3.440 \(\int \frac{\left (a+b x^2\right )^2}{x^{7/2} \left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=439 \[ -\frac{13 a^2 d^2-10 a b c d+5 b^2 c^2}{20 c^2 d \sqrt{x} \left (c+d x^2\right )^2}-\frac{2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}+\frac{\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{17/4} d^{3/4}}-\frac{\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{17/4} d^{3/4}}-\frac{\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{17/4} d^{3/4}}+\frac{\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{17/4} d^{3/4}}+\frac{5 b^2 c^2-9 a d (10 b c-13 a d)}{16 c^4 d \sqrt{x}}-\frac{5 b^2 c^2-9 a d (10 b c-13 a d)}{80 c^3 d \sqrt{x} \left (c+d x^2\right )} \]

[Out]

(5*b^2*c^2 - 9*a*d*(10*b*c - 13*a*d))/(16*c^4*d*Sqrt[x]) - (2*a^2)/(5*c*x^(5/2)*
(c + d*x^2)^2) - (5*b^2*c^2 - 10*a*b*c*d + 13*a^2*d^2)/(20*c^2*d*Sqrt[x]*(c + d*
x^2)^2) - (5*b^2*c^2 - 9*a*d*(10*b*c - 13*a*d))/(80*c^3*d*Sqrt[x]*(c + d*x^2)) -
((5*b^2*c^2 - 9*a*d*(10*b*c - 13*a*d))*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(
1/4)])/(32*Sqrt[2]*c^(17/4)*d^(3/4)) + ((5*b^2*c^2 - 9*a*d*(10*b*c - 13*a*d))*Ar
cTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(17/4)*d^(3/4)) + ((5
*b^2*c^2 - 9*a*d*(10*b*c - 13*a*d))*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x
] + Sqrt[d]*x])/(64*Sqrt[2]*c^(17/4)*d^(3/4)) - ((5*b^2*c^2 - 9*a*d*(10*b*c - 13
*a*d))*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c
^(17/4)*d^(3/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.979314, antiderivative size = 438, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458 \[ -\frac{13 a^2 d^2-10 a b c d+5 b^2 c^2}{20 c^2 d \sqrt{x} \left (c+d x^2\right )^2}-\frac{2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}-\frac{\frac{5 b^2}{d}-\frac{9 a (10 b c-13 a d)}{c^2}}{80 c \sqrt{x} \left (c+d x^2\right )}+\frac{\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{17/4} d^{3/4}}-\frac{\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{17/4} d^{3/4}}-\frac{\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{17/4} d^{3/4}}+\frac{\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{17/4} d^{3/4}}+\frac{5 b^2 c^2-9 a d (10 b c-13 a d)}{16 c^4 d \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/(x^(7/2)*(c + d*x^2)^3),x]

[Out]

(5*b^2*c^2 - 9*a*d*(10*b*c - 13*a*d))/(16*c^4*d*Sqrt[x]) - (2*a^2)/(5*c*x^(5/2)*
(c + d*x^2)^2) - (5*b^2*c^2 - 10*a*b*c*d + 13*a^2*d^2)/(20*c^2*d*Sqrt[x]*(c + d*
x^2)^2) - ((5*b^2)/d - (9*a*(10*b*c - 13*a*d))/c^2)/(80*c*Sqrt[x]*(c + d*x^2)) -
((5*b^2*c^2 - 9*a*d*(10*b*c - 13*a*d))*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(
1/4)])/(32*Sqrt[2]*c^(17/4)*d^(3/4)) + ((5*b^2*c^2 - 9*a*d*(10*b*c - 13*a*d))*Ar
cTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(17/4)*d^(3/4)) + ((5
*b^2*c^2 - 9*a*d*(10*b*c - 13*a*d))*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x
] + Sqrt[d]*x])/(64*Sqrt[2]*c^(17/4)*d^(3/4)) - ((5*b^2*c^2 - 9*a*d*(10*b*c - 13
*a*d))*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c
^(17/4)*d^(3/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 117.404, size = 420, normalized size = 0.96 \[ - \frac{2 a^{2}}{5 c x^{\frac{5}{2}} \left (c + d x^{2}\right )^{2}} - \frac{a d \left (13 a d - 10 b c\right ) + 5 b^{2} c^{2}}{20 c^{2} d \sqrt{x} \left (c + d x^{2}\right )^{2}} - \frac{9 a d \left (13 a d - 10 b c\right ) + 5 b^{2} c^{2}}{80 c^{3} d \sqrt{x} \left (c + d x^{2}\right )} + \frac{9 a d \left (13 a d - 10 b c\right ) + 5 b^{2} c^{2}}{16 c^{4} d \sqrt{x}} + \frac{\sqrt{2} \left (9 a d \left (13 a d - 10 b c\right ) + 5 b^{2} c^{2}\right ) \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{128 c^{\frac{17}{4}} d^{\frac{3}{4}}} - \frac{\sqrt{2} \left (9 a d \left (13 a d - 10 b c\right ) + 5 b^{2} c^{2}\right ) \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{128 c^{\frac{17}{4}} d^{\frac{3}{4}}} - \frac{\sqrt{2} \left (9 a d \left (13 a d - 10 b c\right ) + 5 b^{2} c^{2}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{64 c^{\frac{17}{4}} d^{\frac{3}{4}}} + \frac{\sqrt{2} \left (9 a d \left (13 a d - 10 b c\right ) + 5 b^{2} c^{2}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{64 c^{\frac{17}{4}} d^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2/x**(7/2)/(d*x**2+c)**3,x)

[Out]

-2*a**2/(5*c*x**(5/2)*(c + d*x**2)**2) - (a*d*(13*a*d - 10*b*c) + 5*b**2*c**2)/(
20*c**2*d*sqrt(x)*(c + d*x**2)**2) - (9*a*d*(13*a*d - 10*b*c) + 5*b**2*c**2)/(80
*c**3*d*sqrt(x)*(c + d*x**2)) + (9*a*d*(13*a*d - 10*b*c) + 5*b**2*c**2)/(16*c**4
*d*sqrt(x)) + sqrt(2)*(9*a*d*(13*a*d - 10*b*c) + 5*b**2*c**2)*log(-sqrt(2)*c**(1
/4)*d**(1/4)*sqrt(x) + sqrt(c) + sqrt(d)*x)/(128*c**(17/4)*d**(3/4)) - sqrt(2)*(
9*a*d*(13*a*d - 10*b*c) + 5*b**2*c**2)*log(sqrt(2)*c**(1/4)*d**(1/4)*sqrt(x) + s
qrt(c) + sqrt(d)*x)/(128*c**(17/4)*d**(3/4)) - sqrt(2)*(9*a*d*(13*a*d - 10*b*c)
+ 5*b**2*c**2)*atan(1 - sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(64*c**(17/4)*d**(3/4
)) + sqrt(2)*(9*a*d*(13*a*d - 10*b*c) + 5*b**2*c**2)*atan(1 + sqrt(2)*d**(1/4)*s
qrt(x)/c**(1/4))/(64*c**(17/4)*d**(3/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.767862, size = 382, normalized size = 0.87 \[ \frac{\frac{40 \sqrt [4]{c} x^{3/2} \left (21 a^2 d^2-26 a b c d+5 b^2 c^2\right )}{c+d x^2}+\frac{5 \sqrt{2} \left (117 a^2 d^2-90 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{d^{3/4}}-\frac{5 \sqrt{2} \left (117 a^2 d^2-90 a b c d+5 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{d^{3/4}}-\frac{10 \sqrt{2} \left (117 a^2 d^2-90 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{d^{3/4}}+\frac{10 \sqrt{2} \left (117 a^2 d^2-90 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{d^{3/4}}-\frac{256 a^2 c^{5/4}}{x^{5/2}}+\frac{160 c^{5/4} x^{3/2} (b c-a d)^2}{\left (c+d x^2\right )^2}+\frac{1280 a \sqrt [4]{c} (3 a d-2 b c)}{\sqrt{x}}}{640 c^{17/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2/(x^(7/2)*(c + d*x^2)^3),x]

[Out]

((-256*a^2*c^(5/4))/x^(5/2) + (1280*a*c^(1/4)*(-2*b*c + 3*a*d))/Sqrt[x] + (160*c
^(5/4)*(b*c - a*d)^2*x^(3/2))/(c + d*x^2)^2 + (40*c^(1/4)*(5*b^2*c^2 - 26*a*b*c*
d + 21*a^2*d^2)*x^(3/2))/(c + d*x^2) - (10*Sqrt[2]*(5*b^2*c^2 - 90*a*b*c*d + 117
*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(3/4) + (10*Sqrt[2]*(
5*b^2*c^2 - 90*a*b*c*d + 117*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/
4)])/d^(3/4) + (5*Sqrt[2]*(5*b^2*c^2 - 90*a*b*c*d + 117*a^2*d^2)*Log[Sqrt[c] - S
qrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(3/4) - (5*Sqrt[2]*(5*b^2*c^2 - 9
0*a*b*c*d + 117*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]
*x])/d^(3/4))/(640*c^(17/4))

_______________________________________________________________________________________

Maple [A]  time = 0.034, size = 590, normalized size = 1.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/x^(7/2)/(d*x^2+c)^3,x)

[Out]

21/16/c^4/(d*x^2+c)^2*x^(7/2)*a^2*d^3-13/8/c^3/(d*x^2+c)^2*x^(7/2)*a*b*d^2+5/16/
c^2/(d*x^2+c)^2*x^(7/2)*b^2*d+25/16/c^3/(d*x^2+c)^2*x^(3/2)*a^2*d^2-17/8/c^2/(d*
x^2+c)^2*x^(3/2)*a*b*d+9/16/c/(d*x^2+c)^2*x^(3/2)*b^2+117/128/c^4*d/(c/d)^(1/4)*
2^(1/2)*a^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)))+117/64/c^4*d/(c/d)^(1/4)*2^(1/2)*a^2*arctan(2^(1/2)/(c/d
)^(1/4)*x^(1/2)+1)+117/64/c^4*d/(c/d)^(1/4)*2^(1/2)*a^2*arctan(2^(1/2)/(c/d)^(1/
4)*x^(1/2)-1)-45/64/c^3/(c/d)^(1/4)*2^(1/2)*a*b*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)))-45/32/c^3/(c/d)^(1/4
)*2^(1/2)*a*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)-45/32/c^3/(c/d)^(1/4)*2^(1/2
)*a*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)+5/128/c^2/d/(c/d)^(1/4)*2^(1/2)*b^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)))+5/64/c^2/d/(c/d)^(1/4)*2^(1/2)*b^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2
)+1)+5/64/c^2/d/(c/d)^(1/4)*2^(1/2)*b^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-2/
5*a^2/c^3/x^(5/2)+6*a^2/c^4/x^(1/2)*d-4*a/c^3/x^(1/2)*b

_______________________________________________________________________________________

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/((d*x^2 + c)^3*x^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.280623, size = 2168, normalized size = 4.94 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^3*x^(7/2)),x, algorithm="fricas")

[Out]

1/320*(20*(5*b^2*c^2*d - 90*a*b*c*d^2 + 117*a^2*d^3)*x^6 - 128*a^2*c^3 + 36*(5*b
^2*c^3 - 90*a*b*c^2*d + 117*a^2*c*d^2)*x^4 - 128*(10*a*b*c^3 - 13*a^2*c^2*d)*x^2
+ 20*(c^4*d^2*x^6 + 2*c^5*d*x^4 + c^6*x^2)*sqrt(x)*(-(625*b^8*c^8 - 45000*a*b^7
*c^7*d + 1273500*a^2*b^6*c^6*d^2 - 17739000*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*
c^4*d^4 - 415092600*a^5*b^3*c^3*d^5 + 697317660*a^6*b^2*c^2*d^6 - 576580680*a^7*
b*c*d^7 + 187388721*a^8*d^8)/(c^17*d^3))^(1/4)*arctan(c^13*d^2*(-(625*b^8*c^8 -
45000*a*b^7*c^7*d + 1273500*a^2*b^6*c^6*d^2 - 17739000*a^3*b^5*c^5*d^3 + 1245253
50*a^4*b^4*c^4*d^4 - 415092600*a^5*b^3*c^3*d^5 + 697317660*a^6*b^2*c^2*d^6 - 576
580680*a^7*b*c*d^7 + 187388721*a^8*d^8)/(c^17*d^3))^(3/4)/((125*b^6*c^6 - 6750*a
*b^5*c^5*d + 130275*a^2*b^4*c^4*d^2 - 1044900*a^3*b^3*c^3*d^3 + 3048435*a^4*b^2*
c^2*d^4 - 3696030*a^5*b*c*d^5 + 1601613*a^6*d^6)*sqrt(x) + sqrt((15625*b^12*c^12
- 1687500*a*b^11*c^11*d + 78131250*a^2*b^10*c^10*d^2 - 2019937500*a^3*b^9*c^9*d
^3 + 31839834375*a^4*b^8*c^8*d^4 - 314326575000*a^5*b^7*c^7*d^5 + 1936382557500*
a^6*b^6*c^6*d^6 - 7355241855000*a^7*b^5*c^5*d^7 + 17434219710375*a^8*b^4*c^4*d^8
- 25881265273500*a^9*b^3*c^3*d^9 + 23425464012210*a^10*b^2*c^2*d^10 - 118392193
92780*a^11*b*c*d^11 + 2565164201769*a^12*d^12)*x - (625*b^8*c^17*d - 45000*a*b^7
*c^16*d^2 + 1273500*a^2*b^6*c^15*d^3 - 17739000*a^3*b^5*c^14*d^4 + 124525350*a^4
*b^4*c^13*d^5 - 415092600*a^5*b^3*c^12*d^6 + 697317660*a^6*b^2*c^11*d^7 - 576580
680*a^7*b*c^10*d^8 + 187388721*a^8*c^9*d^9)*sqrt(-(625*b^8*c^8 - 45000*a*b^7*c^7
*d + 1273500*a^2*b^6*c^6*d^2 - 17739000*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*
d^4 - 415092600*a^5*b^3*c^3*d^5 + 697317660*a^6*b^2*c^2*d^6 - 576580680*a^7*b*c*
d^7 + 187388721*a^8*d^8)/(c^17*d^3))))) + 5*(c^4*d^2*x^6 + 2*c^5*d*x^4 + c^6*x^2
)*sqrt(x)*(-(625*b^8*c^8 - 45000*a*b^7*c^7*d + 1273500*a^2*b^6*c^6*d^2 - 1773900
0*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 415092600*a^5*b^3*c^3*d^5 + 6973
17660*a^6*b^2*c^2*d^6 - 576580680*a^7*b*c*d^7 + 187388721*a^8*d^8)/(c^17*d^3))^(
1/4)*log(c^13*d^2*(-(625*b^8*c^8 - 45000*a*b^7*c^7*d + 1273500*a^2*b^6*c^6*d^2 -
17739000*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 415092600*a^5*b^3*c^3*d^
5 + 697317660*a^6*b^2*c^2*d^6 - 576580680*a^7*b*c*d^7 + 187388721*a^8*d^8)/(c^17
*d^3))^(3/4) + (125*b^6*c^6 - 6750*a*b^5*c^5*d + 130275*a^2*b^4*c^4*d^2 - 104490
0*a^3*b^3*c^3*d^3 + 3048435*a^4*b^2*c^2*d^4 - 3696030*a^5*b*c*d^5 + 1601613*a^6*
d^6)*sqrt(x)) - 5*(c^4*d^2*x^6 + 2*c^5*d*x^4 + c^6*x^2)*sqrt(x)*(-(625*b^8*c^8 -
45000*a*b^7*c^7*d + 1273500*a^2*b^6*c^6*d^2 - 17739000*a^3*b^5*c^5*d^3 + 124525
350*a^4*b^4*c^4*d^4 - 415092600*a^5*b^3*c^3*d^5 + 697317660*a^6*b^2*c^2*d^6 - 57
6580680*a^7*b*c*d^7 + 187388721*a^8*d^8)/(c^17*d^3))^(1/4)*log(-c^13*d^2*(-(625*
b^8*c^8 - 45000*a*b^7*c^7*d + 1273500*a^2*b^6*c^6*d^2 - 17739000*a^3*b^5*c^5*d^3
+ 124525350*a^4*b^4*c^4*d^4 - 415092600*a^5*b^3*c^3*d^5 + 697317660*a^6*b^2*c^2
*d^6 - 576580680*a^7*b*c*d^7 + 187388721*a^8*d^8)/(c^17*d^3))^(3/4) + (125*b^6*c
^6 - 6750*a*b^5*c^5*d + 130275*a^2*b^4*c^4*d^2 - 1044900*a^3*b^3*c^3*d^3 + 30484
35*a^4*b^2*c^2*d^4 - 3696030*a^5*b*c*d^5 + 1601613*a^6*d^6)*sqrt(x)))/((c^4*d^2*
x^6 + 2*c^5*d*x^4 + c^6*x^2)*sqrt(x))

_______________________________________________________________________________________

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**(7/2)/(d*x**2+c)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.31651, size = 599, normalized size = 1.36 \[ \frac{5 \, b^{2} c^{2} d x^{\frac{7}{2}} - 26 \, a b c d^{2} x^{\frac{7}{2}} + 21 \, a^{2} d^{3} x^{\frac{7}{2}} + 9 \, b^{2} c^{3} x^{\frac{3}{2}} - 34 \, a b c^{2} d x^{\frac{3}{2}} + 25 \, a^{2} c d^{2} x^{\frac{3}{2}}}{16 \,{\left (d x^{2} + c\right )}^{2} c^{4}} - \frac{2 \,{\left (10 \, a b c x^{2} - 15 \, a^{2} d x^{2} + a^{2} c\right )}}{5 \, c^{4} x^{\frac{5}{2}}} + \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{64 \, c^{5} d^{3}} + \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{64 \, c^{5} d^{3}} - \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{128 \, c^{5} d^{3}} + \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{128 \, c^{5} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^3*x^(7/2)),x, algorithm="giac")

[Out]

1/16*(5*b^2*c^2*d*x^(7/2) - 26*a*b*c*d^2*x^(7/2) + 21*a^2*d^3*x^(7/2) + 9*b^2*c^
3*x^(3/2) - 34*a*b*c^2*d*x^(3/2) + 25*a^2*c*d^2*x^(3/2))/((d*x^2 + c)^2*c^4) - 2
/5*(10*a*b*c*x^2 - 15*a^2*d*x^2 + a^2*c)/(c^4*x^(5/2)) + 1/64*sqrt(2)*(5*(c*d^3)
^(3/4)*b^2*c^2 - 90*(c*d^3)^(3/4)*a*b*c*d + 117*(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^5*d^3) + 1/64*sqrt(2
)*(5*(c*d^3)^(3/4)*b^2*c^2 - 90*(c*d^3)^(3/4)*a*b*c*d + 117*(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^5*d^3)
- 1/128*sqrt(2)*(5*(c*d^3)^(3/4)*b^2*c^2 - 90*(c*d^3)^(3/4)*a*b*c*d + 117*(c*d^3
)^(3/4)*a^2*d^2)*ln(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^5*d^3) + 1/1
28*sqrt(2)*(5*(c*d^3)^(3/4)*b^2*c^2 - 90*(c*d^3)^(3/4)*a*b*c*d + 117*(c*d^3)^(3/
4)*a^2*d^2)*ln(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^5*d^3)