3.1560 \(\int (a+b x)^{3/2} \sqrt [3]{c+d x} \, dx\)

Optimal. Leaf size=457 \[ -\frac{108\ 3^{3/4} \sqrt{2-\sqrt{3}} (b c-a d)^3 \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right )}{935 b^{4/3} d^3 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{108 \sqrt{a+b x} \sqrt [3]{c+d x} (b c-a d)^2}{935 b d^2}+\frac{12 (a+b x)^{3/2} \sqrt [3]{c+d x} (b c-a d)}{187 b d}+\frac{6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b} \]

[Out]

(-108*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(1/3))/(935*b*d^2) + (12*(b*c - a*d)
*(a + b*x)^(3/2)*(c + d*x)^(1/3))/(187*b*d) + (6*(a + b*x)^(5/2)*(c + d*x)^(1/3)
)/(17*b) - (108*3^(3/4)*Sqrt[2 - Sqrt[3]]*(b*c - a*d)^3*((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))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*
(c + d*x)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*
(c + d*x)^(1/3))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))], -
7 + 4*Sqrt[3]])/(935*b^(4/3)*d^3*Sqrt[a + b*x]*Sqrt[-(((b*c - a*d)^(1/3)*((b*c -
a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/
3)*(c + d*x)^(1/3))^2)])

_______________________________________________________________________________________

Rubi [A]  time = 1.1164, antiderivative size = 457, 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 \[ -\frac{108\ 3^{3/4} \sqrt{2-\sqrt{3}} (b c-a d)^3 \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right )}{935 b^{4/3} d^3 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{108 \sqrt{a+b x} \sqrt [3]{c+d x} (b c-a d)^2}{935 b d^2}+\frac{12 (a+b x)^{3/2} \sqrt [3]{c+d x} (b c-a d)}{187 b d}+\frac{6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b} \]

Antiderivative was successfully verified.

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

[Out]

(-108*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(1/3))/(935*b*d^2) + (12*(b*c - a*d)
*(a + b*x)^(3/2)*(c + d*x)^(1/3))/(187*b*d) + (6*(a + b*x)^(5/2)*(c + d*x)^(1/3)
)/(17*b) - (108*3^(3/4)*Sqrt[2 - Sqrt[3]]*(b*c - a*d)^3*((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))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*
(c + d*x)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*
(c + d*x)^(1/3))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))], -
7 + 4*Sqrt[3]])/(935*b^(4/3)*d^3*Sqrt[a + b*x]*Sqrt[-(((b*c - a*d)^(1/3)*((b*c -
a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/
3)*(c + d*x)^(1/3))^2)])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 53.5847, size = 389, normalized size = 0.85 \[ \frac{6 \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{4}{3}}}{17 d} + \frac{54 \sqrt{a + b x} \left (c + d x\right )^{\frac{4}{3}} \left (a d - b c\right )}{187 d^{2}} + \frac{162 \sqrt{a + b x} \sqrt [3]{c + d x} \left (a d - b c\right )^{2}}{935 b d^{2}} - \frac{108 \cdot 3^{\frac{3}{4}} \sqrt{\frac{b^{\frac{2}{3}} \left (c + d x\right )^{\frac{2}{3}} - \sqrt [3]{b} \sqrt [3]{c + d x} \sqrt [3]{a d - b c} + \left (a d - b c\right )^{\frac{2}{3}}}{\left (\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (a d - b c\right )^{3} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x} - \left (-1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}}{\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{935 b^{\frac{4}{3}} d^{3} \sqrt{\frac{\sqrt [3]{a d - b c} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )}{\left (\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}\right )^{2}}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*(a + b*x)**(3/2)*(c + d*x)**(4/3)/(17*d) + 54*sqrt(a + b*x)*(c + d*x)**(4/3)*(
a*d - b*c)/(187*d**2) + 162*sqrt(a + b*x)*(c + d*x)**(1/3)*(a*d - b*c)**2/(935*b
*d**2) - 108*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)*(c + d*x)**(1/3) + (1 + sqr
t(3))*(a*d - b*c)**(1/3))**2)*sqrt(sqrt(3) + 2)*(a*d - b*c)**3*(b**(1/3)*(c + d*
x)**(1/3) + (a*d - b*c)**(1/3))*elliptic_f(asin((b**(1/3)*(c + d*x)**(1/3) - (-1
+ sqrt(3))*(a*d - b*c)**(1/3))/(b**(1/3)*(c + d*x)**(1/3) + (1 + sqrt(3))*(a*d
- b*c)**(1/3))), -7 - 4*sqrt(3))/(935*b**(4/3)*d**3*sqrt((a*d - b*c)**(1/3)*(b**
(1/3)*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))/(b**(1/3)*(c + d*x)**(1/3) + (1 + s
qrt(3))*(a*d - b*c)**(1/3))**2)*sqrt(a - b*c/d + b*(c + d*x)/d))

_______________________________________________________________________________________

Mathematica [C]  time = 0.285936, size = 142, normalized size = 0.31 \[ -\frac{6 \sqrt [3]{c+d x} \left (-d (a+b x) \left (27 a^2 d^2+2 a b d (23 c+50 d x)+b^2 \left (-18 c^2+10 c d x+55 d^2 x^2\right )\right )-27 (b c-a d)^3 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\frac{b (c+d x)}{b c-a d}\right )\right )}{935 b d^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(-6*(c + d*x)^(1/3)*(-(d*(a + b*x)*(27*a^2*d^2 + 2*a*b*d*(23*c + 50*d*x) + b^2*(
-18*c^2 + 10*c*d*x + 55*d^2*x^2))) - 27*(b*c - a*d)^3*Sqrt[(d*(a + b*x))/(-(b*c)
+ a*d)]*Hypergeometric2F1[1/3, 1/2, 4/3, (b*(c + d*x))/(b*c - a*d)]))/(935*b*d^
3*Sqrt[a + b*x])

_______________________________________________________________________________________

Maple [F]  time = 0.063, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{{\frac{3}{2}}}\sqrt [3]{dx+c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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