3.106 \(\int \frac{\left (a+b x^4\right )^{11/4}}{\left (c+d x^4\right )^2} \, dx\)
Optimal. Leaf size=280 \[ -\frac{b^{7/4} (8 b c-11 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 d^3}-\frac{b^{7/4} (8 b c-11 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 d^3}+\frac{(b c-a d)^{7/4} (3 a d+8 b c) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} d^3}+\frac{(b c-a d)^{7/4} (3 a d+8 b c) \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} d^3}+\frac{b x \left (a+b x^4\right )^{3/4} (2 b c-a d)}{4 c d^2}-\frac{x \left (a+b x^4\right )^{7/4} (b c-a d)}{4 c d \left (c+d x^4\right )} \]
[Out]
(b*(2*b*c - a*d)*x*(a + b*x^4)^(3/4))/(4*c*d^2) - ((b*c - a*d)*x*(a + b*x^4)^(7/
4))/(4*c*d*(c + d*x^4)) - (b^(7/4)*(8*b*c - 11*a*d)*ArcTan[(b^(1/4)*x)/(a + b*x^
4)^(1/4)])/(8*d^3) + ((b*c - a*d)^(7/4)*(8*b*c + 3*a*d)*ArcTan[((b*c - a*d)^(1/4
)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(8*c^(7/4)*d^3) - (b^(7/4)*(8*b*c - 11*a*d)*A
rcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(8*d^3) + ((b*c - a*d)^(7/4)*(8*b*c + 3*a
*d)*ArcTanh[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(8*c^(7/4)*d^3)
_______________________________________________________________________________________
Rubi [A] time = 0.893411, antiderivative size = 280, normalized size of antiderivative = 1.,
number of steps used = 11, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476
\[ -\frac{b^{7/4} (8 b c-11 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 d^3}-\frac{b^{7/4} (8 b c-11 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 d^3}+\frac{(b c-a d)^{7/4} (3 a d+8 b c) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} d^3}+\frac{(b c-a d)^{7/4} (3 a d+8 b c) \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} d^3}+\frac{b x \left (a+b x^4\right )^{3/4} (2 b c-a d)}{4 c d^2}-\frac{x \left (a+b x^4\right )^{7/4} (b c-a d)}{4 c d \left (c+d x^4\right )} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^4)^(11/4)/(c + d*x^4)^2,x]
[Out]
(b*(2*b*c - a*d)*x*(a + b*x^4)^(3/4))/(4*c*d^2) - ((b*c - a*d)*x*(a + b*x^4)^(7/
4))/(4*c*d*(c + d*x^4)) - (b^(7/4)*(8*b*c - 11*a*d)*ArcTan[(b^(1/4)*x)/(a + b*x^
4)^(1/4)])/(8*d^3) + ((b*c - a*d)^(7/4)*(8*b*c + 3*a*d)*ArcTan[((b*c - a*d)^(1/4
)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(8*c^(7/4)*d^3) - (b^(7/4)*(8*b*c - 11*a*d)*A
rcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(8*d^3) + ((b*c - a*d)^(7/4)*(8*b*c + 3*a
*d)*ArcTanh[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(8*c^(7/4)*d^3)
_______________________________________________________________________________________
Rubi in Sympy [A] time = 101.332, size = 255, normalized size = 0.91 \[ \frac{b^{\frac{7}{4}} \left (11 a d - 8 b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8 d^{3}} + \frac{b^{\frac{7}{4}} \left (11 a d - 8 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8 d^{3}} - \frac{b x \left (a + b x^{4}\right )^{\frac{3}{4}} \left (a d - 2 b c\right )}{4 c d^{2}} + \frac{x \left (a + b x^{4}\right )^{\frac{7}{4}} \left (a d - b c\right )}{4 c d \left (c + d x^{4}\right )} + \frac{\left (- a d + b c\right )^{\frac{7}{4}} \left (3 a d + 8 b c\right ) \operatorname{atan}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{8 c^{\frac{7}{4}} d^{3}} + \frac{\left (- a d + b c\right )^{\frac{7}{4}} \left (3 a d + 8 b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{8 c^{\frac{7}{4}} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**4+a)**(11/4)/(d*x**4+c)**2,x)
[Out]
b**(7/4)*(11*a*d - 8*b*c)*atan(b**(1/4)*x/(a + b*x**4)**(1/4))/(8*d**3) + b**(7/
4)*(11*a*d - 8*b*c)*atanh(b**(1/4)*x/(a + b*x**4)**(1/4))/(8*d**3) - b*x*(a + b*
x**4)**(3/4)*(a*d - 2*b*c)/(4*c*d**2) + x*(a + b*x**4)**(7/4)*(a*d - b*c)/(4*c*d
*(c + d*x**4)) + (-a*d + b*c)**(7/4)*(3*a*d + 8*b*c)*atan(x*(-a*d + b*c)**(1/4)/
(c**(1/4)*(a + b*x**4)**(1/4)))/(8*c**(7/4)*d**3) + (-a*d + b*c)**(7/4)*(3*a*d +
8*b*c)*atanh(x*(-a*d + b*c)**(1/4)/(c**(1/4)*(a + b*x**4)**(1/4)))/(8*c**(7/4)*
d**3)
_______________________________________________________________________________________
Mathematica [C] time = 4.04845, size = 735, normalized size = 2.62 \[ \frac{\frac{5 \left (3 a^3 d^3 x^4 \log \left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )+3 a^3 c d^2 \log \left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )-a \left (c+d x^4\right ) \left (3 a^2 d^2+2 a b c d-2 b^2 c^2\right ) \log \left (\sqrt [4]{c}-\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}\right )+2 a \left (c+d x^4\right ) \left (3 a^2 d^2+2 a b c d-2 b^2 c^2\right ) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a x^4+b}}\right )+4 a^2 c^{3/4} d^2 x \left (a+b x^4\right )^{3/4} \sqrt [4]{b c-a d}+2 a^2 b c^2 d \log \left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )+2 a^2 b c d^2 x^4 \log \left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )+8 b^2 c^{11/4} x \left (a+b x^4\right )^{3/4} \sqrt [4]{b c-a d}+4 b^2 c^{7/4} d x^5 \left (a+b x^4\right )^{3/4} \sqrt [4]{b c-a d}-2 a b^2 c^3 \log \left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )-2 a b^2 c^2 d x^4 \log \left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )-8 a b c^{7/4} d x \left (a+b x^4\right )^{3/4} \sqrt [4]{b c-a d}\right )}{\sqrt [4]{b c-a d}}-\frac{36 a b^2 c^{11/4} x^5 (11 a d-8 b c) F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\sqrt [4]{a+b x^4} \left (x^4 \left (4 a d F_1\left (\frac{9}{4};\frac{1}{4},2;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+b c F_1\left (\frac{9}{4};\frac{5}{4},1;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-9 a c F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}}{80 c^{7/4} d^2 \left (c+d x^4\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x^4)^(11/4)/(c + d*x^4)^2,x]
[Out]
((-36*a*b^2*c^(11/4)*(-8*b*c + 11*a*d)*x^5*AppellF1[5/4, 1/4, 1, 9/4, -((b*x^4)/
a), -((d*x^4)/c)])/((a + b*x^4)^(1/4)*(-9*a*c*AppellF1[5/4, 1/4, 1, 9/4, -((b*x^
4)/a), -((d*x^4)/c)] + x^4*(4*a*d*AppellF1[9/4, 1/4, 2, 13/4, -((b*x^4)/a), -((d
*x^4)/c)] + b*c*AppellF1[9/4, 5/4, 1, 13/4, -((b*x^4)/a), -((d*x^4)/c)]))) + (5*
(8*b^2*c^(11/4)*(b*c - a*d)^(1/4)*x*(a + b*x^4)^(3/4) - 8*a*b*c^(7/4)*d*(b*c - a
*d)^(1/4)*x*(a + b*x^4)^(3/4) + 4*a^2*c^(3/4)*d^2*(b*c - a*d)^(1/4)*x*(a + b*x^4
)^(3/4) + 4*b^2*c^(7/4)*d*(b*c - a*d)^(1/4)*x^5*(a + b*x^4)^(3/4) + 2*a*(-2*b^2*
c^2 + 2*a*b*c*d + 3*a^2*d^2)*(c + d*x^4)*ArcTan[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(
b + a*x^4)^(1/4))] - a*(-2*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*(c + d*x^4)*Log[c^(1
/4) - ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)] - 2*a*b^2*c^3*Log[c^(1/4) + ((b*c
- a*d)^(1/4)*x)/(b + a*x^4)^(1/4)] + 2*a^2*b*c^2*d*Log[c^(1/4) + ((b*c - a*d)^(
1/4)*x)/(b + a*x^4)^(1/4)] + 3*a^3*c*d^2*Log[c^(1/4) + ((b*c - a*d)^(1/4)*x)/(b
+ a*x^4)^(1/4)] - 2*a*b^2*c^2*d*x^4*Log[c^(1/4) + ((b*c - a*d)^(1/4)*x)/(b + a*x
^4)^(1/4)] + 2*a^2*b*c*d^2*x^4*Log[c^(1/4) + ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(
1/4)] + 3*a^3*d^3*x^4*Log[c^(1/4) + ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)]))/(
b*c - a*d)^(1/4))/(80*c^(7/4)*d^2*(c + d*x^4))
_______________________________________________________________________________________
Maple [F] time = 0.096, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{11}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^4+a)^(11/4)/(d*x^4+c)^2,x)
[Out]
int((b*x^4+a)^(11/4)/(d*x^4+c)^2,x)
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{11}{4}}}{{\left (d x^{4} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(11/4)/(d*x^4 + c)^2,x, algorithm="maxima")
[Out]
integrate((b*x^4 + a)^(11/4)/(d*x^4 + c)^2, x)
_______________________________________________________________________________________
Fricas [A] time = 19.9538, size = 3868, normalized size = 13.81 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(11/4)/(d*x^4 + c)^2,x, algorithm="fricas")
[Out]
-1/16*(4*(c*d^3*x^4 + c^2*d^2)*((4096*b^11*c^11 - 22528*a*b^10*c^10*d + 46464*a^
2*b^9*c^9*d^2 - 37664*a^3*b^8*c^8*d^3 - 5071*a^4*b^7*c^7*d^4 + 25641*a^5*b^6*c^6
*d^5 - 7931*a^6*b^5*c^5*d^6 - 6259*a^7*b^4*c^4*d^7 + 2739*a^8*b^3*c^3*d^8 + 891*
a^9*b^2*c^2*d^9 - 297*a^10*b*c*d^10 - 81*a^11*d^11)/(c^7*d^12))^(1/4)*arctan(c^5
*d^9*x*((4096*b^11*c^11 - 22528*a*b^10*c^10*d + 46464*a^2*b^9*c^9*d^2 - 37664*a^
3*b^8*c^8*d^3 - 5071*a^4*b^7*c^7*d^4 + 25641*a^5*b^6*c^6*d^5 - 7931*a^6*b^5*c^5*
d^6 - 6259*a^7*b^4*c^4*d^7 + 2739*a^8*b^3*c^3*d^8 + 891*a^9*b^2*c^2*d^9 - 297*a^
10*b*c*d^10 - 81*a^11*d^11)/(c^7*d^12))^(3/4)/(x*sqrt(((4096*b^11*c^14*d^6 - 225
28*a*b^10*c^13*d^7 + 46464*a^2*b^9*c^12*d^8 - 37664*a^3*b^8*c^11*d^9 - 5071*a^4*
b^7*c^10*d^10 + 25641*a^5*b^6*c^9*d^11 - 7931*a^6*b^5*c^8*d^12 - 6259*a^7*b^4*c^
7*d^13 + 2739*a^8*b^3*c^6*d^14 + 891*a^9*b^2*c^5*d^15 - 297*a^10*b*c^4*d^16 - 81
*a^11*c^3*d^17)*x^2*sqrt((4096*b^11*c^11 - 22528*a*b^10*c^10*d + 46464*a^2*b^9*c
^9*d^2 - 37664*a^3*b^8*c^8*d^3 - 5071*a^4*b^7*c^7*d^4 + 25641*a^5*b^6*c^6*d^5 -
7931*a^6*b^5*c^5*d^6 - 6259*a^7*b^4*c^4*d^7 + 2739*a^8*b^3*c^3*d^8 + 891*a^9*b^2
*c^2*d^9 - 297*a^10*b*c*d^10 - 81*a^11*d^11)/(c^7*d^12)) + (262144*b^16*c^16 - 2
031616*a*b^15*c^15*d + 6451200*a^2*b^14*c^14*d^2 - 10168320*a^3*b^13*c^13*d^3 +
6467520*a^4*b^12*c^12*d^4 + 3123216*a^5*b^11*c^11*d^5 - 7258119*a^6*b^10*c^10*d^
6 + 2307030*a^7*b^9*c^9*d^7 + 2428965*a^8*b^8*c^8*d^8 - 1607320*a^9*b^7*c^7*d^9
- 387134*a^10*b^6*c^6*d^10 + 436356*a^11*b^5*c^5*d^11 + 40770*a^12*b^4*c^4*d^12
- 63720*a^13*b^3*c^3*d^13 - 6075*a^14*b^2*c^2*d^14 + 4374*a^15*b*c*d^15 + 729*a^
16*d^16)*sqrt(b*x^4 + a))/x^2) - (512*b^8*c^8 - 1984*a*b^7*c^7*d + 2456*a^2*b^6*
c^6*d^2 - 413*a^3*b^5*c^5*d^3 - 1175*a^4*b^4*c^4*d^4 + 478*a^5*b^3*c^3*d^5 + 234
*a^6*b^2*c^2*d^6 - 81*a^7*b*c*d^7 - 27*a^8*d^8)*(b*x^4 + a)^(1/4))) - 4*(c*d^3*x
^4 + c^2*d^2)*((4096*b^11*c^4 - 22528*a*b^10*c^3*d + 46464*a^2*b^9*c^2*d^2 - 425
92*a^3*b^8*c*d^3 + 14641*a^4*b^7*d^4)/d^12)^(1/4)*arctan(d^9*x*((4096*b^11*c^4 -
22528*a*b^10*c^3*d + 46464*a^2*b^9*c^2*d^2 - 42592*a^3*b^8*c*d^3 + 14641*a^4*b^
7*d^4)/d^12)^(3/4)/(x*sqrt(((4096*b^11*c^4*d^6 - 22528*a*b^10*c^3*d^7 + 46464*a^
2*b^9*c^2*d^8 - 42592*a^3*b^8*c*d^9 + 14641*a^4*b^7*d^10)*x^2*sqrt((4096*b^11*c^
4 - 22528*a*b^10*c^3*d + 46464*a^2*b^9*c^2*d^2 - 42592*a^3*b^8*c*d^3 + 14641*a^4
*b^7*d^4)/d^12) + (262144*b^16*c^6 - 2162688*a*b^15*c^5*d + 7434240*a^2*b^14*c^4
*d^2 - 13629440*a^3*b^13*c^3*d^3 + 14055360*a^4*b^12*c^2*d^4 - 7730448*a^5*b^11*
c*d^5 + 1771561*a^6*b^10*d^6)*sqrt(b*x^4 + a))/x^2) - (512*b^8*c^3 - 2112*a*b^7*
c^2*d + 2904*a^2*b^6*c*d^2 - 1331*a^3*b^5*d^3)*(b*x^4 + a)^(1/4))) - (c*d^3*x^4
+ c^2*d^2)*((4096*b^11*c^11 - 22528*a*b^10*c^10*d + 46464*a^2*b^9*c^9*d^2 - 3766
4*a^3*b^8*c^8*d^3 - 5071*a^4*b^7*c^7*d^4 + 25641*a^5*b^6*c^6*d^5 - 7931*a^6*b^5*
c^5*d^6 - 6259*a^7*b^4*c^4*d^7 + 2739*a^8*b^3*c^3*d^8 + 891*a^9*b^2*c^2*d^9 - 29
7*a^10*b*c*d^10 - 81*a^11*d^11)/(c^7*d^12))^(1/4)*log(-(c^5*d^9*x*((4096*b^11*c^
11 - 22528*a*b^10*c^10*d + 46464*a^2*b^9*c^9*d^2 - 37664*a^3*b^8*c^8*d^3 - 5071*
a^4*b^7*c^7*d^4 + 25641*a^5*b^6*c^6*d^5 - 7931*a^6*b^5*c^5*d^6 - 6259*a^7*b^4*c^
4*d^7 + 2739*a^8*b^3*c^3*d^8 + 891*a^9*b^2*c^2*d^9 - 297*a^10*b*c*d^10 - 81*a^11
*d^11)/(c^7*d^12))^(3/4) + (512*b^8*c^8 - 1984*a*b^7*c^7*d + 2456*a^2*b^6*c^6*d^
2 - 413*a^3*b^5*c^5*d^3 - 1175*a^4*b^4*c^4*d^4 + 478*a^5*b^3*c^3*d^5 + 234*a^6*b
^2*c^2*d^6 - 81*a^7*b*c*d^7 - 27*a^8*d^8)*(b*x^4 + a)^(1/4))/x) + (c*d^3*x^4 + c
^2*d^2)*((4096*b^11*c^11 - 22528*a*b^10*c^10*d + 46464*a^2*b^9*c^9*d^2 - 37664*a
^3*b^8*c^8*d^3 - 5071*a^4*b^7*c^7*d^4 + 25641*a^5*b^6*c^6*d^5 - 7931*a^6*b^5*c^5
*d^6 - 6259*a^7*b^4*c^4*d^7 + 2739*a^8*b^3*c^3*d^8 + 891*a^9*b^2*c^2*d^9 - 297*a
^10*b*c*d^10 - 81*a^11*d^11)/(c^7*d^12))^(1/4)*log((c^5*d^9*x*((4096*b^11*c^11 -
22528*a*b^10*c^10*d + 46464*a^2*b^9*c^9*d^2 - 37664*a^3*b^8*c^8*d^3 - 5071*a^4*
b^7*c^7*d^4 + 25641*a^5*b^6*c^6*d^5 - 7931*a^6*b^5*c^5*d^6 - 6259*a^7*b^4*c^4*d^
7 + 2739*a^8*b^3*c^3*d^8 + 891*a^9*b^2*c^2*d^9 - 297*a^10*b*c*d^10 - 81*a^11*d^1
1)/(c^7*d^12))^(3/4) - (512*b^8*c^8 - 1984*a*b^7*c^7*d + 2456*a^2*b^6*c^6*d^2 -
413*a^3*b^5*c^5*d^3 - 1175*a^4*b^4*c^4*d^4 + 478*a^5*b^3*c^3*d^5 + 234*a^6*b^2*c
^2*d^6 - 81*a^7*b*c*d^7 - 27*a^8*d^8)*(b*x^4 + a)^(1/4))/x) + (c*d^3*x^4 + c^2*d
^2)*((4096*b^11*c^4 - 22528*a*b^10*c^3*d + 46464*a^2*b^9*c^2*d^2 - 42592*a^3*b^8
*c*d^3 + 14641*a^4*b^7*d^4)/d^12)^(1/4)*log(-(d^9*x*((4096*b^11*c^4 - 22528*a*b^
10*c^3*d + 46464*a^2*b^9*c^2*d^2 - 42592*a^3*b^8*c*d^3 + 14641*a^4*b^7*d^4)/d^12
)^(3/4) + (512*b^8*c^3 - 2112*a*b^7*c^2*d + 2904*a^2*b^6*c*d^2 - 1331*a^3*b^5*d^
3)*(b*x^4 + a)^(1/4))/x) - (c*d^3*x^4 + c^2*d^2)*((4096*b^11*c^4 - 22528*a*b^10*
c^3*d + 46464*a^2*b^9*c^2*d^2 - 42592*a^3*b^8*c*d^3 + 14641*a^4*b^7*d^4)/d^12)^(
1/4)*log((d^9*x*((4096*b^11*c^4 - 22528*a*b^10*c^3*d + 46464*a^2*b^9*c^2*d^2 - 4
2592*a^3*b^8*c*d^3 + 14641*a^4*b^7*d^4)/d^12)^(3/4) - (512*b^8*c^3 - 2112*a*b^7*
c^2*d + 2904*a^2*b^6*c*d^2 - 1331*a^3*b^5*d^3)*(b*x^4 + a)^(1/4))/x) - 4*(b^2*c*
d*x^5 + (2*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)*(b*x^4 + a)^(3/4))/(c*d^3*x^4 + c^2
*d^2)
_______________________________________________________________________________________
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**4+a)**(11/4)/(d*x**4+c)**2,x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{11}{4}}}{{\left (d x^{4} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(11/4)/(d*x^4 + c)^2,x, algorithm="giac")
[Out]
integrate((b*x^4 + a)^(11/4)/(d*x^4 + c)^2, x)