3.1756 \(\int \frac{(a+b x)^{3/2}}{(c+d x)^{5/6}} \, dx\)

Optimal. Leaf size=405 \[ \frac{27\ 3^{3/4} \sqrt [6]{c+d x} (b c-a d)^{5/3} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{40 d^3 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{27 \sqrt{a+b x} \sqrt [6]{c+d x} (b c-a d)}{20 d^2}+\frac{3 (a+b x)^{3/2} \sqrt [6]{c+d x}}{5 d} \]

[Out]

(-27*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(1/6))/(20*d^2) + (3*(a + b*x)^(3/2)*(c
+ d*x)^(1/6))/(5*d) + (27*3^(3/4)*(b*c - a*d)^(5/3)*(c + d*x)^(1/6)*((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))/((b*c - a*d)^(1/3) - (1 + Sqrt
[3])*b^(1/3)*(c + d*x)^(1/3))^2]*EllipticF[ArcCos[((b*c - a*d)^(1/3) - (1 - Sqrt
[3])*b^(1/3)*(c + d*x)^(1/3))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*
x)^(1/3))], (2 + Sqrt[3])/4])/(40*d^3*Sqrt[a + b*x]*Sqrt[-((b^(1/3)*(c + d*x)^(1
/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((b*c - a*d)^(1/3) - (1 + Sqr
t[3])*b^(1/3)*(c + d*x)^(1/3))^2)])

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Rubi [A]  time = 0.648686, antiderivative size = 405, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{27\ 3^{3/4} \sqrt [6]{c+d x} (b c-a d)^{5/3} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{40 d^3 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{27 \sqrt{a+b x} \sqrt [6]{c+d x} (b c-a d)}{20 d^2}+\frac{3 (a+b x)^{3/2} \sqrt [6]{c+d x}}{5 d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(3/2)/(c + d*x)^(5/6),x]

[Out]

(-27*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(1/6))/(20*d^2) + (3*(a + b*x)^(3/2)*(c
+ d*x)^(1/6))/(5*d) + (27*3^(3/4)*(b*c - a*d)^(5/3)*(c + d*x)^(1/6)*((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))/((b*c - a*d)^(1/3) - (1 + Sqrt
[3])*b^(1/3)*(c + d*x)^(1/3))^2]*EllipticF[ArcCos[((b*c - a*d)^(1/3) - (1 - Sqrt
[3])*b^(1/3)*(c + d*x)^(1/3))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*
x)^(1/3))], (2 + Sqrt[3])/4])/(40*d^3*Sqrt[a + b*x]*Sqrt[-((b^(1/3)*(c + d*x)^(1
/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((b*c - a*d)^(1/3) - (1 + Sqr
t[3])*b^(1/3)*(c + d*x)^(1/3))^2)])

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Rubi in Sympy [A]  time = 28.7852, size = 354, normalized size = 0.87 \[ \frac{3 \left (a + b x\right )^{\frac{3}{2}} \sqrt [6]{c + d x}}{5 d} + \frac{27 \sqrt{a + b x} \sqrt [6]{c + d x} \left (a d - b c\right )}{20 d^{2}} + \frac{27 \cdot 3^{\frac{3}{4}} \sqrt{\frac{b^{\frac{2}{3}} \left (c + d x\right )^{\frac{2}{3}} - \sqrt [3]{b} \sqrt [3]{c + d x} \sqrt [3]{a d - b c} + \left (a d - b c\right )^{\frac{2}{3}}}{\left (\sqrt [3]{b} \left (1 + \sqrt{3}\right ) \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )^{2}}} \sqrt [6]{c + d x} \left (a d - b c\right )^{\frac{5}{3}} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right ) F\left (\operatorname{acos}{\left (\frac{\sqrt [3]{b} \left (- \sqrt{3} + 1\right ) \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}}{\sqrt [3]{b} \left (1 + \sqrt{3}\right ) \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{40 d^{3} \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c + d x} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )}{\left (\sqrt [3]{b} \left (1 + \sqrt{3}\right ) \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )^{2}}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)/(d*x+c)**(5/6),x)

[Out]

3*(a + b*x)**(3/2)*(c + d*x)**(1/6)/(5*d) + 27*sqrt(a + b*x)*(c + d*x)**(1/6)*(a
*d - b*c)/(20*d**2) + 27*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)*(1 + sqrt(3))*(
c + d*x)**(1/3) + (a*d - b*c)**(1/3))**2)*(c + d*x)**(1/6)*(a*d - b*c)**(5/3)*(b
**(1/3)*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))*elliptic_f(acos((b**(1/3)*(-sqrt(
3) + 1)*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))/(b**(1/3)*(1 + sqrt(3))*(c + d*x)
**(1/3) + (a*d - b*c)**(1/3))), sqrt(3)/4 + 1/2)/(40*d**3*sqrt(b**(1/3)*(c + d*x
)**(1/3)*(b**(1/3)*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))/(b**(1/3)*(1 + sqrt(3)
)*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))**2)*sqrt(a - b*c/d + b*(c + d*x)/d))

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Mathematica [C]  time = 0.175934, size = 107, normalized size = 0.26 \[ \frac{3 \sqrt [6]{c+d x} \left (27 (b c-a d)^2 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\frac{b (c+d x)}{b c-a d}\right )+d (a+b x) (13 a d-9 b c+4 b d x)\right )}{20 d^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(3/2)/(c + d*x)^(5/6),x]

[Out]

(3*(c + d*x)^(1/6)*(d*(a + b*x)*(-9*b*c + 13*a*d + 4*b*d*x) + 27*(b*c - a*d)^2*S
qrt[(d*(a + b*x))/(-(b*c) + a*d)]*Hypergeometric2F1[1/6, 1/2, 7/6, (b*(c + d*x))
/(b*c - a*d)]))/(20*d^3*Sqrt[a + b*x])

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( dx+c \right ) ^{-{\frac{5}{6}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)/(d*x+c)^(5/6),x)

[Out]

int((b*x+a)^(3/2)/(d*x+c)^(5/6),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{3}{2}}}{{\left (d x + c\right )}^{\frac{5}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)/(d*x + c)^(5/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(3/2)/(d*x + c)^(5/6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{\frac{3}{2}}}{{\left (d x + c\right )}^{\frac{5}{6}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)/(d*x + c)^(5/6),x, algorithm="fricas")

[Out]

integral((b*x + a)^(3/2)/(d*x + c)^(5/6), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{\frac{3}{2}}}{\left (c + d x\right )^{\frac{5}{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(3/2)/(d*x+c)**(5/6),x)

[Out]

Integral((a + b*x)**(3/2)/(c + d*x)**(5/6), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{3}{2}}}{{\left (d x + c\right )}^{\frac{5}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)/(d*x + c)^(5/6),x, algorithm="giac")

[Out]

integrate((b*x + a)^(3/2)/(d*x + c)^(5/6), x)