3.428 \(\int \frac{\sqrt{x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx\)

Optimal. Leaf size=310 \[ -\frac{(b c-a d) (a d+7 b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} d^{11/4}}+\frac{(b c-a d) (a d+7 b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} d^{11/4}}+\frac{(b c-a d) (a d+7 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{5/4} d^{11/4}}-\frac{(b c-a d) (a d+7 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{5/4} d^{11/4}}+\frac{x^{3/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{2 b^2 x^{3/2}}{3 d^2} \]

[Out]

(2*b^2*x^(3/2))/(3*d^2) + ((b*c - a*d)^2*x^(3/2))/(2*c*d^2*(c + d*x^2)) + ((b*c
- a*d)*(7*b*c + a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c
^(5/4)*d^(11/4)) - ((b*c - a*d)*(7*b*c + a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x
])/c^(1/4)])/(4*Sqrt[2]*c^(5/4)*d^(11/4)) - ((b*c - a*d)*(7*b*c + a*d)*Log[Sqrt[
c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(5/4)*d^(11/4))
+ ((b*c - a*d)*(7*b*c + a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqr
t[d]*x])/(8*Sqrt[2]*c^(5/4)*d^(11/4))

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Rubi [A]  time = 0.615305, antiderivative size = 310, 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 \[ -\frac{(b c-a d) (a d+7 b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} d^{11/4}}+\frac{(b c-a d) (a d+7 b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} d^{11/4}}+\frac{(b c-a d) (a d+7 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{5/4} d^{11/4}}-\frac{(b c-a d) (a d+7 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{5/4} d^{11/4}}+\frac{x^{3/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{2 b^2 x^{3/2}}{3 d^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*b^2*x^(3/2))/(3*d^2) + ((b*c - a*d)^2*x^(3/2))/(2*c*d^2*(c + d*x^2)) + ((b*c
- a*d)*(7*b*c + a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c
^(5/4)*d^(11/4)) - ((b*c - a*d)*(7*b*c + a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x
])/c^(1/4)])/(4*Sqrt[2]*c^(5/4)*d^(11/4)) - ((b*c - a*d)*(7*b*c + a*d)*Log[Sqrt[
c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(5/4)*d^(11/4))
+ ((b*c - a*d)*(7*b*c + a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqr
t[d]*x])/(8*Sqrt[2]*c^(5/4)*d^(11/4))

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Rubi in Sympy [A]  time = 103.585, size = 286, normalized size = 0.92 \[ \frac{2 b^{2} x^{\frac{3}{2}}}{3 d^{2}} + \frac{x^{\frac{3}{2}} \left (a d - b c\right )^{2}}{2 c d^{2} \left (c + d x^{2}\right )} + \frac{\sqrt{2} \left (a d - b c\right ) \left (a d + 7 b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{16 c^{\frac{5}{4}} d^{\frac{11}{4}}} - \frac{\sqrt{2} \left (a d - b c\right ) \left (a d + 7 b c\right ) \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{16 c^{\frac{5}{4}} d^{\frac{11}{4}}} - \frac{\sqrt{2} \left (a d - b c\right ) \left (a d + 7 b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{8 c^{\frac{5}{4}} d^{\frac{11}{4}}} + \frac{\sqrt{2} \left (a d - b c\right ) \left (a d + 7 b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{8 c^{\frac{5}{4}} d^{\frac{11}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*x**(1/2)/(d*x**2+c)**2,x)

[Out]

2*b**2*x**(3/2)/(3*d**2) + x**(3/2)*(a*d - b*c)**2/(2*c*d**2*(c + d*x**2)) + sqr
t(2)*(a*d - b*c)*(a*d + 7*b*c)*log(-sqrt(2)*c**(1/4)*d**(1/4)*sqrt(x) + sqrt(c)
+ sqrt(d)*x)/(16*c**(5/4)*d**(11/4)) - sqrt(2)*(a*d - b*c)*(a*d + 7*b*c)*log(sqr
t(2)*c**(1/4)*d**(1/4)*sqrt(x) + sqrt(c) + sqrt(d)*x)/(16*c**(5/4)*d**(11/4)) -
sqrt(2)*(a*d - b*c)*(a*d + 7*b*c)*atan(1 - sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(8
*c**(5/4)*d**(11/4)) + sqrt(2)*(a*d - b*c)*(a*d + 7*b*c)*atan(1 + sqrt(2)*d**(1/
4)*sqrt(x)/c**(1/4))/(8*c**(5/4)*d**(11/4))

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Mathematica [A]  time = 0.301308, size = 319, normalized size = 1.03 \[ \frac{-\frac{3 \sqrt{2} \left (-a^2 d^2-6 a b c d+7 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4}}+\frac{3 \sqrt{2} \left (-a^2 d^2-6 a b c d+7 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4}}+\frac{6 \sqrt{2} \left (-a^2 d^2-6 a b c d+7 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{5/4}}-\frac{6 \sqrt{2} \left (-a^2 d^2-6 a b c d+7 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{5/4}}+\frac{24 d^{3/4} x^{3/2} (b c-a d)^2}{c \left (c+d x^2\right )}+32 b^2 d^{3/4} x^{3/2}}{48 d^{11/4}} \]

Antiderivative was successfully verified.

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

[Out]

(32*b^2*d^(3/4)*x^(3/2) + (24*d^(3/4)*(b*c - a*d)^2*x^(3/2))/(c*(c + d*x^2)) + (
6*Sqrt[2]*(7*b^2*c^2 - 6*a*b*c*d - a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])
/c^(1/4)])/c^(5/4) - (6*Sqrt[2]*(7*b^2*c^2 - 6*a*b*c*d - a^2*d^2)*ArcTan[1 + (Sq
rt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(5/4) - (3*Sqrt[2]*(7*b^2*c^2 - 6*a*b*c*d - a
^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(5/4) + (3
*Sqrt[2]*(7*b^2*c^2 - 6*a*b*c*d - a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)
*Sqrt[x] + Sqrt[d]*x])/c^(5/4))/(48*d^(11/4))

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Maple [B]  time = 0.026, size = 499, normalized size = 1.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*x^(1/2)/(d*x^2+c)^2,x)

[Out]

2/3*b^2*x^(3/2)/d^2+1/2/c*x^(3/2)/(d*x^2+c)*a^2-1/d*x^(3/2)/(d*x^2+c)*a*b+1/2/d^
2*c*x^(3/2)/(d*x^2+c)*b^2+1/8/d/c/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)
*x^(1/2)+1)*a^2+3/4/d^2/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1
)*a*b-7/8/d^3*c/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b^2+1/
8/d/c/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2+3/4/d^2/(c/d
)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a*b-7/8/d^3*c/(c/d)^(1/4)*
2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b^2+1/16/d/c/(c/d)^(1/4)*2^(1/2)*l
n((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+3/8/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*b-7/16/d^3*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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270553, size = 2021, normalized size = 6.52 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(12*(c*d^3*x^2 + c^2*d^2)*(-(2401*b^8*c^8 - 8232*a*b^7*c^7*d + 9212*a^2*b^6
*c^6*d^2 - 2520*a^3*b^5*c^5*d^3 - 1434*a^4*b^4*c^4*d^4 + 360*a^5*b^3*c^3*d^5 + 1
88*a^6*b^2*c^2*d^6 + 24*a^7*b*c*d^7 + a^8*d^8)/(c^5*d^11))^(1/4)*arctan(-c^4*d^8
*(-(2401*b^8*c^8 - 8232*a*b^7*c^7*d + 9212*a^2*b^6*c^6*d^2 - 2520*a^3*b^5*c^5*d^
3 - 1434*a^4*b^4*c^4*d^4 + 360*a^5*b^3*c^3*d^5 + 188*a^6*b^2*c^2*d^6 + 24*a^7*b*
c*d^7 + a^8*d^8)/(c^5*d^11))^(3/4)/((343*b^6*c^6 - 882*a*b^5*c^5*d + 609*a^2*b^4
*c^4*d^2 + 36*a^3*b^3*c^3*d^3 - 87*a^4*b^2*c^2*d^4 - 18*a^5*b*c*d^5 - a^6*d^6)*s
qrt(x) - sqrt((117649*b^12*c^12 - 605052*a*b^11*c^11*d + 1195698*a^2*b^10*c^10*d
^2 - 1049580*a^3*b^9*c^9*d^3 + 247695*a^4*b^8*c^8*d^4 + 184968*a^5*b^7*c^7*d^5 -
73604*a^6*b^6*c^6*d^6 - 26424*a^7*b^5*c^5*d^7 + 5055*a^8*b^4*c^4*d^8 + 3060*a^9
*b^3*c^3*d^9 + 498*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12*d^12)*x - (2401*b
^8*c^11*d^5 - 8232*a*b^7*c^10*d^6 + 9212*a^2*b^6*c^9*d^7 - 2520*a^3*b^5*c^8*d^8
- 1434*a^4*b^4*c^7*d^9 + 360*a^5*b^3*c^6*d^10 + 188*a^6*b^2*c^5*d^11 + 24*a^7*b*
c^4*d^12 + a^8*c^3*d^13)*sqrt(-(2401*b^8*c^8 - 8232*a*b^7*c^7*d + 9212*a^2*b^6*c
^6*d^2 - 2520*a^3*b^5*c^5*d^3 - 1434*a^4*b^4*c^4*d^4 + 360*a^5*b^3*c^3*d^5 + 188
*a^6*b^2*c^2*d^6 + 24*a^7*b*c*d^7 + a^8*d^8)/(c^5*d^11))))) + 3*(c*d^3*x^2 + c^2
*d^2)*(-(2401*b^8*c^8 - 8232*a*b^7*c^7*d + 9212*a^2*b^6*c^6*d^2 - 2520*a^3*b^5*c
^5*d^3 - 1434*a^4*b^4*c^4*d^4 + 360*a^5*b^3*c^3*d^5 + 188*a^6*b^2*c^2*d^6 + 24*a
^7*b*c*d^7 + a^8*d^8)/(c^5*d^11))^(1/4)*log(c^4*d^8*(-(2401*b^8*c^8 - 8232*a*b^7
*c^7*d + 9212*a^2*b^6*c^6*d^2 - 2520*a^3*b^5*c^5*d^3 - 1434*a^4*b^4*c^4*d^4 + 36
0*a^5*b^3*c^3*d^5 + 188*a^6*b^2*c^2*d^6 + 24*a^7*b*c*d^7 + a^8*d^8)/(c^5*d^11))^
(3/4) - (343*b^6*c^6 - 882*a*b^5*c^5*d + 609*a^2*b^4*c^4*d^2 + 36*a^3*b^3*c^3*d^
3 - 87*a^4*b^2*c^2*d^4 - 18*a^5*b*c*d^5 - a^6*d^6)*sqrt(x)) - 3*(c*d^3*x^2 + c^2
*d^2)*(-(2401*b^8*c^8 - 8232*a*b^7*c^7*d + 9212*a^2*b^6*c^6*d^2 - 2520*a^3*b^5*c
^5*d^3 - 1434*a^4*b^4*c^4*d^4 + 360*a^5*b^3*c^3*d^5 + 188*a^6*b^2*c^2*d^6 + 24*a
^7*b*c*d^7 + a^8*d^8)/(c^5*d^11))^(1/4)*log(-c^4*d^8*(-(2401*b^8*c^8 - 8232*a*b^
7*c^7*d + 9212*a^2*b^6*c^6*d^2 - 2520*a^3*b^5*c^5*d^3 - 1434*a^4*b^4*c^4*d^4 + 3
60*a^5*b^3*c^3*d^5 + 188*a^6*b^2*c^2*d^6 + 24*a^7*b*c*d^7 + a^8*d^8)/(c^5*d^11))
^(3/4) - (343*b^6*c^6 - 882*a*b^5*c^5*d + 609*a^2*b^4*c^4*d^2 + 36*a^3*b^3*c^3*d
^3 - 87*a^4*b^2*c^2*d^4 - 18*a^5*b*c*d^5 - a^6*d^6)*sqrt(x)) + 4*(4*b^2*c*d*x^3
+ (7*b^2*c^2 - 6*a*b*c*d + 3*a^2*d^2)*x)*sqrt(x))/(c*d^3*x^2 + c^2*d^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**(1/2)/(d*x**2+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.2549, size = 524, normalized size = 1.69 \[ \frac{2 \, b^{2} x^{\frac{3}{2}}}{3 \, d^{2}} + \frac{b^{2} c^{2} x^{\frac{3}{2}} - 2 \, a b c d x^{\frac{3}{2}} + a^{2} d^{2} x^{\frac{3}{2}}}{2 \,{\left (d x^{2} + c\right )} c d^{2}} - \frac{\sqrt{2}{\left (7 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 6 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{8 \, c^{2} d^{5}} - \frac{\sqrt{2}{\left (7 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 6 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{8 \, c^{2} d^{5}} + \frac{\sqrt{2}{\left (7 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 6 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{16 \, c^{2} d^{5}} - \frac{\sqrt{2}{\left (7 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 6 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{16 \, c^{2} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*b^2*x^(3/2)/d^2 + 1/2*(b^2*c^2*x^(3/2) - 2*a*b*c*d*x^(3/2) + a^2*d^2*x^(3/2)
)/((d*x^2 + c)*c*d^2) - 1/8*sqrt(2)*(7*(c*d^3)^(3/4)*b^2*c^2 - 6*(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))/(c^2*d^5) - 1/8*sqrt(2)*(7*(c*d^3)^(3/4)*b^2*c^2 - 6*(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))/(c^2*d^5) + 1/16*sqrt(2)*(7*(c*d^3)^(3/4)*b^2*c^2 - 6*(
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))/(c^2*d^5) - 1/16*sqrt(2)*(7*(c*d^3)^(3/4)*b^2*c^2 - 6*(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))/(c^2*d^5)