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

Optimal. Leaf size=188 \[ \frac{(b c-a d) (3 a d+5 b c) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{7/4} d^{9/4}}+\frac{(b c-a d) (3 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{7/4} d^{9/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (3 a d+5 b c)}{8 b d^2}+\frac{(a+b x)^{5/4} (c+d x)^{3/4}}{2 b d} \]

[Out]

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

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Rubi [A]  time = 0.246696, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{(b c-a d) (3 a d+5 b c) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{7/4} d^{9/4}}+\frac{(b c-a d) (3 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{7/4} d^{9/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (3 a d+5 b c)}{8 b d^2}+\frac{(a+b x)^{5/4} (c+d x)^{3/4}}{2 b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.3726, size = 170, normalized size = 0.9 \[ \frac{\left (a + b x\right )^{\frac{5}{4}} \left (c + d x\right )^{\frac{3}{4}}}{2 b d} - \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (3 a d + 5 b c\right )}{8 b d^{2}} + \frac{\left (a d - b c\right ) \left (3 a d + 5 b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{d} \sqrt [4]{a + b x}} \right )}}{16 b^{\frac{7}{4}} d^{\frac{9}{4}}} - \frac{\left (a d - b c\right ) \left (3 a d + 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{d} \sqrt [4]{a + b x}} \right )}}{16 b^{\frac{7}{4}} d^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.328832, size = 122, normalized size = 0.65 \[ \frac{(c+d x)^{3/4} \left (\left (-3 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \left (\frac{d (a+b x)}{a d-b c}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )+3 d (a+b x) (a d-5 b c+4 b d x)\right )}{24 b d^3 (a+b x)^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

((c + d*x)^(3/4)*(3*d*(a + b*x)*(-5*b*c + a*d + 4*b*d*x) + (5*b^2*c^2 - 2*a*b*c*
d - 3*a^2*d^2)*((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)]))/(24*b*d^3*(a + b*x)^(3/4))

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{x\sqrt [4]{bx+a}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.27836, size = 1573, normalized size = 8.37 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32*(4*b*d^2*((625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*
b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 +
216*a^7*b*c*d^7 + 81*a^8*d^8)/(b^7*d^9))^(1/4)*arctan(-(b^2*d^3*x + b^2*c*d^2)*(
(625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 6
46*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7
+ 81*a^8*d^8)/(b^7*d^9))^(1/4)/((5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*(b*x + a)^(
1/4)*(d*x + c)^(3/4) - (d*x + c)*sqrt(((25*b^4*c^4 - 20*a*b^3*c^3*d - 26*a^2*b^2
*c^2*d^2 + 12*a^3*b*c*d^3 + 9*a^4*d^4)*sqrt(b*x + a)*sqrt(d*x + c) + (b^4*d^5*x
+ b^4*c*d^4)*sqrt((625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a
^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6
+ 216*a^7*b*c*d^7 + 81*a^8*d^8)/(b^7*d^9)))/(d*x + c)))) + b*d^2*((625*b^8*c^8
- 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^
4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8
)/(b^7*d^9))^(1/4)*log(-((5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*(b*x + a)^(1/4)*(d*
x + c)^(3/4) + (b^2*d^3*x + b^2*c*d^2)*((625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^
2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5
- 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(b^7*d^9))^(1/4))/(d*x +
c)) - b*d^2*((625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^
5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 21
6*a^7*b*c*d^7 + 81*a^8*d^8)/(b^7*d^9))^(1/4)*log(-((5*b^2*c^2 - 2*a*b*c*d - 3*a^
2*d^2)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (b^2*d^3*x + b^2*c*d^2)*((625*b^8*c^8 -
1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4
*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)
/(b^7*d^9))^(1/4))/(d*x + c)) + 4*(4*b*d*x - 5*b*c + a*d)*(b*x + a)^(1/4)*(d*x +
c)^(3/4))/(b*d^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out