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

Optimal. Leaf size=130 \[ -\frac{(3 a d+b c) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}}-\frac{(3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}}+\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{b d} \]

[Out]

((a + b*x)^(1/4)*(c + d*x)^(3/4))/(b*d) - ((b*c + 3*a*d)*ArcTan[(d^(1/4)*(a + b*
x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(2*b^(7/4)*d^(5/4)) - ((b*c + 3*a*d)*ArcTa
nh[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(2*b^(7/4)*d^(5/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.166023, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{(3 a d+b c) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}}-\frac{(3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}}+\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{b d} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1/4)*(c + d*x)^(3/4))/(b*d) - ((b*c + 3*a*d)*ArcTan[(d^(1/4)*(a + b*
x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(2*b^(7/4)*d^(5/4)) - ((b*c + 3*a*d)*ArcTa
nh[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(2*b^(7/4)*d^(5/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.4627, size = 117, normalized size = 0.9 \[ \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}}}{b d} - \frac{\left (3 a d + b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt [4]{a + b x}}{\sqrt [4]{b} \sqrt [4]{c + d x}} \right )}}{2 b^{\frac{7}{4}} d^{\frac{5}{4}}} - \frac{\left (3 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{d} \sqrt [4]{a + b x}}{\sqrt [4]{b} \sqrt [4]{c + d x}} \right )}}{2 b^{\frac{7}{4}} d^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.18563, size = 95, normalized size = 0.73 \[ \frac{(c+d x)^{3/4} \left (3 d (a+b x)-(3 a d+b c) \left (\frac{d (a+b x)}{a d-b c}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )\right )}{3 b d^2 (a+b x)^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{x \left ( bx+a \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.258274, size = 895, normalized size = 6.88 \[ \frac{4 \, b d \left (\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{{\left (b^{2} d^{2} x + b^{2} c d\right )} \left (\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac{1}{4}}}{{\left (b c + 3 \, a d\right )}{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}} +{\left (d x + c\right )} \sqrt{\frac{{\left (b^{2} c^{2} + 6 \, a b c d + 9 \, a^{2} d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (b^{4} d^{3} x + b^{4} c d^{2}\right )} \sqrt{\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}}}{d x + c}}}\right ) - b d \left (\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac{1}{4}} \log \left (\frac{{\left (b c + 3 \, a d\right )}{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}} +{\left (b^{2} d^{2} x + b^{2} c d\right )} \left (\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac{1}{4}}}{d x + c}\right ) + b d \left (\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac{1}{4}} \log \left (\frac{{\left (b c + 3 \, a d\right )}{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}} -{\left (b^{2} d^{2} x + b^{2} c d\right )} \left (\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac{1}{4}}}{d x + c}\right ) + 4 \,{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}{4 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(4*b*d*((b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 8
1*a^4*d^4)/(b^7*d^5))^(1/4)*arctan((b^2*d^2*x + b^2*c*d)*((b^4*c^4 + 12*a*b^3*c^
3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^7*d^5))^(1/4)/((b*c
+ 3*a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (d*x + c)*sqrt(((b^2*c^2 + 6*a*b*c*d
+ 9*a^2*d^2)*sqrt(b*x + a)*sqrt(d*x + c) + (b^4*d^3*x + b^4*c*d^2)*sqrt((b^4*c^4
+ 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^7*d^5)
))/(d*x + c)))) - b*d*((b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*
b*c*d^3 + 81*a^4*d^4)/(b^7*d^5))^(1/4)*log(((b*c + 3*a*d)*(b*x + a)^(1/4)*(d*x +
c)^(3/4) + (b^2*d^2*x + b^2*c*d)*((b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^
2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^7*d^5))^(1/4))/(d*x + c)) + b*d*((b^4*c^4 +
12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^7*d^5))^
(1/4)*log(((b*c + 3*a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (b^2*d^2*x + b^2*c*d)
*((b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)
/(b^7*d^5))^(1/4))/(d*x + c)) + 4*(b*x + a)^(1/4)*(d*x + c)^(3/4))/(b*d)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Timed out