3.451 \(\int \frac{\left (c+d x^2\right )^3}{x^{15/2} \left (a+b x^2\right )} \, dx\)
Optimal. Leaf size=325 \[ \frac{\sqrt [4]{b} (b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{17/4}}-\frac{\sqrt [4]{b} (b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{17/4}}-\frac{\sqrt [4]{b} (b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{17/4}}+\frac{\sqrt [4]{b} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{17/4}}+\frac{2 (b c-a d)^3}{a^4 \sqrt{x}}+\frac{2 c^2 (b c-3 a d)}{9 a^2 x^{9/2}}-\frac{2 c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{5 a^3 x^{5/2}}-\frac{2 c^3}{13 a x^{13/2}} \]
[Out]
(-2*c^3)/(13*a*x^(13/2)) + (2*c^2*(b*c - 3*a*d))/(9*a^2*x^(9/2)) - (2*c*(b^2*c^2
- 3*a*b*c*d + 3*a^2*d^2))/(5*a^3*x^(5/2)) + (2*(b*c - a*d)^3)/(a^4*Sqrt[x]) - (
b^(1/4)*(b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^
(17/4)) + (b^(1/4)*(b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/
(Sqrt[2]*a^(17/4)) + (b^(1/4)*(b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(17/4)) - (b^(1/4)*(b*c - a*d)^3*Log[Sqrt[a
] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(17/4))
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Rubi [A] time = 0.655789, antiderivative size = 325, normalized size of antiderivative = 1.,
number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333
\[ \frac{\sqrt [4]{b} (b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{17/4}}-\frac{\sqrt [4]{b} (b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{17/4}}-\frac{\sqrt [4]{b} (b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{17/4}}+\frac{\sqrt [4]{b} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{17/4}}+\frac{2 (b c-a d)^3}{a^4 \sqrt{x}}+\frac{2 c^2 (b c-3 a d)}{9 a^2 x^{9/2}}-\frac{2 c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{5 a^3 x^{5/2}}-\frac{2 c^3}{13 a x^{13/2}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2)^3/(x^(15/2)*(a + b*x^2)),x]
[Out]
(-2*c^3)/(13*a*x^(13/2)) + (2*c^2*(b*c - 3*a*d))/(9*a^2*x^(9/2)) - (2*c*(b^2*c^2
- 3*a*b*c*d + 3*a^2*d^2))/(5*a^3*x^(5/2)) + (2*(b*c - a*d)^3)/(a^4*Sqrt[x]) - (
b^(1/4)*(b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^
(17/4)) + (b^(1/4)*(b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/
(Sqrt[2]*a^(17/4)) + (b^(1/4)*(b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(17/4)) - (b^(1/4)*(b*c - a*d)^3*Log[Sqrt[a
] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(17/4))
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Rubi in Sympy [A] time = 170.11, size = 309, normalized size = 0.95 \[ - \frac{2 c^{3}}{13 a x^{\frac{13}{2}}} - \frac{2 c^{2} \left (3 a d - b c\right )}{9 a^{2} x^{\frac{9}{2}}} - \frac{2 c \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{5 a^{3} x^{\frac{5}{2}}} - \frac{2 \left (a d - b c\right )^{3}}{a^{4} \sqrt{x}} - \frac{\sqrt{2} \sqrt [4]{b} \left (a d - b c\right )^{3} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 a^{\frac{17}{4}}} + \frac{\sqrt{2} \sqrt [4]{b} \left (a d - b c\right )^{3} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 a^{\frac{17}{4}}} + \frac{\sqrt{2} \sqrt [4]{b} \left (a d - b c\right )^{3} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{17}{4}}} - \frac{\sqrt{2} \sqrt [4]{b} \left (a d - b c\right )^{3} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{17}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)**3/x**(15/2)/(b*x**2+a),x)
[Out]
-2*c**3/(13*a*x**(13/2)) - 2*c**2*(3*a*d - b*c)/(9*a**2*x**(9/2)) - 2*c*(3*a**2*
d**2 - 3*a*b*c*d + b**2*c**2)/(5*a**3*x**(5/2)) - 2*(a*d - b*c)**3/(a**4*sqrt(x)
) - sqrt(2)*b**(1/4)*(a*d - b*c)**3*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqr
t(a) + sqrt(b)*x)/(4*a**(17/4)) + sqrt(2)*b**(1/4)*(a*d - b*c)**3*log(sqrt(2)*a*
*(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(4*a**(17/4)) + sqrt(2)*b**(1/4)*
(a*d - b*c)**3*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(2*a**(17/4)) - sqrt(
2)*b**(1/4)*(a*d - b*c)**3*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(2*a**(17
/4))
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Mathematica [A] time = 0.285348, size = 314, normalized size = 0.97 \[ \frac{-\frac{520 a^{9/4} c^2 (3 a d-b c)}{x^{9/2}}-\frac{360 a^{13/4} c^3}{x^{13/2}}-\frac{936 a^{5/4} c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{x^{5/2}}-\frac{4680 \sqrt [4]{a} (a d-b c)^3}{\sqrt{x}}+585 \sqrt{2} \sqrt [4]{b} (b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+585 \sqrt{2} \sqrt [4]{b} (a d-b c)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-1170 \sqrt{2} \sqrt [4]{b} (b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+1170 \sqrt{2} \sqrt [4]{b} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{2340 a^{17/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2)^3/(x^(15/2)*(a + b*x^2)),x]
[Out]
((-360*a^(13/4)*c^3)/x^(13/2) - (520*a^(9/4)*c^2*(-(b*c) + 3*a*d))/x^(9/2) - (93
6*a^(5/4)*c*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2))/x^(5/2) - (4680*a^(1/4)*(-(b*c) +
a*d)^3)/Sqrt[x] - 1170*Sqrt[2]*b^(1/4)*(b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4
)*Sqrt[x])/a^(1/4)] + 1170*Sqrt[2]*b^(1/4)*(b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(
1/4)*Sqrt[x])/a^(1/4)] + 585*Sqrt[2]*b^(1/4)*(b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]
*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 585*Sqrt[2]*b^(1/4)*(-(b*c) + a*d)^3*Log
[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2340*a^(17/4))
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Maple [B] time = 0.024, size = 712, normalized size = 2.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)^3/x^(15/2)/(b*x^2+a),x)
[Out]
-2/13*c^3/a/x^(13/2)-2/a/x^(1/2)*d^3+6/a^2/x^(1/2)*c*d^2*b-6/a^3/x^(1/2)*c^2*d*b
^2+2/a^4/x^(1/2)*c^3*b^3-6/5*c/a/x^(5/2)*d^2+6/5*c^2/a^2/x^(5/2)*b*d-2/5*c^3/a^3
/x^(5/2)*b^2-2/3*c^2/a/x^(9/2)*d+2/9*c^3/a^2/x^(9/2)*b-1/2/a/(a/b)^(1/4)*2^(1/2)
*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*d^3+3/2/a^2/(a/b)^(1/4)*2^(1/2)*arctan(2^
(1/2)/(a/b)^(1/4)*x^(1/2)-1)*b*c*d^2-3/2/a^3/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/
(a/b)^(1/4)*x^(1/2)-1)*b^2*c^2*d+1/2/a^4/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b
)^(1/4)*x^(1/2)-1)*b^3*c^3-1/4/a/(a/b)^(1/4)*2^(1/2)*ln((x-(a/b)^(1/4)*x^(1/2)*2
^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))*d^3+3/4/a^2/(a/
b)^(1/4)*2^(1/2)*ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x
^(1/2)*2^(1/2)+(a/b)^(1/2)))*b*c*d^2-3/4/a^3/(a/b)^(1/4)*2^(1/2)*ln((x-(a/b)^(1/
4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))*b^2
*c^2*d+1/4/a^4/(a/b)^(1/4)*2^(1/2)*ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)
)/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))*b^3*c^3-1/2/a/(a/b)^(1/4)*2^(1/2)
*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*d^3+3/2/a^2/(a/b)^(1/4)*2^(1/2)*arctan(2^
(1/2)/(a/b)^(1/4)*x^(1/2)+1)*b*c*d^2-3/2/a^3/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/
(a/b)^(1/4)*x^(1/2)+1)*b^2*c^2*d+1/2/a^4/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b
)^(1/4)*x^(1/2)+1)*b^3*c^3
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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((d*x^2 + c)^3/((b*x^2 + a)*x^(15/2)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.267904, size = 2927, normalized size = 9.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3/((b*x^2 + a)*x^(15/2)),x, algorithm="fricas")
[Out]
-1/1170*(2340*a^4*x^(13/2)*(-(b^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^
2 - 220*a^3*b^10*c^9*d^3 + 495*a^4*b^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b
^7*c^6*d^6 - 792*a^7*b^6*c^5*d^7 + 495*a^8*b^5*c^4*d^8 - 220*a^9*b^4*c^3*d^9 + 6
6*a^10*b^3*c^2*d^10 - 12*a^11*b^2*c*d^11 + a^12*b*d^12)/a^17)^(1/4)*arctan(-a^13
*(-(b^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^2 - 220*a^3*b^10*c^9*d^3 +
495*a^4*b^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b^7*c^6*d^6 - 792*a^7*b^6*c
^5*d^7 + 495*a^8*b^5*c^4*d^8 - 220*a^9*b^4*c^3*d^9 + 66*a^10*b^3*c^2*d^10 - 12*a
^11*b^2*c*d^11 + a^12*b*d^12)/a^17)^(3/4)/((b^10*c^9 - 9*a*b^9*c^8*d + 36*a^2*b^
8*c^7*d^2 - 84*a^3*b^7*c^6*d^3 + 126*a^4*b^6*c^5*d^4 - 126*a^5*b^5*c^4*d^5 + 84*
a^6*b^4*c^3*d^6 - 36*a^7*b^3*c^2*d^7 + 9*a^8*b^2*c*d^8 - a^9*b*d^9)*sqrt(x) - sq
rt((b^20*c^18 - 18*a*b^19*c^17*d + 153*a^2*b^18*c^16*d^2 - 816*a^3*b^17*c^15*d^3
+ 3060*a^4*b^16*c^14*d^4 - 8568*a^5*b^15*c^13*d^5 + 18564*a^6*b^14*c^12*d^6 - 3
1824*a^7*b^13*c^11*d^7 + 43758*a^8*b^12*c^10*d^8 - 48620*a^9*b^11*c^9*d^9 + 4375
8*a^10*b^10*c^8*d^10 - 31824*a^11*b^9*c^7*d^11 + 18564*a^12*b^8*c^6*d^12 - 8568*
a^13*b^7*c^5*d^13 + 3060*a^14*b^6*c^4*d^14 - 816*a^15*b^5*c^3*d^15 + 153*a^16*b^
4*c^2*d^16 - 18*a^17*b^3*c*d^17 + a^18*b^2*d^18)*x - (a^9*b^13*c^12 - 12*a^10*b^
12*c^11*d + 66*a^11*b^11*c^10*d^2 - 220*a^12*b^10*c^9*d^3 + 495*a^13*b^9*c^8*d^4
- 792*a^14*b^8*c^7*d^5 + 924*a^15*b^7*c^6*d^6 - 792*a^16*b^6*c^5*d^7 + 495*a^17
*b^5*c^4*d^8 - 220*a^18*b^4*c^3*d^9 + 66*a^19*b^3*c^2*d^10 - 12*a^20*b^2*c*d^11
+ a^21*b*d^12)*sqrt(-(b^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^2 - 220*
a^3*b^10*c^9*d^3 + 495*a^4*b^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b^7*c^6*d
^6 - 792*a^7*b^6*c^5*d^7 + 495*a^8*b^5*c^4*d^8 - 220*a^9*b^4*c^3*d^9 + 66*a^10*b
^3*c^2*d^10 - 12*a^11*b^2*c*d^11 + a^12*b*d^12)/a^17)))) + 585*a^4*x^(13/2)*(-(b
^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^2 - 220*a^3*b^10*c^9*d^3 + 495*
a^4*b^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b^7*c^6*d^6 - 792*a^7*b^6*c^5*d^
7 + 495*a^8*b^5*c^4*d^8 - 220*a^9*b^4*c^3*d^9 + 66*a^10*b^3*c^2*d^10 - 12*a^11*b
^2*c*d^11 + a^12*b*d^12)/a^17)^(1/4)*log(a^13*(-(b^13*c^12 - 12*a*b^12*c^11*d +
66*a^2*b^11*c^10*d^2 - 220*a^3*b^10*c^9*d^3 + 495*a^4*b^9*c^8*d^4 - 792*a^5*b^8*
c^7*d^5 + 924*a^6*b^7*c^6*d^6 - 792*a^7*b^6*c^5*d^7 + 495*a^8*b^5*c^4*d^8 - 220*
a^9*b^4*c^3*d^9 + 66*a^10*b^3*c^2*d^10 - 12*a^11*b^2*c*d^11 + a^12*b*d^12)/a^17)
^(3/4) - (b^10*c^9 - 9*a*b^9*c^8*d + 36*a^2*b^8*c^7*d^2 - 84*a^3*b^7*c^6*d^3 + 1
26*a^4*b^6*c^5*d^4 - 126*a^5*b^5*c^4*d^5 + 84*a^6*b^4*c^3*d^6 - 36*a^7*b^3*c^2*d
^7 + 9*a^8*b^2*c*d^8 - a^9*b*d^9)*sqrt(x)) - 585*a^4*x^(13/2)*(-(b^13*c^12 - 12*
a*b^12*c^11*d + 66*a^2*b^11*c^10*d^2 - 220*a^3*b^10*c^9*d^3 + 495*a^4*b^9*c^8*d^
4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b^7*c^6*d^6 - 792*a^7*b^6*c^5*d^7 + 495*a^8*b^
5*c^4*d^8 - 220*a^9*b^4*c^3*d^9 + 66*a^10*b^3*c^2*d^10 - 12*a^11*b^2*c*d^11 + a^
12*b*d^12)/a^17)^(1/4)*log(-a^13*(-(b^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c
^10*d^2 - 220*a^3*b^10*c^9*d^3 + 495*a^4*b^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924
*a^6*b^7*c^6*d^6 - 792*a^7*b^6*c^5*d^7 + 495*a^8*b^5*c^4*d^8 - 220*a^9*b^4*c^3*d
^9 + 66*a^10*b^3*c^2*d^10 - 12*a^11*b^2*c*d^11 + a^12*b*d^12)/a^17)^(3/4) - (b^1
0*c^9 - 9*a*b^9*c^8*d + 36*a^2*b^8*c^7*d^2 - 84*a^3*b^7*c^6*d^3 + 126*a^4*b^6*c^
5*d^4 - 126*a^5*b^5*c^4*d^5 + 84*a^6*b^4*c^3*d^6 - 36*a^7*b^3*c^2*d^7 + 9*a^8*b^
2*c*d^8 - a^9*b*d^9)*sqrt(x)) - 2340*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 -
a^3*d^3)*x^6 + 180*a^3*c^3 + 468*(a*b^2*c^3 - 3*a^2*b*c^2*d + 3*a^3*c*d^2)*x^4 -
260*(a^2*b*c^3 - 3*a^3*c^2*d)*x^2)/(a^4*x^(13/2))
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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((d*x**2+c)**3/x**(15/2)/(b*x**2+a),x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.321783, size = 724, normalized size = 2.23 \[ \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{3}{4}} a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a^{5} b^{2}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{3}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a^{5} b^{2}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{3}{4}} a^{3} d^{3}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, a^{5} b^{2}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{3}{4}} a^{3} d^{3}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, a^{5} b^{2}} + \frac{2 \,{\left (585 \, b^{3} c^{3} x^{6} - 1755 \, a b^{2} c^{2} d x^{6} + 1755 \, a^{2} b c d^{2} x^{6} - 585 \, a^{3} d^{3} x^{6} - 117 \, a b^{2} c^{3} x^{4} + 351 \, a^{2} b c^{2} d x^{4} - 351 \, a^{3} c d^{2} x^{4} + 65 \, a^{2} b c^{3} x^{2} - 195 \, a^{3} c^{2} d x^{2} - 45 \, a^{3} c^{3}\right )}}{585 \, a^{4} x^{\frac{13}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3/((b*x^2 + a)*x^(15/2)),x, algorithm="giac")
[Out]
1/2*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/
4)*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4)
+ 2*sqrt(x))/(a/b)^(1/4))/(a^5*b^2) + 1/2*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*
b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*ar
ctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^5*b^2) - 1/4
*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*
a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a
/b))/(a^5*b^2) + 1/4*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*
d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*ln(-sqrt(2)*sqrt(x)*(a/
b)^(1/4) + x + sqrt(a/b))/(a^5*b^2) + 2/585*(585*b^3*c^3*x^6 - 1755*a*b^2*c^2*d*
x^6 + 1755*a^2*b*c*d^2*x^6 - 585*a^3*d^3*x^6 - 117*a*b^2*c^3*x^4 + 351*a^2*b*c^2
*d*x^4 - 351*a^3*c*d^2*x^4 + 65*a^2*b*c^3*x^2 - 195*a^3*c^2*d*x^2 - 45*a^3*c^3)/
(a^4*x^(13/2))