3.1656 \(\int \frac{\sqrt{a+b x}}{(c+d x)^{3/4}} \, dx\)

Optimal. Leaf size=111 \[ \frac{4 \sqrt{a+b x} \sqrt [4]{c+d x}}{3 d}-\frac{8 (b c-a d)^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 \sqrt [4]{b} d^2 \sqrt{a+b x}} \]

[Out]

(4*Sqrt[a + b*x]*(c + d*x)^(1/4))/(3*d) - (8*(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^(1/4)*d^2*Sqrt[a + b*x])

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Rubi [A]  time = 0.167426, 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 \[ \frac{4 \sqrt{a+b x} \sqrt [4]{c+d x}}{3 d}-\frac{8 (b c-a d)^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 \sqrt [4]{b} d^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[a + b*x]*(c + d*x)^(1/4))/(3*d) - (8*(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^(1/4)*d^2*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 21.1155, size = 168, normalized size = 1.51 \[ \frac{4 \sqrt{a + b x} \sqrt [4]{c + d x}}{3 d} + \frac{4 \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{5}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{3 \sqrt [4]{b} d^{2} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/2)/(d*x+c)**(3/4),x)

[Out]

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

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Mathematica [C]  time = 0.161622, size = 77, normalized size = 0.69 \[ \frac{4 \sqrt{a+b x} \sqrt [4]{c+d x} \left (\frac{2 \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )}{\sqrt{\frac{d (a+b x)}{a d-b c}}}+1\right )}{3 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{1\sqrt{bx+a} \left ( dx+c \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/2)/(d*x+c)^(3/4),x)

[Out]

int((b*x+a)^(1/2)/(d*x+c)^(3/4),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{3}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(b*x + a)/(d*x + c)^(3/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(1/2)/(d*x+c)**(3/4),x)

[Out]

Integral(sqrt(a + b*x)/(c + d*x)**(3/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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