3.1631 \(\int \sqrt{a+b x} \sqrt [4]{c+d x} \, dx\)

Optimal. Leaf size=147 \[ -\frac{8 (b c-a d)^{9/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{21 b^{5/4} d^2 \sqrt{a+b x}}+\frac{4 \sqrt{a+b x} \sqrt [4]{c+d x} (b c-a d)}{21 b d}+\frac{4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 b} \]

[Out]

(4*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(1/4))/(21*b*d) + (4*(a + b*x)^(3/2)*(c +
d*x)^(1/4))/(7*b) - (8*(b*c - a*d)^(9/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Ell
ipticF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(21*b^(5/4)*d^2
*Sqrt[a + b*x])

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Rubi [A]  time = 0.234486, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{8 (b c-a d)^{9/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{21 b^{5/4} d^2 \sqrt{a+b x}}+\frac{4 \sqrt{a+b x} \sqrt [4]{c+d x} (b c-a d)}{21 b d}+\frac{4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 b} \]

Antiderivative was successfully verified.

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

[Out]

(4*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(1/4))/(21*b*d) + (4*(a + b*x)^(3/2)*(c +
d*x)^(1/4))/(7*b) - (8*(b*c - a*d)^(9/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Ell
ipticF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(21*b^(5/4)*d^2
*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 30.2235, size = 199, normalized size = 1.35 \[ \frac{4 \sqrt{a + b x} \left (c + d x\right )^{\frac{5}{4}}}{7 d} + \frac{8 \sqrt{a + b x} \sqrt [4]{c + d x} \left (a d - b c\right )}{21 b d} - \frac{4 \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{9}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{21 b^{\frac{5}{4}} d^{2} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

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)**(5/4)/(7*d) + 8*sqrt(a + b*x)*(c + d*x)**(1/4)*(a*d -
b*c)/(21*b*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)**(9/4)*(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)/(21*b**(5/4)*d**2*sqrt(a - b*c/d + b*(c + d*x)/d))

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Mathematica [C]  time = 0.19848, size = 109, normalized size = 0.74 \[ \frac{4 \sqrt [4]{c+d x} \left (d (a+b x) (2 a d+b (c+3 d x))-2 (b c-a d)^2 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )\right )}{21 b d^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(4*(c + d*x)^(1/4)*(d*(a + b*x)*(2*a*d + b*(c + 3*d*x)) - 2*(b*c - a*d)^2*Sqrt[(
d*(a + b*x))/(-(b*c) + a*d)]*Hypergeometric2F1[1/4, 1/2, 5/4, (b*(c + d*x))/(b*c
- a*d)]))/(21*b*d^2*Sqrt[a + b*x])

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Maple [F]  time = 0.035, size = 0, normalized size = 0. \[ \int \sqrt{bx+a}\sqrt [4]{dx+c}\, dx \]

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. \[ \int \sqrt{b x + a}{\left (d x + c\right )}^{\frac{1}{4}}\,{d x} \]

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. \[{\rm integral}\left (\sqrt{b x + a}{\left (d x + c\right )}^{\frac{1}{4}}, x\right ) \]

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. \[ \int \sqrt{a + b x} \sqrt [4]{c + d x}\, dx \]

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. \[ \int \sqrt{b x + a}{\left (d x + c\right )}^{\frac{1}{4}}\,{d x} \]

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)