3.877 \(\int \frac{\sqrt [4]{a+b x}}{x^3 \sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=194 \[ \frac{(b c-a d) (5 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{7/4} c^{9/4}}+\frac{(b c-a d) (5 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{7/4} c^{9/4}}+\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (5 a d+3 b c)}{8 a c^2 x}-\frac{(a+b x)^{5/4} (c+d x)^{3/4}}{2 a c x^2} \]

[Out]

((3*b*c + 5*a*d)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(8*a*c^2*x) - ((a + b*x)^(5/4)
*(c + d*x)^(3/4))/(2*a*c*x^2) + ((b*c - a*d)*(3*b*c + 5*a*d)*ArcTan[(c^(1/4)*(a
+ b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(16*a^(7/4)*c^(9/4)) + ((b*c - a*d)*(3
*b*c + 5*a*d)*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(16*
a^(7/4)*c^(9/4))

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Rubi [A]  time = 0.309189, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{(b c-a d) (5 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{7/4} c^{9/4}}+\frac{(b c-a d) (5 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{7/4} c^{9/4}}+\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (5 a d+3 b c)}{8 a c^2 x}-\frac{(a+b x)^{5/4} (c+d x)^{3/4}}{2 a c x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(1/4)/(x^3*(c + d*x)^(1/4)),x]

[Out]

((3*b*c + 5*a*d)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(8*a*c^2*x) - ((a + b*x)^(5/4)
*(c + d*x)^(3/4))/(2*a*c*x^2) + ((b*c - a*d)*(3*b*c + 5*a*d)*ArcTan[(c^(1/4)*(a
+ b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(16*a^(7/4)*c^(9/4)) + ((b*c - a*d)*(3
*b*c + 5*a*d)*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(16*
a^(7/4)*c^(9/4))

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Rubi in Sympy [A]  time = 27.8204, size = 175, normalized size = 0.9 \[ - \frac{\left (a + b x\right )^{\frac{5}{4}} \left (c + d x\right )^{\frac{3}{4}}}{2 a c x^{2}} + \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (5 a d + 3 b c\right )}{8 a c^{2} x} - \frac{\left (a d - b c\right ) \left (5 a d + 3 b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt [4]{a + b x}}{\sqrt [4]{a} \sqrt [4]{c + d x}} \right )}}{16 a^{\frac{7}{4}} c^{\frac{9}{4}}} - \frac{\left (a d - b c\right ) \left (5 a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt [4]{a + b x}}{\sqrt [4]{a} \sqrt [4]{c + d x}} \right )}}{16 a^{\frac{7}{4}} c^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/4)/x**3/(d*x+c)**(1/4),x)

[Out]

-(a + b*x)**(5/4)*(c + d*x)**(3/4)/(2*a*c*x**2) + (a + b*x)**(1/4)*(c + d*x)**(3
/4)*(5*a*d + 3*b*c)/(8*a*c**2*x) - (a*d - b*c)*(5*a*d + 3*b*c)*atan(c**(1/4)*(a
+ b*x)**(1/4)/(a**(1/4)*(c + d*x)**(1/4)))/(16*a**(7/4)*c**(9/4)) - (a*d - b*c)*
(5*a*d + 3*b*c)*atanh(c**(1/4)*(a + b*x)**(1/4)/(a**(1/4)*(c + d*x)**(1/4)))/(16
*a**(7/4)*c**(9/4))

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Mathematica [C]  time = 0.39033, size = 211, normalized size = 1.09 \[ \frac{\frac{2 b d x^3 \left (5 a^2 d^2-2 a b c d-3 b^2 c^2\right ) F_1\left (1;\frac{3}{4},\frac{1}{4};2;-\frac{a}{b x},-\frac{c}{d x}\right )}{-8 b d x F_1\left (1;\frac{3}{4},\frac{1}{4};2;-\frac{a}{b x},-\frac{c}{d x}\right )+b c F_1\left (2;\frac{3}{4},\frac{5}{4};3;-\frac{a}{b x},-\frac{c}{d x}\right )+3 a d F_1\left (2;\frac{7}{4},\frac{1}{4};3;-\frac{a}{b x},-\frac{c}{d x}\right )}+(a+b x) (c+d x) (-4 a c+5 a d x-b c x)}{8 a c^2 x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)^(1/4)/(x^3*(c + d*x)^(1/4)),x]

[Out]

((a + b*x)*(c + d*x)*(-4*a*c - b*c*x + 5*a*d*x) + (2*b*d*(-3*b^2*c^2 - 2*a*b*c*d
+ 5*a^2*d^2)*x^3*AppellF1[1, 3/4, 1/4, 2, -(a/(b*x)), -(c/(d*x))])/(-8*b*d*x*Ap
pellF1[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))]))
/(8*a*c^2*x^2*(a + b*x)^(3/4)*(c + d*x)^(1/4))

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}}\sqrt [4]{bx+a}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/4)/x^3/(d*x+c)^(1/4),x)

[Out]

int((b*x+a)^(1/4)/x^3/(d*x+c)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{1}{4}}}{{\left (d x + c\right )}^{\frac{1}{4}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x^3),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x^3), x)

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Fricas [A]  time = 0.278792, size = 1592, normalized size = 8.21 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x^3),x, algorithm="fricas")

[Out]

1/32*(4*a*c^2*x^2*((81*b^8*c^8 + 216*a*b^7*c^7*d - 324*a^2*b^6*c^6*d^2 - 984*a^3
*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 + 1640*a^5*b^3*c^3*d^5 - 900*a^6*b^2*c^2*d^6
- 1000*a^7*b*c*d^7 + 625*a^8*d^8)/(a^7*c^9))^(1/4)*arctan(-(a^2*c^2*d*x + a^2*c^
3)*((81*b^8*c^8 + 216*a*b^7*c^7*d - 324*a^2*b^6*c^6*d^2 - 984*a^3*b^5*c^5*d^3 +
646*a^4*b^4*c^4*d^4 + 1640*a^5*b^3*c^3*d^5 - 900*a^6*b^2*c^2*d^6 - 1000*a^7*b*c*
d^7 + 625*a^8*d^8)/(a^7*c^9))^(1/4)/((3*b^2*c^2 + 2*a*b*c*d - 5*a^2*d^2)*(b*x +
a)^(1/4)*(d*x + c)^(3/4) - (d*x + c)*sqrt(((9*b^4*c^4 + 12*a*b^3*c^3*d - 26*a^2*
b^2*c^2*d^2 - 20*a^3*b*c*d^3 + 25*a^4*d^4)*sqrt(b*x + a)*sqrt(d*x + c) + (a^4*c^
4*d*x + a^4*c^5)*sqrt((81*b^8*c^8 + 216*a*b^7*c^7*d - 324*a^2*b^6*c^6*d^2 - 984*
a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 + 1640*a^5*b^3*c^3*d^5 - 900*a^6*b^2*c^2*d
^6 - 1000*a^7*b*c*d^7 + 625*a^8*d^8)/(a^7*c^9)))/(d*x + c)))) + a*c^2*x^2*((81*b
^8*c^8 + 216*a*b^7*c^7*d - 324*a^2*b^6*c^6*d^2 - 984*a^3*b^5*c^5*d^3 + 646*a^4*b
^4*c^4*d^4 + 1640*a^5*b^3*c^3*d^5 - 900*a^6*b^2*c^2*d^6 - 1000*a^7*b*c*d^7 + 625
*a^8*d^8)/(a^7*c^9))^(1/4)*log(-((3*b^2*c^2 + 2*a*b*c*d - 5*a^2*d^2)*(b*x + a)^(
1/4)*(d*x + c)^(3/4) + (a^2*c^2*d*x + a^2*c^3)*((81*b^8*c^8 + 216*a*b^7*c^7*d -
324*a^2*b^6*c^6*d^2 - 984*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 + 1640*a^5*b^3*c
^3*d^5 - 900*a^6*b^2*c^2*d^6 - 1000*a^7*b*c*d^7 + 625*a^8*d^8)/(a^7*c^9))^(1/4))
/(d*x + c)) - a*c^2*x^2*((81*b^8*c^8 + 216*a*b^7*c^7*d - 324*a^2*b^6*c^6*d^2 - 9
84*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 + 1640*a^5*b^3*c^3*d^5 - 900*a^6*b^2*c^
2*d^6 - 1000*a^7*b*c*d^7 + 625*a^8*d^8)/(a^7*c^9))^(1/4)*log(-((3*b^2*c^2 + 2*a*
b*c*d - 5*a^2*d^2)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (a^2*c^2*d*x + a^2*c^3)*((8
1*b^8*c^8 + 216*a*b^7*c^7*d - 324*a^2*b^6*c^6*d^2 - 984*a^3*b^5*c^5*d^3 + 646*a^
4*b^4*c^4*d^4 + 1640*a^5*b^3*c^3*d^5 - 900*a^6*b^2*c^2*d^6 - 1000*a^7*b*c*d^7 +
625*a^8*d^8)/(a^7*c^9))^(1/4))/(d*x + c)) - 4*(4*a*c + (b*c - 5*a*d)*x)*(b*x + a
)^(1/4)*(d*x + c)^(3/4))/(a*c^2*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [4]{a + b x}}{x^{3} \sqrt [4]{c + d x}}\, dx \]

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)

[Out]

Integral((a + b*x)**(1/4)/(x**3*(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^3),x, algorithm="giac")

[Out]

Timed out