3.878 \(\int \frac{\sqrt [4]{a+b x}}{x^4 \sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=266 \[ \frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (7 b c-15 a d) (3 a d+b c)}{96 a^2 c^3 x}-\frac{(b c-a d) \left (15 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{11/4} c^{13/4}}-\frac{(b c-a d) \left (15 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{11/4} c^{13/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (b c-9 a d)}{24 a c^2 x^2}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 c x^3} \]

[Out]

-((a + b*x)^(1/4)*(c + d*x)^(3/4))/(3*c*x^3) - ((b*c - 9*a*d)*(a + b*x)^(1/4)*(c
+ d*x)^(3/4))/(24*a*c^2*x^2) + ((7*b*c - 15*a*d)*(b*c + 3*a*d)*(a + b*x)^(1/4)*
(c + d*x)^(3/4))/(96*a^2*c^3*x) - ((b*c - a*d)*(7*b^2*c^2 + 10*a*b*c*d + 15*a^2*
d^2)*ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(64*a^(11/4)*c
^(13/4)) - ((b*c - a*d)*(7*b^2*c^2 + 10*a*b*c*d + 15*a^2*d^2)*ArcTanh[(c^(1/4)*(
a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(64*a^(11/4)*c^(13/4))

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Rubi [A]  time = 0.621686, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (7 b c-15 a d) (3 a d+b c)}{96 a^2 c^3 x}-\frac{(b c-a d) \left (15 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{11/4} c^{13/4}}-\frac{(b c-a d) \left (15 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{11/4} c^{13/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (b c-9 a d)}{24 a c^2 x^2}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 c x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(1/4)/(x^4*(c + d*x)^(1/4)),x]

[Out]

-((a + b*x)^(1/4)*(c + d*x)^(3/4))/(3*c*x^3) - ((b*c - 9*a*d)*(a + b*x)^(1/4)*(c
+ d*x)^(3/4))/(24*a*c^2*x^2) + ((7*b*c - 15*a*d)*(b*c + 3*a*d)*(a + b*x)^(1/4)*
(c + d*x)^(3/4))/(96*a^2*c^3*x) - ((b*c - a*d)*(7*b^2*c^2 + 10*a*b*c*d + 15*a^2*
d^2)*ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(64*a^(11/4)*c
^(13/4)) - ((b*c - a*d)*(7*b^2*c^2 + 10*a*b*c*d + 15*a^2*d^2)*ArcTanh[(c^(1/4)*(
a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(64*a^(11/4)*c^(13/4))

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Rubi in Sympy [A]  time = 77.8157, size = 250, normalized size = 0.94 \[ - \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}}}{3 c x^{3}} + \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (9 a d - b c\right )}{24 a c^{2} x^{2}} - \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (3 a d + b c\right ) \left (15 a d - 7 b c\right )}{96 a^{2} c^{3} x} + \frac{\left (a d - b c\right ) \left (15 a^{2} d^{2} + 10 a b c d + 7 b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt [4]{a + b x}}{\sqrt [4]{a} \sqrt [4]{c + d x}} \right )}}{64 a^{\frac{11}{4}} c^{\frac{13}{4}}} + \frac{\left (a d - b c\right ) \left (15 a^{2} d^{2} + 10 a b c d + 7 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt [4]{a + b x}}{\sqrt [4]{a} \sqrt [4]{c + d x}} \right )}}{64 a^{\frac{11}{4}} c^{\frac{13}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/4)/x**4/(d*x+c)**(1/4),x)

[Out]

-(a + b*x)**(1/4)*(c + d*x)**(3/4)/(3*c*x**3) + (a + b*x)**(1/4)*(c + d*x)**(3/4
)*(9*a*d - b*c)/(24*a*c**2*x**2) - (a + b*x)**(1/4)*(c + d*x)**(3/4)*(3*a*d + b*
c)*(15*a*d - 7*b*c)/(96*a**2*c**3*x) + (a*d - b*c)*(15*a**2*d**2 + 10*a*b*c*d +
7*b**2*c**2)*atan(c**(1/4)*(a + b*x)**(1/4)/(a**(1/4)*(c + d*x)**(1/4)))/(64*a**
(11/4)*c**(13/4)) + (a*d - b*c)*(15*a**2*d**2 + 10*a*b*c*d + 7*b**2*c**2)*atanh(
c**(1/4)*(a + b*x)**(1/4)/(a**(1/4)*(c + d*x)**(1/4)))/(64*a**(11/4)*c**(13/4))

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Mathematica [C]  time = 0.425905, size = 260, normalized size = 0.98 \[ \frac{\frac{6 b d x^4 \left (-15 a^3 d^3+5 a^2 b c d^2+3 a b^2 c^2 d+7 b^3 c^3\right ) F_1\left (1;\frac{3}{4},\frac{1}{4};2;-\frac{a}{b x},-\frac{c}{d x}\right )}{-8 b d x F_1\left (1;\frac{3}{4},\frac{1}{4};2;-\frac{a}{b x},-\frac{c}{d x}\right )+b c F_1\left (2;\frac{3}{4},\frac{5}{4};3;-\frac{a}{b x},-\frac{c}{d x}\right )+3 a d F_1\left (2;\frac{7}{4},\frac{1}{4};3;-\frac{a}{b x},-\frac{c}{d x}\right )}-(a+b x) (c+d x) \left (a^2 \left (32 c^2-36 c d x+45 d^2 x^2\right )+2 a b c x (2 c-3 d x)-7 b^2 c^2 x^2\right )}{96 a^2 c^3 x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)^(1/4)/(x^4*(c + d*x)^(1/4)),x]

[Out]

(-((a + b*x)*(c + d*x)*(-7*b^2*c^2*x^2 + 2*a*b*c*x*(2*c - 3*d*x) + a^2*(32*c^2 -
36*c*d*x + 45*d^2*x^2))) + (6*b*d*(7*b^3*c^3 + 3*a*b^2*c^2*d + 5*a^2*b*c*d^2 -
15*a^3*d^3)*x^4*AppellF1[1, 3/4, 1/4, 2, -(a/(b*x)), -(c/(d*x))])/(-8*b*d*x*Appe
llF1[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))]))/(
96*a^2*c^3*x^3*(a + b*x)^(3/4)*(c + d*x)^(1/4))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}}\sqrt [4]{bx+a}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/4)/x^4/(d*x+c)^(1/4),x)

[Out]

int((b*x+a)^(1/4)/x^4/(d*x+c)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{1}{4}}}{{\left (d x + c\right )}^{\frac{1}{4}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x^4),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x^4), x)

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Fricas [A]  time = 0.354935, size = 2277, normalized size = 8.56 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x^4),x, algorithm="fricas")

[Out]

-1/384*(12*a^2*c^3*x^3*((2401*b^12*c^12 + 4116*a*b^11*c^11*d + 9506*a^2*b^10*c^1
0*d^2 - 11004*a^3*b^9*c^9*d^3 - 15249*a^4*b^8*c^8*d^4 - 48600*a^5*b^7*c^7*d^5 +
31580*a^6*b^6*c^6*d^6 + 18600*a^7*b^5*c^5*d^7 + 93775*a^8*b^4*c^4*d^8 - 61500*a^
9*b^3*c^3*d^9 - 6750*a^10*b^2*c^2*d^10 - 67500*a^11*b*c*d^11 + 50625*a^12*d^12)/
(a^11*c^13))^(1/4)*arctan(-(a^3*c^3*d*x + a^3*c^4)*((2401*b^12*c^12 + 4116*a*b^1
1*c^11*d + 9506*a^2*b^10*c^10*d^2 - 11004*a^3*b^9*c^9*d^3 - 15249*a^4*b^8*c^8*d^
4 - 48600*a^5*b^7*c^7*d^5 + 31580*a^6*b^6*c^6*d^6 + 18600*a^7*b^5*c^5*d^7 + 9377
5*a^8*b^4*c^4*d^8 - 61500*a^9*b^3*c^3*d^9 - 6750*a^10*b^2*c^2*d^10 - 67500*a^11*
b*c*d^11 + 50625*a^12*d^12)/(a^11*c^13))^(1/4)/((7*b^3*c^3 + 3*a*b^2*c^2*d + 5*a
^2*b*c*d^2 - 15*a^3*d^3)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (d*x + c)*sqrt(((49*b
^6*c^6 + 42*a*b^5*c^5*d + 79*a^2*b^4*c^4*d^2 - 180*a^3*b^3*c^3*d^3 - 65*a^4*b^2*
c^2*d^4 - 150*a^5*b*c*d^5 + 225*a^6*d^6)*sqrt(b*x + a)*sqrt(d*x + c) + (a^6*c^6*
d*x + a^6*c^7)*sqrt((2401*b^12*c^12 + 4116*a*b^11*c^11*d + 9506*a^2*b^10*c^10*d^
2 - 11004*a^3*b^9*c^9*d^3 - 15249*a^4*b^8*c^8*d^4 - 48600*a^5*b^7*c^7*d^5 + 3158
0*a^6*b^6*c^6*d^6 + 18600*a^7*b^5*c^5*d^7 + 93775*a^8*b^4*c^4*d^8 - 61500*a^9*b^
3*c^3*d^9 - 6750*a^10*b^2*c^2*d^10 - 67500*a^11*b*c*d^11 + 50625*a^12*d^12)/(a^1
1*c^13)))/(d*x + c)))) + 3*a^2*c^3*x^3*((2401*b^12*c^12 + 4116*a*b^11*c^11*d + 9
506*a^2*b^10*c^10*d^2 - 11004*a^3*b^9*c^9*d^3 - 15249*a^4*b^8*c^8*d^4 - 48600*a^
5*b^7*c^7*d^5 + 31580*a^6*b^6*c^6*d^6 + 18600*a^7*b^5*c^5*d^7 + 93775*a^8*b^4*c^
4*d^8 - 61500*a^9*b^3*c^3*d^9 - 6750*a^10*b^2*c^2*d^10 - 67500*a^11*b*c*d^11 + 5
0625*a^12*d^12)/(a^11*c^13))^(1/4)*log(-((7*b^3*c^3 + 3*a*b^2*c^2*d + 5*a^2*b*c*
d^2 - 15*a^3*d^3)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (a^3*c^3*d*x + a^3*c^4)*((24
01*b^12*c^12 + 4116*a*b^11*c^11*d + 9506*a^2*b^10*c^10*d^2 - 11004*a^3*b^9*c^9*d
^3 - 15249*a^4*b^8*c^8*d^4 - 48600*a^5*b^7*c^7*d^5 + 31580*a^6*b^6*c^6*d^6 + 186
00*a^7*b^5*c^5*d^7 + 93775*a^8*b^4*c^4*d^8 - 61500*a^9*b^3*c^3*d^9 - 6750*a^10*b
^2*c^2*d^10 - 67500*a^11*b*c*d^11 + 50625*a^12*d^12)/(a^11*c^13))^(1/4))/(d*x +
c)) - 3*a^2*c^3*x^3*((2401*b^12*c^12 + 4116*a*b^11*c^11*d + 9506*a^2*b^10*c^10*d
^2 - 11004*a^3*b^9*c^9*d^3 - 15249*a^4*b^8*c^8*d^4 - 48600*a^5*b^7*c^7*d^5 + 315
80*a^6*b^6*c^6*d^6 + 18600*a^7*b^5*c^5*d^7 + 93775*a^8*b^4*c^4*d^8 - 61500*a^9*b
^3*c^3*d^9 - 6750*a^10*b^2*c^2*d^10 - 67500*a^11*b*c*d^11 + 50625*a^12*d^12)/(a^
11*c^13))^(1/4)*log(-((7*b^3*c^3 + 3*a*b^2*c^2*d + 5*a^2*b*c*d^2 - 15*a^3*d^3)*(
b*x + a)^(1/4)*(d*x + c)^(3/4) - (a^3*c^3*d*x + a^3*c^4)*((2401*b^12*c^12 + 4116
*a*b^11*c^11*d + 9506*a^2*b^10*c^10*d^2 - 11004*a^3*b^9*c^9*d^3 - 15249*a^4*b^8*
c^8*d^4 - 48600*a^5*b^7*c^7*d^5 + 31580*a^6*b^6*c^6*d^6 + 18600*a^7*b^5*c^5*d^7
+ 93775*a^8*b^4*c^4*d^8 - 61500*a^9*b^3*c^3*d^9 - 6750*a^10*b^2*c^2*d^10 - 67500
*a^11*b*c*d^11 + 50625*a^12*d^12)/(a^11*c^13))^(1/4))/(d*x + c)) + 4*(32*a^2*c^2
- (7*b^2*c^2 + 6*a*b*c*d - 45*a^2*d^2)*x^2 + 4*(a*b*c^2 - 9*a^2*c*d)*x)*(b*x +
a)^(1/4)*(d*x + c)^(3/4))/(a^2*c^3*x^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [4]{a + b x}}{x^{4} \sqrt [4]{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(1/4)/x**4/(d*x+c)**(1/4),x)

[Out]

Integral((a + b*x)**(1/4)/(x**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)/((d*x + c)^(1/4)*x^4),x, algorithm="giac")

[Out]

Timed out