3.879 \(\int \frac{\sqrt [4]{a+b x}}{x^5 \sqrt [4]{c+d x}} \, dx\)
Optimal. Leaf size=368 \[ \frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (-117 a^2 d^2+10 a b c d+11 b^2 c^2\right )}{384 a^2 c^3 x^2}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (-585 a^3 d^3+63 a^2 b c d^2+61 a b^2 c^2 d+77 b^3 c^3\right )}{1536 a^3 c^4 x}+\frac{(b c-a d) \left (195 a^3 d^3+135 a^2 b c d^2+105 a b^2 c^2 d+77 b^3 c^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{1024 a^{15/4} c^{17/4}}+\frac{(b c-a d) \left (195 a^3 d^3+135 a^2 b c d^2+105 a b^2 c^2 d+77 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{1024 a^{15/4} c^{17/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (b c-13 a d)}{48 a c^2 x^3}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4} \]
[Out]
-((a + b*x)^(1/4)*(c + d*x)^(3/4))/(4*c*x^4) - ((b*c - 13*a*d)*(a + b*x)^(1/4)*(
c + d*x)^(3/4))/(48*a*c^2*x^3) + ((11*b^2*c^2 + 10*a*b*c*d - 117*a^2*d^2)*(a + b
*x)^(1/4)*(c + d*x)^(3/4))/(384*a^2*c^3*x^2) - ((77*b^3*c^3 + 61*a*b^2*c^2*d + 6
3*a^2*b*c*d^2 - 585*a^3*d^3)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(1536*a^3*c^4*x) +
((b*c - a*d)*(77*b^3*c^3 + 105*a*b^2*c^2*d + 135*a^2*b*c*d^2 + 195*a^3*d^3)*Arc
Tan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(1024*a^(15/4)*c^(17/4
)) + ((b*c - a*d)*(77*b^3*c^3 + 105*a*b^2*c^2*d + 135*a^2*b*c*d^2 + 195*a^3*d^3)
*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(1024*a^(15/4)*c^
(17/4))
_______________________________________________________________________________________
Rubi [A] time = 0.934639, antiderivative size = 368, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318
\[ \frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (-117 a^2 d^2+10 a b c d+11 b^2 c^2\right )}{384 a^2 c^3 x^2}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (-585 a^3 d^3+63 a^2 b c d^2+61 a b^2 c^2 d+77 b^3 c^3\right )}{1536 a^3 c^4 x}+\frac{(b c-a d) \left (195 a^3 d^3+135 a^2 b c d^2+105 a b^2 c^2 d+77 b^3 c^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{1024 a^{15/4} c^{17/4}}+\frac{(b c-a d) \left (195 a^3 d^3+135 a^2 b c d^2+105 a b^2 c^2 d+77 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{1024 a^{15/4} c^{17/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (b c-13 a d)}{48 a c^2 x^3}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^(1/4)/(x^5*(c + d*x)^(1/4)),x]
[Out]
-((a + b*x)^(1/4)*(c + d*x)^(3/4))/(4*c*x^4) - ((b*c - 13*a*d)*(a + b*x)^(1/4)*(
c + d*x)^(3/4))/(48*a*c^2*x^3) + ((11*b^2*c^2 + 10*a*b*c*d - 117*a^2*d^2)*(a + b
*x)^(1/4)*(c + d*x)^(3/4))/(384*a^2*c^3*x^2) - ((77*b^3*c^3 + 61*a*b^2*c^2*d + 6
3*a^2*b*c*d^2 - 585*a^3*d^3)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(1536*a^3*c^4*x) +
((b*c - a*d)*(77*b^3*c^3 + 105*a*b^2*c^2*d + 135*a^2*b*c*d^2 + 195*a^3*d^3)*Arc
Tan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(1024*a^(15/4)*c^(17/4
)) + ((b*c - a*d)*(77*b^3*c^3 + 105*a*b^2*c^2*d + 135*a^2*b*c*d^2 + 195*a^3*d^3)
*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(1024*a^(15/4)*c^
(17/4))
_______________________________________________________________________________________
Rubi in Sympy [A] time = 158.041, size = 357, normalized size = 0.97 \[ - \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}}}{4 c x^{4}} + \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (13 a d - b c\right )}{48 a c^{2} x^{3}} - \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (117 a^{2} d^{2} - 10 a b c d - 11 b^{2} c^{2}\right )}{384 a^{2} c^{3} x^{2}} + \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (585 a^{3} d^{3} - 63 a^{2} b c d^{2} - 61 a b^{2} c^{2} d - 77 b^{3} c^{3}\right )}{1536 a^{3} c^{4} x} - \frac{\left (a d - b c\right ) \left (195 a^{3} d^{3} + 135 a^{2} b c d^{2} + 105 a b^{2} c^{2} d + 77 b^{3} c^{3}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt [4]{a + b x}}{\sqrt [4]{a} \sqrt [4]{c + d x}} \right )}}{1024 a^{\frac{15}{4}} c^{\frac{17}{4}}} - \frac{\left (a d - b c\right ) \left (195 a^{3} d^{3} + 135 a^{2} b c d^{2} + 105 a b^{2} c^{2} d + 77 b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt [4]{a + b x}}{\sqrt [4]{a} \sqrt [4]{c + d x}} \right )}}{1024 a^{\frac{15}{4}} c^{\frac{17}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(1/4)/x**5/(d*x+c)**(1/4),x)
[Out]
-(a + b*x)**(1/4)*(c + d*x)**(3/4)/(4*c*x**4) + (a + b*x)**(1/4)*(c + d*x)**(3/4
)*(13*a*d - b*c)/(48*a*c**2*x**3) - (a + b*x)**(1/4)*(c + d*x)**(3/4)*(117*a**2*
d**2 - 10*a*b*c*d - 11*b**2*c**2)/(384*a**2*c**3*x**2) + (a + b*x)**(1/4)*(c + d
*x)**(3/4)*(585*a**3*d**3 - 63*a**2*b*c*d**2 - 61*a*b**2*c**2*d - 77*b**3*c**3)/
(1536*a**3*c**4*x) - (a*d - b*c)*(195*a**3*d**3 + 135*a**2*b*c*d**2 + 105*a*b**2
*c**2*d + 77*b**3*c**3)*atan(c**(1/4)*(a + b*x)**(1/4)/(a**(1/4)*(c + d*x)**(1/4
)))/(1024*a**(15/4)*c**(17/4)) - (a*d - b*c)*(195*a**3*d**3 + 135*a**2*b*c*d**2
+ 105*a*b**2*c**2*d + 77*b**3*c**3)*atanh(c**(1/4)*(a + b*x)**(1/4)/(a**(1/4)*(c
+ d*x)**(1/4)))/(1024*a**(15/4)*c**(17/4))
_______________________________________________________________________________________
Mathematica [C] time = 0.510293, size = 315, normalized size = 0.86 \[ \frac{(a+b x) (c+d x) \left (a^3 \left (-384 c^3+416 c^2 d x-468 c d^2 x^2+585 d^3 x^3\right )+a^2 b c x \left (-32 c^2+40 c d x-63 d^2 x^2\right )+a b^2 c^2 x^2 (44 c-61 d x)-77 b^3 c^3 x^3\right )-\frac{6 b d x^5 \left (-195 a^4 d^4+60 a^3 b c d^3+30 a^2 b^2 c^2 d^2+28 a b^3 c^3 d+77 b^4 c^4\right ) F_1\left (1;\frac{3}{4},\frac{1}{4};2;-\frac{a}{b x},-\frac{c}{d x}\right )}{-8 b d x F_1\left (1;\frac{3}{4},\frac{1}{4};2;-\frac{a}{b x},-\frac{c}{d x}\right )+b c F_1\left (2;\frac{3}{4},\frac{5}{4};3;-\frac{a}{b x},-\frac{c}{d x}\right )+3 a d F_1\left (2;\frac{7}{4},\frac{1}{4};3;-\frac{a}{b x},-\frac{c}{d x}\right )}}{1536 a^3 c^4 x^4 (a+b x)^{3/4} \sqrt [4]{c+d x}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x)^(1/4)/(x^5*(c + d*x)^(1/4)),x]
[Out]
((a + b*x)*(c + d*x)*(-77*b^3*c^3*x^3 + a*b^2*c^2*x^2*(44*c - 61*d*x) + a^2*b*c*
x*(-32*c^2 + 40*c*d*x - 63*d^2*x^2) + a^3*(-384*c^3 + 416*c^2*d*x - 468*c*d^2*x^
2 + 585*d^3*x^3)) - (6*b*d*(77*b^4*c^4 + 28*a*b^3*c^3*d + 30*a^2*b^2*c^2*d^2 + 6
0*a^3*b*c*d^3 - 195*a^4*d^4)*x^5*AppellF1[1, 3/4, 1/4, 2, -(a/(b*x)), -(c/(d*x))
])/(-8*b*d*x*AppellF1[1, 3/4, 1/4, 2, -(a/(b*x)), -(c/(d*x))] + b*c*AppellF1[2,
3/4, 5/4, 3, -(a/(b*x)), -(c/(d*x))] + 3*a*d*AppellF1[2, 7/4, 1/4, 3, -(a/(b*x))
, -(c/(d*x))]))/(1536*a^3*c^4*x^4*(a + b*x)^(3/4)*(c + d*x)^(1/4))
_______________________________________________________________________________________
Maple [F] time = 0.06, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5}}\sqrt [4]{bx+a}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(1/4)/x^5/(d*x+c)^(1/4),x)
[Out]
int((b*x+a)^(1/4)/x^5/(d*x+c)^(1/4),x)
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{1}{4}}}{{\left (d x + c\right )}^{\frac{1}{4}} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x^5),x, algorithm="maxima")
[Out]
integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x^5), x)
_______________________________________________________________________________________
Fricas [A] time = 0.666209, size = 2970, normalized size = 8.07 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x^5),x, algorithm="fricas")
[Out]
1/6144*(12*a^3*c^4*x^4*((35153041*b^16*c^16 + 51131696*a*b^15*c^15*d + 82673976*
a^2*b^14*c^14*d^2 + 176093456*a^3*b^13*c^13*d^3 - 182203364*a^4*b^12*c^12*d^4 -
191017680*a^5*b^11*c^11*d^5 - 318453240*a^6*b^10*c^10*d^6 - 989262960*a^7*b^9*c^
9*d^7 + 665778150*a^8*b^8*c^8*d^8 + 275389200*a^9*b^7*c^7*d^9 + 370974600*a^10*b
^6*c^6*d^10 + 2155086000*a^11*b^5*c^5*d^11 - 1551622500*a^12*b^4*c^4*d^12 - 1776
06000*a^13*b^3*c^3*d^13 - 68445000*a^14*b^2*c^2*d^14 - 1779570000*a^15*b*c*d^15
+ 1445900625*a^16*d^16)/(a^15*c^17))^(1/4)*arctan(-(a^4*c^4*d*x + a^4*c^5)*((351
53041*b^16*c^16 + 51131696*a*b^15*c^15*d + 82673976*a^2*b^14*c^14*d^2 + 17609345
6*a^3*b^13*c^13*d^3 - 182203364*a^4*b^12*c^12*d^4 - 191017680*a^5*b^11*c^11*d^5
- 318453240*a^6*b^10*c^10*d^6 - 989262960*a^7*b^9*c^9*d^7 + 665778150*a^8*b^8*c^
8*d^8 + 275389200*a^9*b^7*c^7*d^9 + 370974600*a^10*b^6*c^6*d^10 + 2155086000*a^1
1*b^5*c^5*d^11 - 1551622500*a^12*b^4*c^4*d^12 - 177606000*a^13*b^3*c^3*d^13 - 68
445000*a^14*b^2*c^2*d^14 - 1779570000*a^15*b*c*d^15 + 1445900625*a^16*d^16)/(a^1
5*c^17))^(1/4)/((77*b^4*c^4 + 28*a*b^3*c^3*d + 30*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d
^3 - 195*a^4*d^4)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (d*x + c)*sqrt(((5929*b^8*c^
8 + 4312*a*b^7*c^7*d + 5404*a^2*b^6*c^6*d^2 + 10920*a^3*b^5*c^5*d^3 - 25770*a^4*
b^4*c^4*d^4 - 7320*a^5*b^3*c^3*d^5 - 8100*a^6*b^2*c^2*d^6 - 23400*a^7*b*c*d^7 +
38025*a^8*d^8)*sqrt(b*x + a)*sqrt(d*x + c) + (a^8*c^8*d*x + a^8*c^9)*sqrt((35153
041*b^16*c^16 + 51131696*a*b^15*c^15*d + 82673976*a^2*b^14*c^14*d^2 + 176093456*
a^3*b^13*c^13*d^3 - 182203364*a^4*b^12*c^12*d^4 - 191017680*a^5*b^11*c^11*d^5 -
318453240*a^6*b^10*c^10*d^6 - 989262960*a^7*b^9*c^9*d^7 + 665778150*a^8*b^8*c^8*
d^8 + 275389200*a^9*b^7*c^7*d^9 + 370974600*a^10*b^6*c^6*d^10 + 2155086000*a^11*
b^5*c^5*d^11 - 1551622500*a^12*b^4*c^4*d^12 - 177606000*a^13*b^3*c^3*d^13 - 6844
5000*a^14*b^2*c^2*d^14 - 1779570000*a^15*b*c*d^15 + 1445900625*a^16*d^16)/(a^15*
c^17)))/(d*x + c)))) + 3*a^3*c^4*x^4*((35153041*b^16*c^16 + 51131696*a*b^15*c^15
*d + 82673976*a^2*b^14*c^14*d^2 + 176093456*a^3*b^13*c^13*d^3 - 182203364*a^4*b^
12*c^12*d^4 - 191017680*a^5*b^11*c^11*d^5 - 318453240*a^6*b^10*c^10*d^6 - 989262
960*a^7*b^9*c^9*d^7 + 665778150*a^8*b^8*c^8*d^8 + 275389200*a^9*b^7*c^7*d^9 + 37
0974600*a^10*b^6*c^6*d^10 + 2155086000*a^11*b^5*c^5*d^11 - 1551622500*a^12*b^4*c
^4*d^12 - 177606000*a^13*b^3*c^3*d^13 - 68445000*a^14*b^2*c^2*d^14 - 1779570000*
a^15*b*c*d^15 + 1445900625*a^16*d^16)/(a^15*c^17))^(1/4)*log(-((77*b^4*c^4 + 28*
a*b^3*c^3*d + 30*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 195*a^4*d^4)*(b*x + a)^(1/4)
*(d*x + c)^(3/4) + (a^4*c^4*d*x + a^4*c^5)*((35153041*b^16*c^16 + 51131696*a*b^1
5*c^15*d + 82673976*a^2*b^14*c^14*d^2 + 176093456*a^3*b^13*c^13*d^3 - 182203364*
a^4*b^12*c^12*d^4 - 191017680*a^5*b^11*c^11*d^5 - 318453240*a^6*b^10*c^10*d^6 -
989262960*a^7*b^9*c^9*d^7 + 665778150*a^8*b^8*c^8*d^8 + 275389200*a^9*b^7*c^7*d^
9 + 370974600*a^10*b^6*c^6*d^10 + 2155086000*a^11*b^5*c^5*d^11 - 1551622500*a^12
*b^4*c^4*d^12 - 177606000*a^13*b^3*c^3*d^13 - 68445000*a^14*b^2*c^2*d^14 - 17795
70000*a^15*b*c*d^15 + 1445900625*a^16*d^16)/(a^15*c^17))^(1/4))/(d*x + c)) - 3*a
^3*c^4*x^4*((35153041*b^16*c^16 + 51131696*a*b^15*c^15*d + 82673976*a^2*b^14*c^1
4*d^2 + 176093456*a^3*b^13*c^13*d^3 - 182203364*a^4*b^12*c^12*d^4 - 191017680*a^
5*b^11*c^11*d^5 - 318453240*a^6*b^10*c^10*d^6 - 989262960*a^7*b^9*c^9*d^7 + 6657
78150*a^8*b^8*c^8*d^8 + 275389200*a^9*b^7*c^7*d^9 + 370974600*a^10*b^6*c^6*d^10
+ 2155086000*a^11*b^5*c^5*d^11 - 1551622500*a^12*b^4*c^4*d^12 - 177606000*a^13*b
^3*c^3*d^13 - 68445000*a^14*b^2*c^2*d^14 - 1779570000*a^15*b*c*d^15 + 1445900625
*a^16*d^16)/(a^15*c^17))^(1/4)*log(-((77*b^4*c^4 + 28*a*b^3*c^3*d + 30*a^2*b^2*c
^2*d^2 + 60*a^3*b*c*d^3 - 195*a^4*d^4)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (a^4*c^
4*d*x + a^4*c^5)*((35153041*b^16*c^16 + 51131696*a*b^15*c^15*d + 82673976*a^2*b^
14*c^14*d^2 + 176093456*a^3*b^13*c^13*d^3 - 182203364*a^4*b^12*c^12*d^4 - 191017
680*a^5*b^11*c^11*d^5 - 318453240*a^6*b^10*c^10*d^6 - 989262960*a^7*b^9*c^9*d^7
+ 665778150*a^8*b^8*c^8*d^8 + 275389200*a^9*b^7*c^7*d^9 + 370974600*a^10*b^6*c^6
*d^10 + 2155086000*a^11*b^5*c^5*d^11 - 1551622500*a^12*b^4*c^4*d^12 - 177606000*
a^13*b^3*c^3*d^13 - 68445000*a^14*b^2*c^2*d^14 - 1779570000*a^15*b*c*d^15 + 1445
900625*a^16*d^16)/(a^15*c^17))^(1/4))/(d*x + c)) - 4*(384*a^3*c^3 + (77*b^3*c^3
+ 61*a*b^2*c^2*d + 63*a^2*b*c*d^2 - 585*a^3*d^3)*x^3 - 4*(11*a*b^2*c^3 + 10*a^2*
b*c^2*d - 117*a^3*c*d^2)*x^2 + 32*(a^2*b*c^3 - 13*a^3*c^2*d)*x)*(b*x + a)^(1/4)*
(d*x + c)^(3/4))/(a^3*c^4*x^4)
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [4]{a + b x}}{x^{5} \sqrt [4]{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(1/4)/x**5/(d*x+c)**(1/4),x)
[Out]
Integral((a + b*x)**(1/4)/(x**5*(c + d*x)**(1/4)), x)
_______________________________________________________________________________________
GIAC/XCAS [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 + a)^(1/4)/((d*x + c)^(1/4)*x^5),x, algorithm="giac")
[Out]
Timed out