3.1745 \(\int \frac{(c+d x)^{5/6}}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=817 \[ -\frac{15 \sqrt [4]{3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) (b c-a d)^{4/3}}{8 b^{5/3} d \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{5\ 3^{3/4} \left (1-\sqrt{3}\right ) \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) (b c-a d)^{4/3}}{16 b^{5/3} d \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{15 \left (1+\sqrt{3}\right ) \sqrt{a+b x} \sqrt [6]{c+d x} (b c-a d)}{8 b^{5/3} \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac{3 \sqrt{a+b x} (c+d x)^{5/6}}{4 b} \]

[Out]

(3*Sqrt[a + b*x]*(c + d*x)^(5/6))/(4*b) - (15*(1 + Sqrt[3])*(b*c - a*d)*Sqrt[a +
b*x]*(c + d*x)^(1/6))/(8*b^(5/3)*((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c
+ d*x)^(1/3))) - (15*3^(1/4)*(b*c - a*d)^(4/3)*(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])/(8*b^(5/3)*d*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)]) - (5*3^(3/4)*(1 - Sqrt[3])*(b*c - a*d)^(4/3)*
(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])/(16*b^(5/3)*d*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 = 1.44344, antiderivative size = 817, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{15 \sqrt [4]{3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) (b c-a d)^{4/3}}{8 b^{5/3} d \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{5\ 3^{3/4} \left (1-\sqrt{3}\right ) \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) (b c-a d)^{4/3}}{16 b^{5/3} d \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{15 \left (1+\sqrt{3}\right ) \sqrt{a+b x} \sqrt [6]{c+d x} (b c-a d)}{8 b^{5/3} \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac{3 \sqrt{a+b x} (c+d x)^{5/6}}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

(3*Sqrt[a + b*x]*(c + d*x)^(5/6))/(4*b) - (15*(1 + Sqrt[3])*(b*c - a*d)*Sqrt[a +
b*x]*(c + d*x)^(1/6))/(8*b^(5/3)*((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c
+ d*x)^(1/3))) - (15*3^(1/4)*(b*c - a*d)^(4/3)*(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])/(8*b^(5/3)*d*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)]) - (5*3^(3/4)*(1 - Sqrt[3])*(b*c - a*d)^(4/3)*
(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])/(16*b^(5/3)*d*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 = 66.6389, size = 721, normalized size = 0.88 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(a + b*x)*(c + d*x)**(5/6)/(4*b) - (15/8 + 15*sqrt(3)/8)*(c + d*x)**(1/6)*
(a*d - b*c)*sqrt(a - b*c/d + b*(c + d*x)/d)/(b**(5/3)*(b**(1/3)*(1 + sqrt(3))*(c
+ d*x)**(1/3) + (a*d - b*c)**(1/3))) + 15*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)**(4/3)*(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)/(8*b**(5/3)*d
*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)) + 5*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)**(4/3)*(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 + sqr
t(3))*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))), sqrt(3)/4 + 1/2)/(16*b**(5/3)*d*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))

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Mathematica [C]  time = 0.160231, size = 93, normalized size = 0.11 \[ \frac{3 (c+d x)^{5/6} \left ((b c-a d) \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\frac{b (c+d x)}{b c-a d}\right )+d (a+b x)\right )}{4 b d \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(c + d*x)^(5/6)*(d*(a + b*x) + (b*c - a*d)*Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]
*Hypergeometric2F1[1/2, 5/6, 11/6, (b*(c + d*x))/(b*c - a*d)]))/(4*b*d*Sqrt[a +
b*x])

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{5}{6}}}{\frac{1}{\sqrt{bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out