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

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

[Out]

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

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Rubi [A]  time = 0.0808533, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{3/4} \sqrt [4]{d}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{3/4} \sqrt [4]{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.7416, size = 80, normalized size = 0.94 \[ - \frac{2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{d} \sqrt [4]{a + b x}} \right )}}{b^{\frac{3}{4}} \sqrt [4]{d}} + \frac{2 \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{d} \sqrt [4]{a + b x}} \right )}}{b^{\frac{3}{4}} \sqrt [4]{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0574104, size = 73, normalized size = 0.86 \[ \frac{4 (c+d x)^{3/4} \left (\frac{d (a+b x)}{a d-b c}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )}{3 d (a+b x)^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(4*((d*(a + b*x))/(-(b*c) + a*d))^(3/4)*(c + d*x)^(3/4)*Hypergeometric2F1[3/4, 3
/4, 7/4, (b*(c + d*x))/(b*c - a*d)])/(3*d*(a + b*x)^(3/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.243785, size = 286, normalized size = 3.36 \[ -4 \, \left (\frac{1}{b^{3} d}\right )^{\frac{1}{4}} \arctan \left (\frac{{\left (b d x + b c\right )} \left (\frac{1}{b^{3} d}\right )^{\frac{1}{4}}}{{\left (d x + c\right )} \sqrt{\frac{{\left (b^{2} d x + b^{2} c\right )} \sqrt{\frac{1}{b^{3} d}} + \sqrt{b x + a} \sqrt{d x + c}}{d x + c}} +{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}\right ) + \left (\frac{1}{b^{3} d}\right )^{\frac{1}{4}} \log \left (\frac{{\left (b d x + b c\right )} \left (\frac{1}{b^{3} d}\right )^{\frac{1}{4}} +{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}{d x + c}\right ) - \left (\frac{1}{b^{3} d}\right )^{\frac{1}{4}} \log \left (-\frac{{\left (b d x + b c\right )} \left (\frac{1}{b^{3} d}\right )^{\frac{1}{4}} -{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}{d x + c}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*(1/(b^3*d))^(1/4)*arctan((b*d*x + b*c)*(1/(b^3*d))^(1/4)/((d*x + c)*sqrt(((b^
2*d*x + b^2*c)*sqrt(1/(b^3*d)) + sqrt(b*x + a)*sqrt(d*x + c))/(d*x + c)) + (b*x
+ a)^(1/4)*(d*x + c)^(3/4))) + (1/(b^3*d))^(1/4)*log(((b*d*x + b*c)*(1/(b^3*d))^
(1/4) + (b*x + a)^(1/4)*(d*x + c)^(3/4))/(d*x + c)) - (1/(b^3*d))^(1/4)*log(-((b
*d*x + b*c)*(1/(b^3*d))^(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. \[ \int \frac{1}{\left (a + b x\right )^{\frac{3}{4}} \sqrt [4]{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((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, algorithm="giac")

[Out]

Timed out