3.1651 \(\int \frac{\sqrt{a+b x}}{\sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=196 \[ \frac{8 (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 b^{3/4} d^2 \sqrt{a+b x}}-\frac{8 (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 b^{3/4} d^2 \sqrt{a+b x}}+\frac{4 \sqrt{a+b x} (c+d x)^{3/4}}{5 d} \]

[Out]

(4*Sqrt[a + b*x]*(c + d*x)^(3/4))/(5*d) - (8*(b*c - a*d)^(7/4)*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])/(5*b^(3/4)*d^2*Sqrt[a + b*x]) + (8*(b*c - a*d)^(7/4)*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]
)/(5*b^(3/4)*d^2*Sqrt[a + b*x])

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Rubi [A]  time = 0.653413, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ \frac{8 (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 b^{3/4} d^2 \sqrt{a+b x}}-\frac{8 (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 b^{3/4} d^2 \sqrt{a+b x}}+\frac{4 \sqrt{a+b x} (c+d x)^{3/4}}{5 d} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[a + b*x]*(c + d*x)^(3/4))/(5*d) - (8*(b*c - a*d)^(7/4)*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])/(5*b^(3/4)*d^2*Sqrt[a + b*x]) + (8*(b*c - a*d)^(7/4)*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]
)/(5*b^(3/4)*d^2*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 66.5413, size = 389, normalized size = 1.98 \[ \frac{4 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{4}}}{5 d} + \frac{8 \sqrt [4]{c + d x} \sqrt{a d - b c} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}}{5 \sqrt{b} d \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )} - \frac{8 \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{7}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{3}{4}} d^{2} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{4 \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{7}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{3}{4}} d^{2} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*sqrt(a + b*x)*(c + d*x)**(3/4)/(5*d) + 8*(c + d*x)**(1/4)*sqrt(a*d - b*c)*sqrt
(a - b*c/d + b*(c + d*x)/d)/(5*sqrt(b)*d*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c)
+ 1)) - 8*sqrt((a*d - b*c + b*(c + d*x))/((a*d - b*c)*(sqrt(b)*sqrt(c + d*x)/sqr
t(a*d - b*c) + 1)**2))*(a*d - b*c)**(7/4)*(sqrt(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)/(5*b
**(3/4)*d**2*sqrt(a - b*c/d + b*(c + d*x)/d)) + 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)**(7/4
)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)*elliptic_f(2*atan(b**(1/4)*(c + d*
x)**(1/4)/(a*d - b*c)**(1/4)), 1/2)/(5*b**(3/4)*d**2*sqrt(a - b*c/d + b*(c + d*x
)/d))

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Mathematica [C]  time = 0.173131, size = 77, normalized size = 0.39 \[ \frac{4 \sqrt{a+b x} (c+d x)^{3/4} \left (\frac{2 \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )}{\sqrt{\frac{d (a+b x)}{a d-b c}}}+3\right )}{15 d} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[a + b*x]*(c + d*x)^(3/4)*(3 + (2*Hypergeometric2F1[1/2, 3/4, 7/4, (b*(c
+ d*x))/(b*c - a*d)])/Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]))/(15*d)

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Maple [F]  time = 0.032, size = 0, normalized size = 0. \[ \int{1\sqrt{bx+a}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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