3.895 \(\int \frac{1}{x^4 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx\)
Optimal. Leaf size=288 \[ \frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (9 a d+11 b c)}{24 a^2 c^2 x^2}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (45 a^2 d^2+54 a b c d+77 b^2 c^2\right )}{96 a^3 c^3 x}+\frac{\left (15 a^3 d^3+15 a^2 b c d^2+21 a b^2 c^2 d+77 b^3 c^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{15/4} c^{13/4}}+\frac{\left (15 a^3 d^3+15 a^2 b c d^2+21 a b^2 c^2 d+77 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{15/4} c^{13/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3} \]
[Out]
-((a + b*x)^(1/4)*(c + d*x)^(3/4))/(3*a*c*x^3) + ((11*b*c + 9*a*d)*(a + b*x)^(1/
4)*(c + d*x)^(3/4))/(24*a^2*c^2*x^2) - ((77*b^2*c^2 + 54*a*b*c*d + 45*a^2*d^2)*(
a + b*x)^(1/4)*(c + d*x)^(3/4))/(96*a^3*c^3*x) + ((77*b^3*c^3 + 21*a*b^2*c^2*d +
15*a^2*b*c*d^2 + 15*a^3*d^3)*ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x
)^(1/4))])/(64*a^(15/4)*c^(13/4)) + ((77*b^3*c^3 + 21*a*b^2*c^2*d + 15*a^2*b*c*d
^2 + 15*a^3*d^3)*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(
64*a^(15/4)*c^(13/4))
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Rubi [A] time = 0.621975, antiderivative size = 288, 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
\[ \frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (9 a d+11 b c)}{24 a^2 c^2 x^2}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (45 a^2 d^2+54 a b c d+77 b^2 c^2\right )}{96 a^3 c^3 x}+\frac{\left (15 a^3 d^3+15 a^2 b c d^2+21 a b^2 c^2 d+77 b^3 c^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{15/4} c^{13/4}}+\frac{\left (15 a^3 d^3+15 a^2 b c d^2+21 a b^2 c^2 d+77 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{15/4} c^{13/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3} \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*(a + b*x)^(3/4)*(c + d*x)^(1/4)),x]
[Out]
-((a + b*x)^(1/4)*(c + d*x)^(3/4))/(3*a*c*x^3) + ((11*b*c + 9*a*d)*(a + b*x)^(1/
4)*(c + d*x)^(3/4))/(24*a^2*c^2*x^2) - ((77*b^2*c^2 + 54*a*b*c*d + 45*a^2*d^2)*(
a + b*x)^(1/4)*(c + d*x)^(3/4))/(96*a^3*c^3*x) + ((77*b^3*c^3 + 21*a*b^2*c^2*d +
15*a^2*b*c*d^2 + 15*a^3*d^3)*ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x
)^(1/4))])/(64*a^(15/4)*c^(13/4)) + ((77*b^3*c^3 + 21*a*b^2*c^2*d + 15*a^2*b*c*d
^2 + 15*a^3*d^3)*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(
64*a^(15/4)*c^(13/4))
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Rubi in Sympy [A] time = 94.2009, size = 279, normalized size = 0.97 \[ - \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}}}{3 a c x^{3}} + \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (9 a d + 11 b c\right )}{24 a^{2} c^{2} x^{2}} - \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (45 a^{2} d^{2} + 54 a b c d + 77 b^{2} c^{2}\right )}{96 a^{3} c^{3} x} + \frac{\left (15 a^{3} d^{3} + 15 a^{2} b c d^{2} + 21 a b^{2} c^{2} d + 77 b^{3} c^{3}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt [4]{a + b x}}{\sqrt [4]{a} \sqrt [4]{c + d x}} \right )}}{64 a^{\frac{15}{4}} c^{\frac{13}{4}}} + \frac{\left (15 a^{3} d^{3} + 15 a^{2} b c d^{2} + 21 a b^{2} c^{2} d + 77 b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt [4]{a + b x}}{\sqrt [4]{a} \sqrt [4]{c + d x}} \right )}}{64 a^{\frac{15}{4}} c^{\frac{13}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(b*x+a)**(3/4)/(d*x+c)**(1/4),x)
[Out]
-(a + b*x)**(1/4)*(c + d*x)**(3/4)/(3*a*c*x**3) + (a + b*x)**(1/4)*(c + d*x)**(3
/4)*(9*a*d + 11*b*c)/(24*a**2*c**2*x**2) - (a + b*x)**(1/4)*(c + d*x)**(3/4)*(45
*a**2*d**2 + 54*a*b*c*d + 77*b**2*c**2)/(96*a**3*c**3*x) + (15*a**3*d**3 + 15*a*
*2*b*c*d**2 + 21*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)))/(64*a**(15/4)*c**(13/4)) + (15*a**3*d**3 + 15*a**2*b*
c*d**2 + 21*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)))/(64*a**(15/4)*c**(13/4))
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Mathematica [C] time = 0.493199, size = 259, normalized size = 0.9 \[ -\frac{(a+b x) (c+d x) \left (a^2 \left (32 c^2-36 c d x+45 d^2 x^2\right )+2 a b c x (27 d x-22 c)+77 b^2 c^2 x^2\right )+\frac{6 b d x^4 \left (15 a^3 d^3+15 a^2 b c d^2+21 a b^2 c^2 d+77 b^3 c^3\right ) F_1\left (1;\frac{3}{4},\frac{1}{4};2;-\frac{a}{b x},-\frac{c}{d x}\right )}{-8 b d x F_1\left (1;\frac{3}{4},\frac{1}{4};2;-\frac{a}{b x},-\frac{c}{d x}\right )+b c F_1\left (2;\frac{3}{4},\frac{5}{4};3;-\frac{a}{b x},-\frac{c}{d x}\right )+3 a d F_1\left (2;\frac{7}{4},\frac{1}{4};3;-\frac{a}{b x},-\frac{c}{d x}\right )}}{96 a^3 c^3 x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^4*(a + b*x)^(3/4)*(c + d*x)^(1/4)),x]
[Out]
-((a + b*x)*(c + d*x)*(77*b^2*c^2*x^2 + 2*a*b*c*x*(-22*c + 27*d*x) + a^2*(32*c^2
- 36*c*d*x + 45*d^2*x^2)) + (6*b*d*(77*b^3*c^3 + 21*a*b^2*c^2*d + 15*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*
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))]
))/(96*a^3*c^3*x^3*(a + b*x)^(3/4)*(c + d*x)^(1/4))
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Maple [F] time = 0.058, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}} \left ( bx+a \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(b*x+a)^(3/4)/(d*x+c)^(1/4),x)
[Out]
int(1/x^4/(b*x+a)^(3/4)/(d*x+c)^(1/4),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/4)*(d*x + c)^(1/4)*x^4),x, algorithm="maxima")
[Out]
integrate(1/((b*x + a)^(3/4)*(d*x + c)^(1/4)*x^4), x)
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Fricas [A] time = 0.36611, size = 2272, normalized size = 7.89 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/4)*(d*x + c)^(1/4)*x^4),x, algorithm="fricas")
[Out]
-1/384*(12*a^3*c^3*x^3*((35153041*b^12*c^12 + 38348772*a*b^11*c^11*d + 43080114*
a^2*b^10*c^10*d^2 + 52655988*a^3*b^9*c^9*d^3 + 36722511*a^4*b^8*c^8*d^4 + 270421
20*a^5*b^7*c^7*d^5 + 18926460*a^6*b^6*c^6*d^6 + 8958600*a^7*b^5*c^5*d^7 + 461497
5*a^8*b^4*c^4*d^8 + 2092500*a^9*b^3*c^3*d^9 + 587250*a^10*b^2*c^2*d^10 + 202500*
a^11*b*c*d^11 + 50625*a^12*d^12)/(a^15*c^13))^(1/4)*arctan((a^4*c^3*d*x + a^4*c^
4)*((35153041*b^12*c^12 + 38348772*a*b^11*c^11*d + 43080114*a^2*b^10*c^10*d^2 +
52655988*a^3*b^9*c^9*d^3 + 36722511*a^4*b^8*c^8*d^4 + 27042120*a^5*b^7*c^7*d^5 +
18926460*a^6*b^6*c^6*d^6 + 8958600*a^7*b^5*c^5*d^7 + 4614975*a^8*b^4*c^4*d^8 +
2092500*a^9*b^3*c^3*d^9 + 587250*a^10*b^2*c^2*d^10 + 202500*a^11*b*c*d^11 + 5062
5*a^12*d^12)/(a^15*c^13))^(1/4)/((77*b^3*c^3 + 21*a*b^2*c^2*d + 15*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(((5929*b^6*c^6 + 3
234*a*b^5*c^5*d + 2751*a^2*b^4*c^4*d^2 + 2940*a^3*b^3*c^3*d^3 + 855*a^4*b^2*c^2*
d^4 + 450*a^5*b*c*d^5 + 225*a^6*d^6)*sqrt(b*x + a)*sqrt(d*x + c) + (a^8*c^6*d*x
+ a^8*c^7)*sqrt((35153041*b^12*c^12 + 38348772*a*b^11*c^11*d + 43080114*a^2*b^10
*c^10*d^2 + 52655988*a^3*b^9*c^9*d^3 + 36722511*a^4*b^8*c^8*d^4 + 27042120*a^5*b
^7*c^7*d^5 + 18926460*a^6*b^6*c^6*d^6 + 8958600*a^7*b^5*c^5*d^7 + 4614975*a^8*b^
4*c^4*d^8 + 2092500*a^9*b^3*c^3*d^9 + 587250*a^10*b^2*c^2*d^10 + 202500*a^11*b*c
*d^11 + 50625*a^12*d^12)/(a^15*c^13)))/(d*x + c)))) - 3*a^3*c^3*x^3*((35153041*b
^12*c^12 + 38348772*a*b^11*c^11*d + 43080114*a^2*b^10*c^10*d^2 + 52655988*a^3*b^
9*c^9*d^3 + 36722511*a^4*b^8*c^8*d^4 + 27042120*a^5*b^7*c^7*d^5 + 18926460*a^6*b
^6*c^6*d^6 + 8958600*a^7*b^5*c^5*d^7 + 4614975*a^8*b^4*c^4*d^8 + 2092500*a^9*b^3
*c^3*d^9 + 587250*a^10*b^2*c^2*d^10 + 202500*a^11*b*c*d^11 + 50625*a^12*d^12)/(a
^15*c^13))^(1/4)*log(((77*b^3*c^3 + 21*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 15*a^3*d^3
)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (a^4*c^3*d*x + a^4*c^4)*((35153041*b^12*c^12
+ 38348772*a*b^11*c^11*d + 43080114*a^2*b^10*c^10*d^2 + 52655988*a^3*b^9*c^9*d^
3 + 36722511*a^4*b^8*c^8*d^4 + 27042120*a^5*b^7*c^7*d^5 + 18926460*a^6*b^6*c^6*d
^6 + 8958600*a^7*b^5*c^5*d^7 + 4614975*a^8*b^4*c^4*d^8 + 2092500*a^9*b^3*c^3*d^9
+ 587250*a^10*b^2*c^2*d^10 + 202500*a^11*b*c*d^11 + 50625*a^12*d^12)/(a^15*c^13
))^(1/4))/(d*x + c)) + 3*a^3*c^3*x^3*((35153041*b^12*c^12 + 38348772*a*b^11*c^11
*d + 43080114*a^2*b^10*c^10*d^2 + 52655988*a^3*b^9*c^9*d^3 + 36722511*a^4*b^8*c^
8*d^4 + 27042120*a^5*b^7*c^7*d^5 + 18926460*a^6*b^6*c^6*d^6 + 8958600*a^7*b^5*c^
5*d^7 + 4614975*a^8*b^4*c^4*d^8 + 2092500*a^9*b^3*c^3*d^9 + 587250*a^10*b^2*c^2*
d^10 + 202500*a^11*b*c*d^11 + 50625*a^12*d^12)/(a^15*c^13))^(1/4)*log(((77*b^3*c
^3 + 21*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 15*a^3*d^3)*(b*x + a)^(1/4)*(d*x + c)^(3/
4) - (a^4*c^3*d*x + a^4*c^4)*((35153041*b^12*c^12 + 38348772*a*b^11*c^11*d + 430
80114*a^2*b^10*c^10*d^2 + 52655988*a^3*b^9*c^9*d^3 + 36722511*a^4*b^8*c^8*d^4 +
27042120*a^5*b^7*c^7*d^5 + 18926460*a^6*b^6*c^6*d^6 + 8958600*a^7*b^5*c^5*d^7 +
4614975*a^8*b^4*c^4*d^8 + 2092500*a^9*b^3*c^3*d^9 + 587250*a^10*b^2*c^2*d^10 + 2
02500*a^11*b*c*d^11 + 50625*a^12*d^12)/(a^15*c^13))^(1/4))/(d*x + c)) + 4*(32*a^
2*c^2 + (77*b^2*c^2 + 54*a*b*c*d + 45*a^2*d^2)*x^2 - 4*(11*a*b*c^2 + 9*a^2*c*d)*
x)*(b*x + a)^(1/4)*(d*x + c)^(3/4))/(a^3*c^3*x^3)
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \left (a + b x\right )^{\frac{3}{4}} \sqrt [4]{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**4/(b*x+a)**(3/4)/(d*x+c)**(1/4),x)
[Out]
Integral(1/(x**4*(a + b*x)**(3/4)*(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(1/((b*x + a)^(3/4)*(d*x + c)^(1/4)*x^4),x, algorithm="giac")
[Out]
Timed out