3.1755 \(\int \frac{(a+b x)^{5/2}}{(c+d x)^{5/6}} \, dx\)

Optimal. Leaf size=440 \[ -\frac{81\ 3^{3/4} \sqrt [6]{c+d x} (b c-a d)^{8/3} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{128 d^4 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{81 \sqrt{a+b x} \sqrt [6]{c+d x} (b c-a d)^2}{64 d^3}-\frac{9 (a+b x)^{3/2} \sqrt [6]{c+d x} (b c-a d)}{16 d^2}+\frac{3 (a+b x)^{5/2} \sqrt [6]{c+d x}}{8 d} \]

[Out]

(81*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(1/6))/(64*d^3) - (9*(b*c - a*d)*(a +
b*x)^(3/2)*(c + d*x)^(1/6))/(16*d^2) + (3*(a + b*x)^(5/2)*(c + d*x)^(1/6))/(8*d)
- (81*3^(3/4)*(b*c - a*d)^(8/3)*(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 + Sqr
t[3])/4])/(128*d^4*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 + Sqrt[3])*b^(1/3)*(c +
d*x)^(1/3))^2)])

_______________________________________________________________________________________

Rubi [A]  time = 0.773243, antiderivative size = 440, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{81\ 3^{3/4} \sqrt [6]{c+d x} (b c-a d)^{8/3} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{128 d^4 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{81 \sqrt{a+b x} \sqrt [6]{c+d x} (b c-a d)^2}{64 d^3}-\frac{9 (a+b x)^{3/2} \sqrt [6]{c+d x} (b c-a d)}{16 d^2}+\frac{3 (a+b x)^{5/2} \sqrt [6]{c+d x}}{8 d} \]

Antiderivative was successfully verified.

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

[Out]

(81*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(1/6))/(64*d^3) - (9*(b*c - a*d)*(a +
b*x)^(3/2)*(c + d*x)^(1/6))/(16*d^2) + (3*(a + b*x)^(5/2)*(c + d*x)^(1/6))/(8*d)
- (81*3^(3/4)*(b*c - a*d)^(8/3)*(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 + Sqr
t[3])/4])/(128*d^4*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 + Sqrt[3])*b^(1/3)*(c +
d*x)^(1/3))^2)])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 38.5319, size = 386, normalized size = 0.88 \[ \frac{3 \left (a + b x\right )^{\frac{5}{2}} \sqrt [6]{c + d x}}{8 d} + \frac{9 \left (a + b x\right )^{\frac{3}{2}} \sqrt [6]{c + d x} \left (a d - b c\right )}{16 d^{2}} + \frac{81 \sqrt{a + b x} \sqrt [6]{c + d x} \left (a d - b c\right )^{2}}{64 d^{3}} + \frac{81 \cdot 3^{\frac{3}{4}} \sqrt{\frac{b^{\frac{2}{3}} \left (c + d x\right )^{\frac{2}{3}} - \sqrt [3]{b} \sqrt [3]{c + d x} \sqrt [3]{a d - b c} + \left (a d - b c\right )^{\frac{2}{3}}}{\left (\sqrt [3]{b} \left (1 + \sqrt{3}\right ) \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )^{2}}} \sqrt [6]{c + d x} \left (a d - b c\right )^{\frac{8}{3}} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right ) F\left (\operatorname{acos}{\left (\frac{\sqrt [3]{b} \left (- \sqrt{3} + 1\right ) \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}}{\sqrt [3]{b} \left (1 + \sqrt{3}\right ) \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{128 d^{4} \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c + d x} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )}{\left (\sqrt [3]{b} \left (1 + \sqrt{3}\right ) \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )^{2}}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(a + b*x)**(5/2)*(c + d*x)**(1/6)/(8*d) + 9*(a + b*x)**(3/2)*(c + d*x)**(1/6)*
(a*d - b*c)/(16*d**2) + 81*sqrt(a + b*x)*(c + d*x)**(1/6)*(a*d - b*c)**2/(64*d**
3) + 81*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)**(8/3)*(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)/(128*d**4*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))

_______________________________________________________________________________________

Mathematica [C]  time = 0.281593, size = 138, normalized size = 0.31 \[ \frac{3 \sqrt [6]{c+d x} \left (d (a+b x) \left (47 a^2 d^2+2 a b d (14 d x-33 c)+b^2 \left (27 c^2-12 c d x+8 d^2 x^2\right )\right )-81 (b c-a d)^3 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\frac{b (c+d x)}{b c-a d}\right )\right )}{64 d^4 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(c + d*x)^(1/6)*(d*(a + b*x)*(47*a^2*d^2 + 2*a*b*d*(-33*c + 14*d*x) + b^2*(27
*c^2 - 12*c*d*x + 8*d^2*x^2)) - 81*(b*c - a*d)^3*Sqrt[(d*(a + b*x))/(-(b*c) + a*
d)]*Hypergeometric2F1[1/6, 1/2, 7/6, (b*(c + d*x))/(b*c - a*d)]))/(64*d^4*Sqrt[a
+ b*x])

_______________________________________________________________________________________

Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{5}{2}}} \left ( dx+c \right ) ^{-{\frac{5}{6}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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