3.434 \(\int \frac{x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=401 \[ -\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} d^{15/4}}+\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} d^{15/4}}+\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{5/4} d^{15/4}}-\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{5/4} d^{15/4}}-\frac{x^{3/2} \left (\frac{3 a^2 d}{c}+42 a b-\frac{77 b^2 c}{d}\right )}{48 c d^2}-\frac{x^{7/2} (b c-a d) (a d+15 b c)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]

[Out]

-((42*a*b - (77*b^2*c)/d + (3*a^2*d)/c)*x^(3/2))/(48*c*d^2) + ((b*c - a*d)^2*x^(
7/2))/(4*c*d^2*(c + d*x^2)^2) - ((b*c - a*d)*(15*b*c + a*d)*x^(7/2))/(16*c^2*d^2
*(c + d*x^2)) + ((77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/
4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*d^(15/4)) - ((77*b^2*c^2 - 42*a*b*c*d
- 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*
d^(15/4)) - ((77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)
*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(5/4)*d^(15/4)) + ((77*b^2*c^2 - 42
*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x]
)/(64*Sqrt[2]*c^(5/4)*d^(15/4))

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Rubi [A]  time = 0.758133, antiderivative size = 401, 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 \[ -\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} d^{15/4}}+\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} d^{15/4}}+\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{5/4} d^{15/4}}-\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{5/4} d^{15/4}}-\frac{x^{3/2} \left (\frac{3 a^2 d}{c}+42 a b-\frac{77 b^2 c}{d}\right )}{48 c d^2}-\frac{x^{7/2} (b c-a d) (a d+15 b c)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^3,x]

[Out]

-((42*a*b - (77*b^2*c)/d + (3*a^2*d)/c)*x^(3/2))/(48*c*d^2) + ((b*c - a*d)^2*x^(
7/2))/(4*c*d^2*(c + d*x^2)^2) - ((b*c - a*d)*(15*b*c + a*d)*x^(7/2))/(16*c^2*d^2
*(c + d*x^2)) + ((77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/
4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*d^(15/4)) - ((77*b^2*c^2 - 42*a*b*c*d
- 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*
d^(15/4)) - ((77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)
*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(5/4)*d^(15/4)) + ((77*b^2*c^2 - 42
*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x]
)/(64*Sqrt[2]*c^(5/4)*d^(15/4))

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Rubi in Sympy [A]  time = 115.602, size = 388, normalized size = 0.97 \[ \frac{x^{\frac{7}{2}} \left (a d - b c\right )^{2}}{4 c d^{2} \left (c + d x^{2}\right )^{2}} + \frac{x^{\frac{7}{2}} \left (a d - b c\right ) \left (a d + 15 b c\right )}{16 c^{2} d^{2} \left (c + d x^{2}\right )} - \frac{x^{\frac{3}{2}} \left (3 a^{2} d^{2} + 42 a b c d - 77 b^{2} c^{2}\right )}{48 c^{2} d^{3}} + \frac{\sqrt{2} \left (3 a^{2} d^{2} + 42 a b c d - 77 b^{2} c^{2}\right ) \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{128 c^{\frac{5}{4}} d^{\frac{15}{4}}} - \frac{\sqrt{2} \left (3 a^{2} d^{2} + 42 a b c d - 77 b^{2} c^{2}\right ) \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{128 c^{\frac{5}{4}} d^{\frac{15}{4}}} - \frac{\sqrt{2} \left (3 a^{2} d^{2} + 42 a b c d - 77 b^{2} c^{2}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{64 c^{\frac{5}{4}} d^{\frac{15}{4}}} + \frac{\sqrt{2} \left (3 a^{2} d^{2} + 42 a b c d - 77 b^{2} c^{2}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{64 c^{\frac{5}{4}} d^{\frac{15}{4}}} \]

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)**3,x)

[Out]

x**(7/2)*(a*d - b*c)**2/(4*c*d**2*(c + d*x**2)**2) + x**(7/2)*(a*d - b*c)*(a*d +
15*b*c)/(16*c**2*d**2*(c + d*x**2)) - x**(3/2)*(3*a**2*d**2 + 42*a*b*c*d - 77*b
**2*c**2)/(48*c**2*d**3) + sqrt(2)*(3*a**2*d**2 + 42*a*b*c*d - 77*b**2*c**2)*log
(-sqrt(2)*c**(1/4)*d**(1/4)*sqrt(x) + sqrt(c) + sqrt(d)*x)/(128*c**(5/4)*d**(15/
4)) - sqrt(2)*(3*a**2*d**2 + 42*a*b*c*d - 77*b**2*c**2)*log(sqrt(2)*c**(1/4)*d**
(1/4)*sqrt(x) + sqrt(c) + sqrt(d)*x)/(128*c**(5/4)*d**(15/4)) - sqrt(2)*(3*a**2*
d**2 + 42*a*b*c*d - 77*b**2*c**2)*atan(1 - sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(6
4*c**(5/4)*d**(15/4)) + sqrt(2)*(3*a**2*d**2 + 42*a*b*c*d - 77*b**2*c**2)*atan(1
+ sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(64*c**(5/4)*d**(15/4))

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Mathematica [A]  time = 0.401856, size = 363, normalized size = 0.91 \[ \frac{\frac{24 d^{3/4} x^{3/2} \left (3 a^2 d^2-22 a b c d+19 b^2 c^2\right )}{c \left (c+d x^2\right )}-\frac{3 \sqrt{2} \left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4}}+\frac{3 \sqrt{2} \left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4}}+\frac{6 \sqrt{2} \left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{5/4}}-\frac{6 \sqrt{2} \left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{5/4}}-\frac{96 d^{3/4} x^{3/2} (b c-a d)^2}{\left (c+d x^2\right )^2}+256 b^2 d^{3/4} x^{3/2}}{384 d^{15/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^3,x]

[Out]

(256*b^2*d^(3/4)*x^(3/2) - (96*d^(3/4)*(b*c - a*d)^2*x^(3/2))/(c + d*x^2)^2 + (2
4*d^(3/4)*(19*b^2*c^2 - 22*a*b*c*d + 3*a^2*d^2)*x^(3/2))/(c*(c + d*x^2)) + (6*Sq
rt[2]*(77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])
/c^(1/4)])/c^(5/4) - (6*Sqrt[2]*(77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*ArcTan[1 +
(Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(5/4) - (3*Sqrt[2]*(77*b^2*c^2 - 42*a*b*c
*d - 3*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]*(77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(
1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(5/4))/(384*d^(15/4))

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Maple [A]  time = 0.029, size = 562, normalized size = 1.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

2/3*x^(3/2)*b^2/d^3+3/16/(d*x^2+c)^2/c*x^(7/2)*a^2-11/8/d/(d*x^2+c)^2*x^(7/2)*a*
b+19/16/d^2/(d*x^2+c)^2*c*x^(7/2)*b^2-1/16/d/(d*x^2+c)^2*x^(3/2)*a^2-7/8/d^2/(d*
x^2+c)^2*x^(3/2)*c*a*b+15/16/d^3/(d*x^2+c)^2*x^(3/2)*b^2*c^2+3/64/d^2/c/(c/d)^(1
/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2+21/32/d^3/(c/d)^(1/4)*2^(1
/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a*b-77/64/d^4*c/(c/d)^(1/4)*2^(1/2)*ar
ctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b^2+3/128/d^2/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
)))*a^2+21/64/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-77/128/d^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+3/64/d^2/c/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)
*x^(1/2)+1)*a^2+21/32/d^3/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)
+1)*a*b-77/64/d^4*c/(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^(5/2)/(d*x^2 + c)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274627, size = 2140, normalized size = 5.34 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*x^(5/2)/(d*x^2 + c)^3,x, algorithm="fricas")

[Out]

1/192*(12*(c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*d^3)*(-(35153041*b^8*c^8 - 76697544*a
*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b
^4*c^4*d^4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2*c^2*d^6 + 4536*a^7*b*c*d^7 +
81*a^8*d^8)/(c^5*d^15))^(1/4)*arctan(-c^4*d^11*(-(35153041*b^8*c^8 - 76697544*a
*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b
^4*c^4*d^4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2*c^2*d^6 + 4536*a^7*b*c*d^7 +
81*a^8*d^8)/(c^5*d^15))^(3/4)/((456533*b^6*c^6 - 747054*a*b^5*c^5*d + 354123*a^
2*b^4*c^4*d^2 - 15876*a^3*b^3*c^3*d^3 - 13797*a^4*b^2*c^2*d^4 - 1134*a^5*b*c*d^5
- 27*a^6*d^6)*sqrt(x) - sqrt((208422380089*b^12*c^12 - 682109607564*a*b^11*c^11
*d + 881427350034*a^2*b^10*c^10*d^2 - 543593843100*a^3*b^9*c^9*d^3 + 13652598613
5*a^4*b^8*c^8*d^4 + 8334677736*a^5*b^7*c^7*d^5 - 7849956996*a^6*b^6*c^6*d^6 - 32
4727704*a^7*b^5*c^5*d^7 + 207241335*a^8*b^4*c^4*d^8 + 32148900*a^9*b^3*c^3*d^9 +
2030994*a^10*b^2*c^2*d^10 + 61236*a^11*b*c*d^11 + 729*a^12*d^12)*x - (35153041*
b^8*c^11*d^7 - 76697544*a*b^7*c^10*d^8 + 57274140*a^2*b^6*c^9*d^9 - 13854456*a^3
*b^5*c^8*d^10 - 1457946*a^4*b^4*c^7*d^11 + 539784*a^5*b^3*c^6*d^12 + 86940*a^6*b
^2*c^5*d^13 + 4536*a^7*b*c^4*d^14 + 81*a^8*c^3*d^15)*sqrt(-(35153041*b^8*c^8 - 7
6697544*a*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457
946*a^4*b^4*c^4*d^4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2*c^2*d^6 + 4536*a^7*
b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))))) + 3*(c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*d^3)*(
-(35153041*b^8*c^8 - 76697544*a*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*
a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b
^2*c^2*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(1/4)*log(c^4*d^11*(-(35
153041*b^8*c^8 - 76697544*a*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*
b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2*c
^2*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(3/4) - (456533*b^6*c^6 - 74
7054*a*b^5*c^5*d + 354123*a^2*b^4*c^4*d^2 - 15876*a^3*b^3*c^3*d^3 - 13797*a^4*b^
2*c^2*d^4 - 1134*a^5*b*c*d^5 - 27*a^6*d^6)*sqrt(x)) - 3*(c*d^5*x^4 + 2*c^2*d^4*x
^2 + c^3*d^3)*(-(35153041*b^8*c^8 - 76697544*a*b^7*c^7*d + 57274140*a^2*b^6*c^6*
d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 + 539784*a^5*b^3*c^3*d^
5 + 86940*a^6*b^2*c^2*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(1/4)*log
(-c^4*d^11*(-(35153041*b^8*c^8 - 76697544*a*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2
- 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 + 539784*a^5*b^3*c^3*d^5 +
86940*a^6*b^2*c^2*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(3/4) - (456
533*b^6*c^6 - 747054*a*b^5*c^5*d + 354123*a^2*b^4*c^4*d^2 - 15876*a^3*b^3*c^3*d^
3 - 13797*a^4*b^2*c^2*d^4 - 1134*a^5*b*c*d^5 - 27*a^6*d^6)*sqrt(x)) + 4*(32*b^2*
c*d^2*x^5 + (121*b^2*c^2*d - 66*a*b*c*d^2 + 9*a^2*d^3)*x^3 + (77*b^2*c^3 - 42*a*
b*c^2*d - 3*a^2*c*d^2)*x)*sqrt(x))/(c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.262009, size = 576, normalized size = 1.44 \[ \frac{2 \, b^{2} x^{\frac{3}{2}}}{3 \, d^{3}} + \frac{19 \, b^{2} c^{2} d x^{\frac{7}{2}} - 22 \, a b c d^{2} x^{\frac{7}{2}} + 3 \, a^{2} d^{3} x^{\frac{7}{2}} + 15 \, b^{2} c^{3} x^{\frac{3}{2}} - 14 \, a b c^{2} d x^{\frac{3}{2}} - a^{2} c d^{2} x^{\frac{3}{2}}}{16 \,{\left (d x^{2} + c\right )}^{2} c d^{3}} - \frac{\sqrt{2}{\left (77 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 42 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{64 \, c^{2} d^{6}} - \frac{\sqrt{2}{\left (77 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 42 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{64 \, c^{2} d^{6}} + \frac{\sqrt{2}{\left (77 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 42 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{128 \, c^{2} d^{6}} - \frac{\sqrt{2}{\left (77 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 42 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{128 \, c^{2} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*x^(5/2)/(d*x^2 + c)^3,x, algorithm="giac")

[Out]

2/3*b^2*x^(3/2)/d^3 + 1/16*(19*b^2*c^2*d*x^(7/2) - 22*a*b*c*d^2*x^(7/2) + 3*a^2*
d^3*x^(7/2) + 15*b^2*c^3*x^(3/2) - 14*a*b*c^2*d*x^(3/2) - a^2*c*d^2*x^(3/2))/((d
*x^2 + c)^2*c*d^3) - 1/64*sqrt(2)*(77*(c*d^3)^(3/4)*b^2*c^2 - 42*(c*d^3)^(3/4)*a
*b*c*d - 3*(c*d^3)^(3/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sq
rt(x))/(c/d)^(1/4))/(c^2*d^6) - 1/64*sqrt(2)*(77*(c*d^3)^(3/4)*b^2*c^2 - 42*(c*d
^3)^(3/4)*a*b*c*d - 3*(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^6) + 1/128*sqrt(2)*(77*(c*d^3)^(3/4)*b^2*
c^2 - 42*(c*d^3)^(3/4)*a*b*c*d - 3*(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^6) - 1/128*sqrt(2)*(77*(c*d^3)^(3/4)*b^2*c^2 -
42*(c*d^3)^(3/4)*a*b*c*d - 3*(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^6)