^12*d
+ 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^
8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8)/d^17) + (b^4*c^6 - 4
*a*b^3*c^5*d + 6*a^2*b^2*c^4*d^2 - 4*a^3*b*c^3*d^3 + a^4*c^2*d^4)*x))) - 585*d^4
*(-(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a
^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8
*c^5*d^8)/d^17)^(1/4)*log(d^4*(-(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2
- 56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^
7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8)/d^17)^(1/4) + (b^2*c^3 - 2*a*b*c^2*d + a^
2*c*d^2)*sqrt(x)) + 585*d^4*(-(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 -
56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*
d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8)/d^17)^(1/4)*log(-d^4*(-(b^8*c^13 - 8*a*b^7*
c^12*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5
*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8)/d^17)^(1/4) +
(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*sqrt(x)) - 4*(45*b^2*d^3*x^6 - 585*b^2*c^3
+ 1170*a*b*c^2*d - 585*a^2*c*d^2 - 65*(b^2*c*d^2 - 2*a*b*d^3)*x^4 + 117*(b^2*c^2
*d - 2*a*b*c*d^2 + a^2*d^3)*x^2)*sqrt(x))/d^4

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.255166, size = 589, normalized size = 1.89 \[ \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{3} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c^{2} d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} c d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{2 \, d^{5}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{3} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c^{2} d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} c d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{2 \, d^{5}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{3} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c^{2} d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} c d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{4 \, d^{5}} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{3} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c^{2} d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} c d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{4 \, d^{5}} + \frac{2 \,{\left (45 \, b^{2} d^{12} x^{\frac{13}{2}} - 65 \, b^{2} c d^{11} x^{\frac{9}{2}} + 130 \, a b d^{12} x^{\frac{9}{2}} + 117 \, b^{2} c^{2} d^{10} x^{\frac{5}{2}} - 234 \, a b c d^{11} x^{\frac{5}{2}} + 117 \, a^{2} d^{12} x^{\frac{5}{2}} - 585 \, b^{2} c^{3} d^{9} \sqrt{x} + 1170 \, a b c^{2} d^{10} \sqrt{x} - 585 \, a^{2} c d^{11} \sqrt{x}\right )}}{585 \, d^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(2)*((c*d^3)^(1/4)*b^2*c^3 - 2*(c*d^3)^(1/4)*a*b*c^2*d + (c*d^3)^(1/4)*a
^2*c*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/d^5
+ 1/2*sqrt(2)*((c*d^3)^(1/4)*b^2*c^3 - 2*(c*d^3)^(1/4)*a*b*c^2*d + (c*d^3)^(1/4)
*a^2*c*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/d
^5 + 1/4*sqrt(2)*((c*d^3)^(1/4)*b^2*c^3 - 2*(c*d^3)^(1/4)*a*b*c^2*d + (c*d^3)^(1
/4)*a^2*c*d^2)*ln(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/d^5 - 1/4*sqrt(2)
*((c*d^3)^(1/4)*b^2*c^3 - 2*(c*d^3)^(1/4)*a*b*c^2*d + (c*d^3)^(1/4)*a^2*c*d^2)*l
n(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/d^5 + 2/585*(45*b^2*d^12*x^(13/2
) - 65*b^2*c*d^11*x^(9/2) + 130*a*b*d^12*x^(9/2) + 117*b^2*c^2*d^10*x^(5/2) - 23
4*a*b*c*d^11*x^(5/2) + 117*a^2*d^12*x^(5/2) - 585*b^2*c^3*d^9*sqrt(x) + 1170*a*b
*c^2*d^10*sqrt(x) - 585*a^2*c*d^11*sqrt(x))/d^13