3.1750 \(\int \frac{(a+b x)^{3/2}}{\sqrt [6]{c+d x}} \, dx\)

Optimal. Leaf size=855 \[ -\frac{81 \sqrt [4]{3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) (b c-a d)^{7/3}}{112 b^{2/3} d^3 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{27\ 3^{3/4} \left (1-\sqrt{3}\right ) \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) (b c-a d)^{7/3}}{224 b^{2/3} d^3 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{81 \left (1+\sqrt{3}\right ) \sqrt{a+b x} \sqrt [6]{c+d x} (b c-a d)^2}{112 b^{2/3} d^2 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}-\frac{27 \sqrt{a+b x} (c+d x)^{5/6} (b c-a d)}{56 d^2}+\frac{3 (a+b x)^{3/2} (c+d x)^{5/6}}{7 d} \]

[Out]

(-27*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(5/6))/(56*d^2) + (3*(a + b*x)^(3/2)*(c
+ d*x)^(5/6))/(7*d) - (81*(1 + Sqrt[3])*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(
1/6))/(112*b^(2/3)*d^2*((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3
))) - (81*3^(1/4)*(b*c - a*d)^(7/3)*(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]*EllipticE[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])/(112*b^(2/3)*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 + Sqrt[3])*b^
(1/3)*(c + d*x)^(1/3))^2)]) - (27*3^(3/4)*(1 - Sqrt[3])*(b*c - a*d)^(7/3)*(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 + Sqr
t[3])*b^(1/3)*(c + d*x)^(1/3))], (2 + Sqrt[3])/4])/(224*b^(2/3)*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 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2)])

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Rubi [A]  time = 1.75054, antiderivative size = 855, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{81 \sqrt [4]{3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) (b c-a d)^{7/3}}{112 b^{2/3} d^3 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{27\ 3^{3/4} \left (1-\sqrt{3}\right ) \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) (b c-a d)^{7/3}}{224 b^{2/3} d^3 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{81 \left (1+\sqrt{3}\right ) \sqrt{a+b x} \sqrt [6]{c+d x} (b c-a d)^2}{112 b^{2/3} d^2 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}-\frac{27 \sqrt{a+b x} (c+d x)^{5/6} (b c-a d)}{56 d^2}+\frac{3 (a+b x)^{3/2} (c+d x)^{5/6}}{7 d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(3/2)/(c + d*x)^(1/6),x]

[Out]

(-27*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(5/6))/(56*d^2) + (3*(a + b*x)^(3/2)*(c
+ d*x)^(5/6))/(7*d) - (81*(1 + Sqrt[3])*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(
1/6))/(112*b^(2/3)*d^2*((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3
))) - (81*3^(1/4)*(b*c - a*d)^(7/3)*(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]*EllipticE[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])/(112*b^(2/3)*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 + Sqrt[3])*b^
(1/3)*(c + d*x)^(1/3))^2)]) - (27*3^(3/4)*(1 - Sqrt[3])*(b*c - a*d)^(7/3)*(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 + Sqr
t[3])*b^(1/3)*(c + d*x)^(1/3))], (2 + Sqrt[3])/4])/(224*b^(2/3)*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 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2)])

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Rubi in Sympy [A]  time = 84.8632, size = 760, normalized size = 0.89 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)/(d*x+c)**(1/6),x)

[Out]

3*(a + b*x)**(3/2)*(c + d*x)**(5/6)/(7*d) + 27*sqrt(a + b*x)*(c + d*x)**(5/6)*(a
*d - b*c)/(56*d**2) + (81/112 + 81*sqrt(3)/112)*(c + d*x)**(1/6)*(a*d - b*c)**2*
sqrt(a - b*c/d + b*(c + d*x)/d)/(b**(2/3)*d**2*(b**(1/3)*(1 + sqrt(3))*(c + d*x)
**(1/3) + (a*d - b*c)**(1/3))) - 81*3**(1/4)*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
)**(7/3)*(b**(1/3)*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))*elliptic_e(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)/(112*b**(2/3)*d**3*s
qrt(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)) - 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)*(-sqrt(3) + 1)*(c + d*x)**(1/6)*(a*d -
b*c)**(7/3)*(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)/(224*b**(2/3)*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 = 0.214092, size = 108, normalized size = 0.13 \[ \frac{3 (c+d x)^{5/6} \left (27 (b c-a d)^2 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\frac{b (c+d x)}{b c-a d}\right )+5 d (a+b x) (17 a d-9 b c+8 b d x)\right )}{280 d^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(3/2)/(c + d*x)^(1/6),x]

[Out]

(3*(c + d*x)^(5/6)*(5*d*(a + b*x)*(-9*b*c + 17*a*d + 8*b*d*x) + 27*(b*c - a*d)^2
*Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]*Hypergeometric2F1[1/2, 5/6, 11/6, (b*(c + d*
x))/(b*c - a*d)]))/(280*d^3*Sqrt[a + b*x])

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt [6]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)/(d*x+c)^(1/6),x)

[Out]

int((b*x+a)^(3/2)/(d*x+c)^(1/6),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{3}{2}}}{{\left (d x + c\right )}^{\frac{1}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)/(d*x + c)^(1/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(3/2)/(d*x + c)^(1/6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{\frac{3}{2}}}{{\left (d x + c\right )}^{\frac{1}{6}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)/(d*x + c)^(1/6),x, algorithm="fricas")

[Out]

integral((b*x + a)^(3/2)/(d*x + c)^(1/6), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(3/2)/(d*x+c)**(1/6),x)

[Out]

Integral((a + b*x)**(3/2)/(c + d*x)**(1/6), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{3}{2}}}{{\left (d x + c\right )}^{\frac{1}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)/(d*x + c)^(1/6),x, algorithm="giac")

[Out]

integrate((b*x + a)^(3/2)/(d*x + c)^(1/6), x)