3.1632 \(\int \frac{\sqrt [4]{c+d x}}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=111 \[ \frac{4 (b c-a d)^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 b^{5/4} d \sqrt{a+b x}}+\frac{4 \sqrt{a+b x} \sqrt [4]{c+d x}}{3 b} \]

[Out]

(4*Sqrt[a + b*x]*(c + d*x)^(1/4))/(3*b) + (4*(b*c - a*d)^(5/4)*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])/(3*b^(5/4)*d*Sqrt[a + b*x])

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Rubi [A]  time = 0.164162, antiderivative size = 111, 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 \[ \frac{4 (b c-a d)^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 b^{5/4} d \sqrt{a+b x}}+\frac{4 \sqrt{a+b x} \sqrt [4]{c+d x}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[a + b*x]*(c + d*x)^(1/4))/(3*b) + (4*(b*c - a*d)^(5/4)*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])/(3*b^(5/4)*d*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 21.0175, size = 167, normalized size = 1.5 \[ \frac{4 \sqrt{a + b x} \sqrt [4]{c + d x}}{3 b} - \frac{2 \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{5}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{3 b^{\frac{5}{4}} d \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*sqrt(a + b*x)*(c + d*x)**(1/4)/(3*b) - 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)**(5/4)*(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)/(3*b**(5/4)*d*sqrt(a - b*c/d + b*(c + d*x)/d))

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Mathematica [C]  time = 0.155237, size = 93, normalized size = 0.84 \[ \frac{4 \sqrt [4]{c+d x} \left ((b c-a d) \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )+d (a+b x)\right )}{3 b d \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(4*(c + d*x)^(1/4)*(d*(a + b*x) + (b*c - a*d)*Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]
*Hypergeometric2F1[1/4, 1/2, 5/4, (b*(c + d*x))/(b*c - a*d)]))/(3*b*d*Sqrt[a + b
*x])

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{1\sqrt [4]{dx+c}{\frac{1}{\sqrt{bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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