3.871 \(\int \frac{x^3 \sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx\)
Optimal. Leaf size=340 \[ \frac{(a+b x)^{5/4} (c+d x)^{3/4} \left (77 a^2 d^2-8 b d x (11 a d+13 b c)+94 a b c d+117 b^2 c^2\right )}{384 b^3 d^3}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (77 a^3 d^3+105 a^2 b c d^2+135 a b^2 c^2 d+195 b^3 c^3\right )}{512 b^3 d^4}+\frac{(b c-a d) \left (77 a^3 d^3+105 a^2 b c d^2+135 a b^2 c^2 d+195 b^3 c^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{1024 b^{15/4} d^{17/4}}+\frac{(b c-a d) \left (77 a^3 d^3+105 a^2 b c d^2+135 a b^2 c^2 d+195 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{1024 b^{15/4} d^{17/4}}+\frac{x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d} \]
[Out]
-((195*b^3*c^3 + 135*a*b^2*c^2*d + 105*a^2*b*c*d^2 + 77*a^3*d^3)*(a + b*x)^(1/4)
*(c + d*x)^(3/4))/(512*b^3*d^4) + (x^2*(a + b*x)^(5/4)*(c + d*x)^(3/4))/(4*b*d)
+ ((a + b*x)^(5/4)*(c + d*x)^(3/4)*(117*b^2*c^2 + 94*a*b*c*d + 77*a^2*d^2 - 8*b*
d*(13*b*c + 11*a*d)*x))/(384*b^3*d^3) + ((b*c - a*d)*(195*b^3*c^3 + 135*a*b^2*c^
2*d + 105*a^2*b*c*d^2 + 77*a^3*d^3)*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c
+ d*x)^(1/4))])/(1024*b^(15/4)*d^(17/4)) + ((b*c - a*d)*(195*b^3*c^3 + 135*a*b^
2*c^2*d + 105*a^2*b*c*d^2 + 77*a^3*d^3)*ArcTanh[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/
4)*(c + d*x)^(1/4))])/(1024*b^(15/4)*d^(17/4))
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Rubi [A] time = 0.609344, antiderivative size = 340, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318
\[ \frac{(a+b x)^{5/4} (c+d x)^{3/4} \left (77 a^2 d^2-8 b d x (11 a d+13 b c)+94 a b c d+117 b^2 c^2\right )}{384 b^3 d^3}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (77 a^3 d^3+105 a^2 b c d^2+135 a b^2 c^2 d+195 b^3 c^3\right )}{512 b^3 d^4}+\frac{(b c-a d) \left (77 a^3 d^3+105 a^2 b c d^2+135 a b^2 c^2 d+195 b^3 c^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{1024 b^{15/4} d^{17/4}}+\frac{(b c-a d) \left (77 a^3 d^3+105 a^2 b c d^2+135 a b^2 c^2 d+195 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{1024 b^{15/4} d^{17/4}}+\frac{x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(a + b*x)^(1/4))/(c + d*x)^(1/4),x]
[Out]
-((195*b^3*c^3 + 135*a*b^2*c^2*d + 105*a^2*b*c*d^2 + 77*a^3*d^3)*(a + b*x)^(1/4)
*(c + d*x)^(3/4))/(512*b^3*d^4) + (x^2*(a + b*x)^(5/4)*(c + d*x)^(3/4))/(4*b*d)
+ ((a + b*x)^(5/4)*(c + d*x)^(3/4)*(117*b^2*c^2 + 94*a*b*c*d + 77*a^2*d^2 - 8*b*
d*(13*b*c + 11*a*d)*x))/(384*b^3*d^3) + ((b*c - a*d)*(195*b^3*c^3 + 135*a*b^2*c^
2*d + 105*a^2*b*c*d^2 + 77*a^3*d^3)*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c
+ d*x)^(1/4))])/(1024*b^(15/4)*d^(17/4)) + ((b*c - a*d)*(195*b^3*c^3 + 135*a*b^
2*c^2*d + 105*a^2*b*c*d^2 + 77*a^3*d^3)*ArcTanh[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/
4)*(c + d*x)^(1/4))])/(1024*b^(15/4)*d^(17/4))
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Rubi in Sympy [A] time = 53.9675, size = 340, normalized size = 1. \[ \frac{x^{2} \left (a + b x\right )^{\frac{5}{4}} \left (c + d x\right )^{\frac{3}{4}}}{4 b d} + \frac{\left (a + b x\right )^{\frac{5}{4}} \left (c + d x\right )^{\frac{3}{4}} \left (\frac{77 a^{2} d^{2}}{16} + \frac{47 a b c d}{8} + \frac{117 b^{2} c^{2}}{16} - \frac{b d x \left (11 a d + 13 b c\right )}{2}\right )}{24 b^{3} d^{3}} - \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (77 a^{3} d^{3} + 105 a^{2} b c d^{2} + 135 a b^{2} c^{2} d + 195 b^{3} c^{3}\right )}{512 b^{3} d^{4}} + \frac{\left (a d - b c\right ) \left (77 a^{3} d^{3} + 105 a^{2} b c d^{2} + 135 a b^{2} c^{2} d + 195 b^{3} c^{3}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{d} \sqrt [4]{a + b x}} \right )}}{1024 b^{\frac{15}{4}} d^{\frac{17}{4}}} - \frac{\left (a d - b c\right ) \left (77 a^{3} d^{3} + 105 a^{2} b c d^{2} + 135 a b^{2} c^{2} d + 195 b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{d} \sqrt [4]{a + b x}} \right )}}{1024 b^{\frac{15}{4}} d^{\frac{17}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(b*x+a)**(1/4)/(d*x+c)**(1/4),x)
[Out]
x**2*(a + b*x)**(5/4)*(c + d*x)**(3/4)/(4*b*d) + (a + b*x)**(5/4)*(c + d*x)**(3/
4)*(77*a**2*d**2/16 + 47*a*b*c*d/8 + 117*b**2*c**2/16 - b*d*x*(11*a*d + 13*b*c)/
2)/(24*b**3*d**3) - (a + b*x)**(1/4)*(c + d*x)**(3/4)*(77*a**3*d**3 + 105*a**2*b
*c*d**2 + 135*a*b**2*c**2*d + 195*b**3*c**3)/(512*b**3*d**4) + (a*d - b*c)*(77*a
**3*d**3 + 105*a**2*b*c*d**2 + 135*a*b**2*c**2*d + 195*b**3*c**3)*atan(b**(1/4)*
(c + d*x)**(1/4)/(d**(1/4)*(a + b*x)**(1/4)))/(1024*b**(15/4)*d**(17/4)) - (a*d
- b*c)*(77*a**3*d**3 + 105*a**2*b*c*d**2 + 135*a*b**2*c**2*d + 195*b**3*c**3)*at
anh(b**(1/4)*(c + d*x)**(1/4)/(d**(1/4)*(a + b*x)**(1/4)))/(1024*b**(15/4)*d**(1
7/4))
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Mathematica [C] time = 0.417013, size = 221, normalized size = 0.65 \[ \frac{(c+d x)^{3/4} \left (d (a+b x) \left (77 a^3 d^3+a^2 b d^2 (61 c-44 d x)+a b^2 d \left (63 c^2-40 c d x+32 d^2 x^2\right )+b^3 \left (-585 c^3+468 c^2 d x-416 c d^2 x^2+384 d^3 x^3\right )\right )-\left (77 a^4 d^4+28 a^3 b c d^3+30 a^2 b^2 c^2 d^2+60 a b^3 c^3 d-195 b^4 c^4\right ) \left (\frac{d (a+b x)}{a d-b c}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )\right )}{1536 b^3 d^5 (a+b x)^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(a + b*x)^(1/4))/(c + d*x)^(1/4),x]
[Out]
((c + d*x)^(3/4)*(d*(a + b*x)*(77*a^3*d^3 + a^2*b*d^2*(61*c - 44*d*x) + a*b^2*d*
(63*c^2 - 40*c*d*x + 32*d^2*x^2) + b^3*(-585*c^3 + 468*c^2*d*x - 416*c*d^2*x^2 +
384*d^3*x^3)) - (-195*b^4*c^4 + 60*a*b^3*c^3*d + 30*a^2*b^2*c^2*d^2 + 28*a^3*b*
c*d^3 + 77*a^4*d^4)*((d*(a + b*x))/(-(b*c) + a*d))^(3/4)*Hypergeometric2F1[3/4,
3/4, 7/4, (b*(c + d*x))/(b*c - a*d)]))/(1536*b^3*d^5*(a + b*x)^(3/4))
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Maple [F] time = 0.048, size = 0, normalized size = 0. \[ \int{{x}^{3}\sqrt [4]{bx+a}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(b*x+a)^(1/4)/(d*x+c)^(1/4),x)
[Out]
int(x^3*(b*x+a)^(1/4)/(d*x+c)^(1/4),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{1}{4}} x^{3}}{{\left (d x + c\right )}^{\frac{1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(1/4)*x^3/(d*x + c)^(1/4),x, algorithm="maxima")
[Out]
integrate((b*x + a)^(1/4)*x^3/(d*x + c)^(1/4), x)
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Fricas [A] time = 0.634914, size = 2952, normalized size = 8.68 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(1/4)*x^3/(d*x + c)^(1/4),x, algorithm="fricas")
[Out]
1/6144*(12*b^3*d^4*((1445900625*b^16*c^16 - 1779570000*a*b^15*c^15*d - 68445000*
a^2*b^14*c^14*d^2 - 177606000*a^3*b^13*c^13*d^3 - 1551622500*a^4*b^12*c^12*d^4 +
2155086000*a^5*b^11*c^11*d^5 + 370974600*a^6*b^10*c^10*d^6 + 275389200*a^7*b^9*
c^9*d^7 + 665778150*a^8*b^8*c^8*d^8 - 989262960*a^9*b^7*c^7*d^9 - 318453240*a^10
*b^6*c^6*d^10 - 191017680*a^11*b^5*c^5*d^11 - 182203364*a^12*b^4*c^4*d^12 + 1760
93456*a^13*b^3*c^3*d^13 + 82673976*a^14*b^2*c^2*d^14 + 51131696*a^15*b*c*d^15 +
35153041*a^16*d^16)/(b^15*d^17))^(1/4)*arctan(-(b^4*d^5*x + b^4*c*d^4)*((1445900
625*b^16*c^16 - 1779570000*a*b^15*c^15*d - 68445000*a^2*b^14*c^14*d^2 - 17760600
0*a^3*b^13*c^13*d^3 - 1551622500*a^4*b^12*c^12*d^4 + 2155086000*a^5*b^11*c^11*d^
5 + 370974600*a^6*b^10*c^10*d^6 + 275389200*a^7*b^9*c^9*d^7 + 665778150*a^8*b^8*
c^8*d^8 - 989262960*a^9*b^7*c^7*d^9 - 318453240*a^10*b^6*c^6*d^10 - 191017680*a^
11*b^5*c^5*d^11 - 182203364*a^12*b^4*c^4*d^12 + 176093456*a^13*b^3*c^3*d^13 + 82
673976*a^14*b^2*c^2*d^14 + 51131696*a^15*b*c*d^15 + 35153041*a^16*d^16)/(b^15*d^
17))^(1/4)/((195*b^4*c^4 - 60*a*b^3*c^3*d - 30*a^2*b^2*c^2*d^2 - 28*a^3*b*c*d^3
- 77*a^4*d^4)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (d*x + c)*sqrt(((38025*b^8*c^8 -
23400*a*b^7*c^7*d - 8100*a^2*b^6*c^6*d^2 - 7320*a^3*b^5*c^5*d^3 - 25770*a^4*b^4
*c^4*d^4 + 10920*a^5*b^3*c^3*d^5 + 5404*a^6*b^2*c^2*d^6 + 4312*a^7*b*c*d^7 + 592
9*a^8*d^8)*sqrt(b*x + a)*sqrt(d*x + c) + (b^8*d^9*x + b^8*c*d^8)*sqrt((144590062
5*b^16*c^16 - 1779570000*a*b^15*c^15*d - 68445000*a^2*b^14*c^14*d^2 - 177606000*
a^3*b^13*c^13*d^3 - 1551622500*a^4*b^12*c^12*d^4 + 2155086000*a^5*b^11*c^11*d^5
+ 370974600*a^6*b^10*c^10*d^6 + 275389200*a^7*b^9*c^9*d^7 + 665778150*a^8*b^8*c^
8*d^8 - 989262960*a^9*b^7*c^7*d^9 - 318453240*a^10*b^6*c^6*d^10 - 191017680*a^11
*b^5*c^5*d^11 - 182203364*a^12*b^4*c^4*d^12 + 176093456*a^13*b^3*c^3*d^13 + 8267
3976*a^14*b^2*c^2*d^14 + 51131696*a^15*b*c*d^15 + 35153041*a^16*d^16)/(b^15*d^17
)))/(d*x + c)))) + 3*b^3*d^4*((1445900625*b^16*c^16 - 1779570000*a*b^15*c^15*d -
68445000*a^2*b^14*c^14*d^2 - 177606000*a^3*b^13*c^13*d^3 - 1551622500*a^4*b^12*
c^12*d^4 + 2155086000*a^5*b^11*c^11*d^5 + 370974600*a^6*b^10*c^10*d^6 + 27538920
0*a^7*b^9*c^9*d^7 + 665778150*a^8*b^8*c^8*d^8 - 989262960*a^9*b^7*c^7*d^9 - 3184
53240*a^10*b^6*c^6*d^10 - 191017680*a^11*b^5*c^5*d^11 - 182203364*a^12*b^4*c^4*d
^12 + 176093456*a^13*b^3*c^3*d^13 + 82673976*a^14*b^2*c^2*d^14 + 51131696*a^15*b
*c*d^15 + 35153041*a^16*d^16)/(b^15*d^17))^(1/4)*log(-((195*b^4*c^4 - 60*a*b^3*c
^3*d - 30*a^2*b^2*c^2*d^2 - 28*a^3*b*c*d^3 - 77*a^4*d^4)*(b*x + a)^(1/4)*(d*x +
c)^(3/4) + (b^4*d^5*x + b^4*c*d^4)*((1445900625*b^16*c^16 - 1779570000*a*b^15*c^
15*d - 68445000*a^2*b^14*c^14*d^2 - 177606000*a^3*b^13*c^13*d^3 - 1551622500*a^4
*b^12*c^12*d^4 + 2155086000*a^5*b^11*c^11*d^5 + 370974600*a^6*b^10*c^10*d^6 + 27
5389200*a^7*b^9*c^9*d^7 + 665778150*a^8*b^8*c^8*d^8 - 989262960*a^9*b^7*c^7*d^9
- 318453240*a^10*b^6*c^6*d^10 - 191017680*a^11*b^5*c^5*d^11 - 182203364*a^12*b^4
*c^4*d^12 + 176093456*a^13*b^3*c^3*d^13 + 82673976*a^14*b^2*c^2*d^14 + 51131696*
a^15*b*c*d^15 + 35153041*a^16*d^16)/(b^15*d^17))^(1/4))/(d*x + c)) - 3*b^3*d^4*(
(1445900625*b^16*c^16 - 1779570000*a*b^15*c^15*d - 68445000*a^2*b^14*c^14*d^2 -
177606000*a^3*b^13*c^13*d^3 - 1551622500*a^4*b^12*c^12*d^4 + 2155086000*a^5*b^11
*c^11*d^5 + 370974600*a^6*b^10*c^10*d^6 + 275389200*a^7*b^9*c^9*d^7 + 665778150*
a^8*b^8*c^8*d^8 - 989262960*a^9*b^7*c^7*d^9 - 318453240*a^10*b^6*c^6*d^10 - 1910
17680*a^11*b^5*c^5*d^11 - 182203364*a^12*b^4*c^4*d^12 + 176093456*a^13*b^3*c^3*d
^13 + 82673976*a^14*b^2*c^2*d^14 + 51131696*a^15*b*c*d^15 + 35153041*a^16*d^16)/
(b^15*d^17))^(1/4)*log(-((195*b^4*c^4 - 60*a*b^3*c^3*d - 30*a^2*b^2*c^2*d^2 - 28
*a^3*b*c*d^3 - 77*a^4*d^4)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (b^4*d^5*x + b^4*c*
d^4)*((1445900625*b^16*c^16 - 1779570000*a*b^15*c^15*d - 68445000*a^2*b^14*c^14*
d^2 - 177606000*a^3*b^13*c^13*d^3 - 1551622500*a^4*b^12*c^12*d^4 + 2155086000*a^
5*b^11*c^11*d^5 + 370974600*a^6*b^10*c^10*d^6 + 275389200*a^7*b^9*c^9*d^7 + 6657
78150*a^8*b^8*c^8*d^8 - 989262960*a^9*b^7*c^7*d^9 - 318453240*a^10*b^6*c^6*d^10
- 191017680*a^11*b^5*c^5*d^11 - 182203364*a^12*b^4*c^4*d^12 + 176093456*a^13*b^3
*c^3*d^13 + 82673976*a^14*b^2*c^2*d^14 + 51131696*a^15*b*c*d^15 + 35153041*a^16*
d^16)/(b^15*d^17))^(1/4))/(d*x + c)) + 4*(384*b^3*d^3*x^3 - 585*b^3*c^3 + 63*a*b
^2*c^2*d + 61*a^2*b*c*d^2 + 77*a^3*d^3 - 32*(13*b^3*c*d^2 - a*b^2*d^3)*x^2 + 4*(
117*b^3*c^2*d - 10*a*b^2*c*d^2 - 11*a^2*b*d^3)*x)*(b*x + a)^(1/4)*(d*x + c)^(3/4
))/(b^3*d^4)
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \sqrt [4]{a + b x}}{\sqrt [4]{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(b*x+a)**(1/4)/(d*x+c)**(1/4),x)
[Out]
Integral(x**3*(a + b*x)**(1/4)/(c + d*x)**(1/4), x)
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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)*x^3/(d*x + c)^(1/4),x, algorithm="giac")
[Out]
Timed out