3.1749 \(\int \frac{(a+b x)^{5/2}}{\sqrt [6]{c+d x}} \, dx\)

Optimal. Leaf size=890 \[ \frac{243 \sqrt [4]{3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) (b c-a d)^{10/3}}{448 b^{2/3} d^4 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{81\ 3^{3/4} \left (1-\sqrt{3}\right ) \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) (b c-a d)^{10/3}}{896 b^{2/3} d^4 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{243 \left (1+\sqrt{3}\right ) \sqrt{a+b x} \sqrt [6]{c+d x} (b c-a d)^3}{448 b^{2/3} d^3 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac{81 \sqrt{a+b x} (c+d x)^{5/6} (b c-a d)^2}{224 d^3}-\frac{9 (a+b x)^{3/2} (c+d x)^{5/6} (b c-a d)}{28 d^2}+\frac{3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d} \]

[Out]

(81*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(5/6))/(224*d^3) - (9*(b*c - a*d)*(a +
b*x)^(3/2)*(c + d*x)^(5/6))/(28*d^2) + (3*(a + b*x)^(5/2)*(c + d*x)^(5/6))/(10*
d) + (243*(1 + Sqrt[3])*(b*c - a*d)^3*Sqrt[a + b*x]*(c + d*x)^(1/6))/(448*b^(2/3
)*d^3*((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))) + (243*3^(1/4
)*(b*c - a*d)^(10/3)*(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])/(4
48*b^(2/3)*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)]) + (81*3^(3/4)*(1 - Sqrt[3])*(b*c - a*d)^(10/3)*(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])/(896*b^(2/3)*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)])

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Rubi [A]  time = 2.06235, antiderivative size = 890, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{243 \sqrt [4]{3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) (b c-a d)^{10/3}}{448 b^{2/3} d^4 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{81\ 3^{3/4} \left (1-\sqrt{3}\right ) \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) (b c-a d)^{10/3}}{896 b^{2/3} d^4 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{243 \left (1+\sqrt{3}\right ) \sqrt{a+b x} \sqrt [6]{c+d x} (b c-a d)^3}{448 b^{2/3} d^3 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac{81 \sqrt{a+b x} (c+d x)^{5/6} (b c-a d)^2}{224 d^3}-\frac{9 (a+b x)^{3/2} (c+d x)^{5/6} (b c-a d)}{28 d^2}+\frac{3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(5/2)/(c + d*x)^(1/6),x]

[Out]

(81*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(5/6))/(224*d^3) - (9*(b*c - a*d)*(a +
b*x)^(3/2)*(c + d*x)^(5/6))/(28*d^2) + (3*(a + b*x)^(5/2)*(c + d*x)^(5/6))/(10*
d) + (243*(1 + Sqrt[3])*(b*c - a*d)^3*Sqrt[a + b*x]*(c + d*x)^(1/6))/(448*b^(2/3
)*d^3*((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))) + (243*3^(1/4
)*(b*c - a*d)^(10/3)*(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])/(4
48*b^(2/3)*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)]) + (81*3^(3/4)*(1 - Sqrt[3])*(b*c - a*d)^(10/3)*(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])/(896*b^(2/3)*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)])

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Rubi in Sympy [A]  time = 105.235, size = 792, 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)**(5/2)/(d*x+c)**(1/6),x)

[Out]

3*(a + b*x)**(5/2)*(c + d*x)**(5/6)/(10*d) + 9*(a + b*x)**(3/2)*(c + d*x)**(5/6)
*(a*d - b*c)/(28*d**2) + 81*sqrt(a + b*x)*(c + d*x)**(5/6)*(a*d - b*c)**2/(224*d
**3) + (243/448 + 243*sqrt(3)/448)*(c + d*x)**(1/6)*(a*d - b*c)**3*sqrt(a - b*c/
d + b*(c + d*x)/d)/(b**(2/3)*d**3*(b**(1/3)*(1 + sqrt(3))*(c + d*x)**(1/3) + (a*
d - b*c)**(1/3))) - 243*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)**(10/3)*(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)/(448*b**(2/3)*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)) - 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)*(-sqrt(3) + 1)*(c + d*x)**(1/6)*(a*d - b*c)**(10/3
)*(b**(1/3)*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))*elliptic_f(acos((b**(1/3)*(-s
qrt(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)/(896*b**(2/3)*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))

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Mathematica [C]  time = 0.297344, size = 138, normalized size = 0.16 \[ \frac{3 (c+d x)^{5/6} \left (d (a+b x) \left (367 a^2 d^2+2 a b d (172 d x-195 c)+b^2 \left (135 c^2-120 c d x+112 d^2 x^2\right )\right )-81 (b c-a d)^3 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\frac{b (c+d x)}{b c-a d}\right )\right )}{1120 d^4 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(5/2)/(c + d*x)^(1/6),x]

[Out]

(3*(c + d*x)^(5/6)*(d*(a + b*x)*(367*a^2*d^2 + 2*a*b*d*(-195*c + 172*d*x) + b^2*
(135*c^2 - 120*c*d*x + 112*d^2*x^2)) - 81*(b*c - a*d)^3*Sqrt[(d*(a + b*x))/(-(b*
c) + a*d)]*Hypergeometric2F1[1/2, 5/6, 11/6, (b*(c + d*x))/(b*c - a*d)]))/(1120*
d^4*Sqrt[a + b*x])

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Maple [F]  time = 0.045, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt [6]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)/(d*x+c)^(1/6),x)

[Out]

int((b*x+a)^(5/2)/(d*x+c)^(1/6),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{5}{2}}}{{\left (d x + c\right )}^{\frac{1}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)/(d*x + c)^(1/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(5/2)/(d*x + c)^(1/6), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)/(d*x + c)^(1/6),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(5/2)/(d*x+c)**(1/6),x)

[Out]

Integral((a + b*x)**(5/2)/(c + d*x)**(1/6), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{5}{2}}}{{\left (d x + c\right )}^{\frac{1}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)/(d*x + c)^(1/6),x, algorithm="giac")

[Out]

integrate((b*x + a)^(5/2)/(d*x + c)^(1/6), x)