3.1652 \(\int \frac{1}{\sqrt{a+b x} \sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=167 \[ \frac{4 (b c-a d)^{3/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} d \sqrt{a+b x}}-\frac{4 (b c-a d)^{3/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} d \sqrt{a+b x}} \]

[Out]

(4*(b*c - a*d)^(3/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticE[ArcSin[(b^(1/4
)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(b^(3/4)*d*Sqrt[a + b*x]) - (4*(b*c
- a*d)^(3/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticF[ArcSin[(b^(1/4)*(c + d
*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(b^(3/4)*d*Sqrt[a + b*x])

_______________________________________________________________________________________

Rubi [A]  time = 0.595253, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ \frac{4 (b c-a d)^{3/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} d \sqrt{a+b x}}-\frac{4 (b c-a d)^{3/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} d \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(4*(b*c - a*d)^(3/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticE[ArcSin[(b^(1/4
)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(b^(3/4)*d*Sqrt[a + b*x]) - (4*(b*c
- a*d)^(3/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticF[ArcSin[(b^(1/4)*(c + d
*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(b^(3/4)*d*Sqrt[a + b*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 50.5507, size = 357, normalized size = 2.14 \[ \frac{4 \sqrt [4]{c + d x} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}}{\sqrt{b} \sqrt{a d - b c} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )} - \frac{4 \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{3}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{b^{\frac{3}{4}} d \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{2 \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{3}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{b^{\frac{3}{4}} d \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(1/2)/(d*x+c)**(1/4),x)

[Out]

4*(c + d*x)**(1/4)*sqrt(a - b*c/d + b*(c + d*x)/d)/(sqrt(b)*sqrt(a*d - b*c)*(sqr
t(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)) - 4*sqrt((a*d - b*c + b*(c + d*x))/((a*
d - b*c)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)**2))*(a*d - b*c)**(3/4)*(sq
rt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)*elliptic_e(2*atan(b**(1/4)*(c + d*x)**(
1/4)/(a*d - b*c)**(1/4)), 1/2)/(b**(3/4)*d*sqrt(a - b*c/d + b*(c + d*x)/d)) + 2*
sqrt((a*d - b*c + b*(c + d*x))/((a*d - b*c)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*
c) + 1)**2))*(a*d - b*c)**(3/4)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)*elli
ptic_f(2*atan(b**(1/4)*(c + d*x)**(1/4)/(a*d - b*c)**(1/4)), 1/2)/(b**(3/4)*d*sq
rt(a - b*c/d + b*(c + d*x)/d))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0594388, size = 73, normalized size = 0.44 \[ \frac{4 (c+d x)^{3/4} \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )}{3 d \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]*(c + d*x)^(3/4)*Hypergeometric2F1[1/2, 3/4
, 7/4, (b*(c + d*x))/(b*c - a*d)])/(3*d*Sqrt[a + b*x])

_______________________________________________________________________________________

Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(1/2)/(d*x+c)^(1/4),x)

[Out]

int(1/(b*x+a)^(1/2)/(d*x+c)^(1/4),x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(1/2)/(d*x+c)**(1/4),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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