3.1738 \(\int (a+b x)^{3/2} \sqrt [6]{c+d x} \, dx\)

Optimal. Leaf size=449 \[ \frac{27\ 3^{3/4} \sqrt [6]{c+d x} (b c-a d)^{8/3} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{640 b d^3 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{27 \sqrt{a+b x} \sqrt [6]{c+d x} (b c-a d)^2}{320 b d^2}+\frac{3 (a+b x)^{3/2} \sqrt [6]{c+d x} (b c-a d)}{80 b d}+\frac{3 (a+b x)^{5/2} \sqrt [6]{c+d x}}{8 b} \]

[Out]

(-27*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(1/6))/(320*b*d^2) + (3*(b*c - a*d)*(
a + b*x)^(3/2)*(c + d*x)^(1/6))/(80*b*d) + (3*(a + b*x)^(5/2)*(c + d*x)^(1/6))/(
8*b) + (27*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 +
Sqrt[3])/4])/(640*b*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 = 0.763009, antiderivative size = 449, 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 \[ \frac{27\ 3^{3/4} \sqrt [6]{c+d x} (b c-a d)^{8/3} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{640 b d^3 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{27 \sqrt{a+b x} \sqrt [6]{c+d x} (b c-a d)^2}{320 b d^2}+\frac{3 (a+b x)^{3/2} \sqrt [6]{c+d x} (b c-a d)}{80 b d}+\frac{3 (a+b x)^{5/2} \sqrt [6]{c+d x}}{8 b} \]

Antiderivative was successfully verified.

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

[Out]

(-27*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(1/6))/(320*b*d^2) + (3*(b*c - a*d)*(
a + b*x)^(3/2)*(c + d*x)^(1/6))/(80*b*d) + (3*(a + b*x)^(5/2)*(c + d*x)^(1/6))/(
8*b) + (27*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 +
Sqrt[3])/4])/(640*b*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 = 40.9115, size = 389, normalized size = 0.87 \[ \frac{3 \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{7}{6}}}{8 d} + \frac{27 \sqrt{a + b x} \left (c + d x\right )^{\frac{7}{6}} \left (a d - b c\right )}{80 d^{2}} + \frac{81 \sqrt{a + b x} \sqrt [6]{c + d x} \left (a d - b c\right )^{2}}{320 b d^{2}} - \frac{27 \cdot 3^{\frac{3}{4}} \sqrt{\frac{b^{\frac{2}{3}} \left (c + d x\right )^{\frac{2}{3}} - \sqrt [3]{b} \sqrt [3]{c + d x} \sqrt [3]{a d - b c} + \left (a d - b c\right )^{\frac{2}{3}}}{\left (\sqrt [3]{b} \left (1 + \sqrt{3}\right ) \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )^{2}}} \sqrt [6]{c + d x} \left (a d - b c\right )^{\frac{8}{3}} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right ) F\left (\operatorname{acos}{\left (\frac{\sqrt [3]{b} \left (- \sqrt{3} + 1\right ) \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}}{\sqrt [3]{b} \left (1 + \sqrt{3}\right ) \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{640 b d^{3} \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c + d x} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )}{\left (\sqrt [3]{b} \left (1 + \sqrt{3}\right ) \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )^{2}}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(a + b*x)**(3/2)*(c + d*x)**(7/6)/(8*d) + 27*sqrt(a + b*x)*(c + d*x)**(7/6)*(a
*d - b*c)/(80*d**2) + 81*sqrt(a + b*x)*(c + d*x)**(1/6)*(a*d - b*c)**2/(320*b*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)**(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)/(640*b*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 = 0.267472, size = 142, normalized size = 0.32 \[ -\frac{3 \sqrt [6]{c+d x} \left (-d (a+b x) \left (27 a^2 d^2+2 a b d (11 c+38 d x)+b^2 \left (-9 c^2+4 c d x+40 d^2 x^2\right )\right )-27 (b c-a d)^3 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\frac{b (c+d x)}{b c-a d}\right )\right )}{320 b d^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(c + d*x)^(1/6)*(-(d*(a + b*x)*(27*a^2*d^2 + 2*a*b*d*(11*c + 38*d*x) + b^2*(
-9*c^2 + 4*c*d*x + 40*d^2*x^2))) - 27*(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)]))/(320*b*d^3*
Sqrt[a + b*x])

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

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. \[ \int{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{1}{6}}\,{d x} \]

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. \[{\rm integral}\left ({\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{1}{6}}, x\right ) \]

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

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. \[ \int{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{1}{6}}\,{d x} \]

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)