3.1572 \(\int \frac{\sqrt{a+b x}}{(c+d x)^{2/3}} \, dx\)

Optimal. Leaf size=381 \[ \frac{6\ 3^{3/4} \sqrt{2-\sqrt{3}} (b c-a d) \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right )}{5 \sqrt [3]{b} d^2 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{6 \sqrt{a+b x} \sqrt [3]{c+d x}}{5 d} \]

[Out]

(6*Sqrt[a + b*x]*(c + d*x)^(1/3))/(5*d) + (6*3^(3/4)*Sqrt[2 - Sqrt[3]]*(b*c - a*
d)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))*Sqrt[((b*c - a*d)^(2/3) + b^(1/
3)*(b*c - a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3))/((1 - Sqrt[3])*(
b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*(
b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b
^(1/3)*(c + d*x)^(1/3))], -7 + 4*Sqrt[3]])/(5*b^(1/3)*d^2*Sqrt[a + b*x]*Sqrt[-((
(b*c - a*d)^(1/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((1 - Sqrt[3])*
(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))^2)])

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Rubi [A]  time = 0.622433, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{6\ 3^{3/4} \sqrt{2-\sqrt{3}} (b c-a d) \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right )}{5 \sqrt [3]{b} d^2 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{6 \sqrt{a+b x} \sqrt [3]{c+d x}}{5 d} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x]/(c + d*x)^(2/3),x]

[Out]

(6*Sqrt[a + b*x]*(c + d*x)^(1/3))/(5*d) + (6*3^(3/4)*Sqrt[2 - Sqrt[3]]*(b*c - a*
d)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))*Sqrt[((b*c - a*d)^(2/3) + b^(1/
3)*(b*c - a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3))/((1 - Sqrt[3])*(
b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*(
b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b
^(1/3)*(c + d*x)^(1/3))], -7 + 4*Sqrt[3]])/(5*b^(1/3)*d^2*Sqrt[a + b*x]*Sqrt[-((
(b*c - a*d)^(1/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((1 - Sqrt[3])*
(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))^2)])

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Rubi in Sympy [A]  time = 26.9449, size = 323, normalized size = 0.85 \[ \frac{6 \sqrt{a + b x} \sqrt [3]{c + d x}}{5 d} + \frac{6 \cdot 3^{\frac{3}{4}} \sqrt{\frac{b^{\frac{2}{3}} \left (c + d x\right )^{\frac{2}{3}} - \sqrt [3]{b} \sqrt [3]{c + d x} \sqrt [3]{a d - b c} + \left (a d - b c\right )^{\frac{2}{3}}}{\left (\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (a d - b c\right ) \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x} - \left (-1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}}{\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{5 \sqrt [3]{b} d^{2} \sqrt{\frac{\sqrt [3]{a d - b c} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )}{\left (\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}\right )^{2}}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*sqrt(a + b*x)*(c + d*x)**(1/3)/(5*d) + 6*3**(3/4)*sqrt((b**(2/3)*(c + d*x)**(2
/3) - b**(1/3)*(c + d*x)**(1/3)*(a*d - b*c)**(1/3) + (a*d - b*c)**(2/3))/(b**(1/
3)*(c + d*x)**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(1/3))**2)*sqrt(sqrt(3) + 2)*(a
*d - b*c)*(b**(1/3)*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))*elliptic_f(asin((b**(
1/3)*(c + d*x)**(1/3) - (-1 + sqrt(3))*(a*d - b*c)**(1/3))/(b**(1/3)*(c + d*x)**
(1/3) + (1 + sqrt(3))*(a*d - b*c)**(1/3))), -7 - 4*sqrt(3))/(5*b**(1/3)*d**2*sqr
t((a*d - b*c)**(1/3)*(b**(1/3)*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))/(b**(1/3)*
(c + d*x)**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(1/3))**2)*sqrt(a - b*c/d + b*(c +
d*x)/d))

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Mathematica [C]  time = 0.162799, size = 77, normalized size = 0.2 \[ \frac{3 \sqrt{a+b x} \sqrt [3]{c+d x} \left (\frac{3 \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\frac{b (c+d x)}{b c-a d}\right )}{\sqrt{\frac{d (a+b x)}{a d-b c}}}+2\right )}{5 d} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x]/(c + d*x)^(2/3),x]

[Out]

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

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{1\sqrt{bx+a} \left ( dx+c \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/(d*x + c)^(2/3),x, algorithm="maxima")

[Out]

integrate(sqrt(b*x + a)/(d*x + c)^(2/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{2}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/(d*x + c)^(2/3),x, algorithm="fricas")

[Out]

integral(sqrt(b*x + a)/(d*x + c)^(2/3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x}}{\left (c + d x\right )^{\frac{2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a + b*x)/(c + d*x)**(2/3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/(d*x + c)^(2/3),x, algorithm="giac")

[Out]

integrate(sqrt(b*x + a)/(d*x + c)^(2/3), x)