3.892 \(\int \frac{1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=85 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{a^{3/4} \sqrt [4]{c}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{a^{3/4} \sqrt [4]{c}} \]

[Out]

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

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Rubi [A]  time = 0.109416, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{a^{3/4} \sqrt [4]{c}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{a^{3/4} \sqrt [4]{c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.9046, size = 82, normalized size = 0.96 \[ - \frac{2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt [4]{a + b x}}{\sqrt [4]{a} \sqrt [4]{c + d x}} \right )}}{a^{\frac{3}{4}} \sqrt [4]{c}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt [4]{a + b x}}{\sqrt [4]{a} \sqrt [4]{c + d x}} \right )}}{a^{\frac{3}{4}} \sqrt [4]{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.26263, size = 146, normalized size = 1.72 \[ \frac{8 b d x F_1\left (1;\frac{3}{4},\frac{1}{4};2;-\frac{a}{b x},-\frac{c}{d x}\right )}{(a+b x)^{3/4} \sqrt [4]{c+d x} \left (-8 b d x F_1\left (1;\frac{3}{4},\frac{1}{4};2;-\frac{a}{b x},-\frac{c}{d x}\right )+b c F_1\left (2;\frac{3}{4},\frac{5}{4};3;-\frac{a}{b x},-\frac{c}{d x}\right )+3 a d F_1\left (2;\frac{7}{4},\frac{1}{4};3;-\frac{a}{b x},-\frac{c}{d x}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(8*b*d*x*AppellF1[1, 3/4, 1/4, 2, -(a/(b*x)), -(c/(d*x))])/((a + b*x)^(3/4)*(c +
d*x)^(1/4)*(-8*b*d*x*AppellF1[1, 3/4, 1/4, 2, -(a/(b*x)), -(c/(d*x))] + b*c*App
ellF1[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))]))

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{1}{x} \left ( bx+a \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.244653, size = 286, normalized size = 3.36 \[ 4 \, \left (\frac{1}{a^{3} c}\right )^{\frac{1}{4}} \arctan \left (\frac{{\left (a d x + a c\right )} \left (\frac{1}{a^{3} c}\right )^{\frac{1}{4}}}{{\left (d x + c\right )} \sqrt{\frac{{\left (a^{2} d x + a^{2} c\right )} \sqrt{\frac{1}{a^{3} c}} + \sqrt{b x + a} \sqrt{d x + c}}{d x + c}} +{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}\right ) - \left (\frac{1}{a^{3} c}\right )^{\frac{1}{4}} \log \left (\frac{{\left (a d x + a c\right )} \left (\frac{1}{a^{3} c}\right )^{\frac{1}{4}} +{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}{d x + c}\right ) + \left (\frac{1}{a^{3} c}\right )^{\frac{1}{4}} \log \left (-\frac{{\left (a d x + a c\right )} \left (\frac{1}{a^{3} c}\right )^{\frac{1}{4}} -{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}{d x + c}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*(1/(a^3*c))^(1/4)*arctan((a*d*x + a*c)*(1/(a^3*c))^(1/4)/((d*x + c)*sqrt(((a^2
*d*x + a^2*c)*sqrt(1/(a^3*c)) + sqrt(b*x + a)*sqrt(d*x + c))/(d*x + c)) + (b*x +
a)^(1/4)*(d*x + c)^(3/4))) - (1/(a^3*c))^(1/4)*log(((a*d*x + a*c)*(1/(a^3*c))^(
1/4) + (b*x + a)^(1/4)*(d*x + c)^(3/4))/(d*x + c)) + (1/(a^3*c))^(1/4)*log(-((a*
d*x + a*c)*(1/(a^3*c))^(1/4) - (b*x + a)^(1/4)*(d*x + c)^(3/4))/(d*x + c))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*(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),x, algorithm="giac")

[Out]

Timed out