3.1610 \(\int \frac{(a+b x)^{4/3}}{(c+d x)^{2/3}} \, dx\)

Optimal. Leaf size=614 \[ \frac{2 \sqrt [3]{2} 3^{3/4} \sqrt{2+\sqrt{3}} (b c-a d)^2 ((a+b x) (c+d x))^{2/3} \sqrt{(a d+b c+2 b d x)^2} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right ) \sqrt{\frac{2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{4/3}}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{5 \sqrt [3]{b} d^{7/3} (a+b x)^{2/3} (c+d x)^{2/3} (a d+b c+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right )}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} \sqrt{(a d+b (c+2 d x))^2}}-\frac{6 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c-a d)}{5 d^2}+\frac{3 (a+b x)^{4/3} \sqrt [3]{c+d x}}{5 d} \]

[Out]

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

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Rubi [A]  time = 1.80139, antiderivative size = 614, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{2 \sqrt [3]{2} 3^{3/4} \sqrt{2+\sqrt{3}} (b c-a d)^2 ((a+b x) (c+d x))^{2/3} \sqrt{(a d+b c+2 b d x)^2} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right ) \sqrt{\frac{2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{4/3}}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{5 \sqrt [3]{b} d^{7/3} (a+b x)^{2/3} (c+d x)^{2/3} (a d+b c+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right )}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} \sqrt{(a d+b (c+2 d x))^2}}-\frac{6 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c-a d)}{5 d^2}+\frac{3 (a+b x)^{4/3} \sqrt [3]{c+d x}}{5 d} \]

Warning: Unable to verify antiderivative.

[In]  Int[(a + b*x)^(4/3)/(c + d*x)^(2/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 70.1928, size = 648, normalized size = 1.06 \[ \frac{3 \left (a + b x\right )^{\frac{4}{3}} \sqrt [3]{c + d x}}{5 d} + \frac{6 \sqrt [3]{a + b x} \sqrt [3]{c + d x} \left (a d - b c\right )}{5 d^{2}} + \frac{2 \sqrt [3]{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{2 \sqrt [3]{2} b^{\frac{2}{3}} d^{\frac{2}{3}} \left (a c + b d x^{2} + x \left (a d + b c\right )\right )^{\frac{2}{3}} - 2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \left (a d - b c\right )^{\frac{2}{3}} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} + \left (a d - b c\right )^{\frac{4}{3}}}{\left (2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} + \left (1 + \sqrt{3}\right ) \left (a d - b c\right )^{\frac{2}{3}}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (a d - b c\right )^{2} \left (2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} + \left (a d - b c\right )^{\frac{2}{3}}\right ) \left (a c + b d x^{2} + x \left (a d + b c\right )\right )^{\frac{2}{3}} \sqrt{\left (a d + b c + 2 b d x\right )^{2}} F\left (\operatorname{asin}{\left (\frac{2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} - \left (-1 + \sqrt{3}\right ) \left (a d - b c\right )^{\frac{2}{3}}}{2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} + \left (1 + \sqrt{3}\right ) \left (a d - b c\right )^{\frac{2}{3}}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{5 \sqrt [3]{b} d^{\frac{7}{3}} \sqrt{\frac{\left (a d - b c\right )^{\frac{2}{3}} \left (2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} + \left (a d - b c\right )^{\frac{2}{3}}\right )}{\left (2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} + \left (1 + \sqrt{3}\right ) \left (a d - b c\right )^{\frac{2}{3}}\right )^{2}}} \left (a + b x\right )^{\frac{2}{3}} \left (c + d x\right )^{\frac{2}{3}} \sqrt{b d \left (4 a c + 4 b d x^{2} + x \left (4 a d + 4 b c\right )\right ) + \left (a d - b c\right )^{2}} \left (a d + b c + 2 b d x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(4/3)/(d*x+c)**(2/3),x)

[Out]

3*(a + b*x)**(4/3)*(c + d*x)**(1/3)/(5*d) + 6*(a + b*x)**(1/3)*(c + d*x)**(1/3)*
(a*d - b*c)/(5*d**2) + 2*2**(1/3)*3**(3/4)*sqrt((2*2**(1/3)*b**(2/3)*d**(2/3)*(a
*c + b*d*x**2 + x*(a*d + b*c))**(2/3) - 2**(2/3)*b**(1/3)*d**(1/3)*(a*d - b*c)**
(2/3)*(a*c + b*d*x**2 + x*(a*d + b*c))**(1/3) + (a*d - b*c)**(4/3))/(2**(2/3)*b*
*(1/3)*d**(1/3)*(a*c + b*d*x**2 + x*(a*d + b*c))**(1/3) + (1 + sqrt(3))*(a*d - b
*c)**(2/3))**2)*sqrt(sqrt(3) + 2)*(a*d - b*c)**2*(2**(2/3)*b**(1/3)*d**(1/3)*(a*
c + b*d*x**2 + x*(a*d + b*c))**(1/3) + (a*d - b*c)**(2/3))*(a*c + b*d*x**2 + x*(
a*d + b*c))**(2/3)*sqrt((a*d + b*c + 2*b*d*x)**2)*elliptic_f(asin((2**(2/3)*b**(
1/3)*d**(1/3)*(a*c + b*d*x**2 + x*(a*d + b*c))**(1/3) - (-1 + sqrt(3))*(a*d - b*
c)**(2/3))/(2**(2/3)*b**(1/3)*d**(1/3)*(a*c + b*d*x**2 + x*(a*d + b*c))**(1/3) +
(1 + sqrt(3))*(a*d - b*c)**(2/3))), -7 - 4*sqrt(3))/(5*b**(1/3)*d**(7/3)*sqrt((
a*d - b*c)**(2/3)*(2**(2/3)*b**(1/3)*d**(1/3)*(a*c + b*d*x**2 + x*(a*d + b*c))**
(1/3) + (a*d - b*c)**(2/3))/(2**(2/3)*b**(1/3)*d**(1/3)*(a*c + b*d*x**2 + x*(a*d
+ b*c))**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(2/3))**2)*(a + b*x)**(2/3)*(c + d*
x)**(2/3)*sqrt(b*d*(4*a*c + 4*b*d*x**2 + x*(4*a*d + 4*b*c)) + (a*d - b*c)**2)*(a
*d + b*c + 2*b*d*x))

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Mathematica [C]  time = 0.188068, size = 106, normalized size = 0.17 \[ \frac{3 \sqrt [3]{c+d x} \left (2 (b c-a d)^2 \left (\frac{d (a+b x)}{a d-b c}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{b (c+d x)}{b c-a d}\right )+d (a+b x) (3 a d-2 b c+b d x)\right )}{5 d^3 (a+b x)^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(4/3)/(c + d*x)^(2/3),x]

[Out]

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

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{4}{3}}} \left ( dx+c \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(4/3)/(d*x+c)^(2/3),x)

[Out]

int((b*x+a)^(4/3)/(d*x+c)^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{4}{3}}}{{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(4/3)/(d*x + c)^(2/3),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(4/3)/(d*x + c)^(2/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{\frac{4}{3}}}{{\left (d x + c\right )}^{\frac{2}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(4/3)/(d*x + c)^(2/3),x, algorithm="fricas")

[Out]

integral((b*x + a)^(4/3)/(d*x + c)^(2/3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{\frac{4}{3}}}{\left (c + d x\right )^{\frac{2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(4/3)/(d*x+c)**(2/3),x)

[Out]

Integral((a + b*x)**(4/3)/(c + d*x)**(2/3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{4}{3}}}{{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(4/3)/(d*x + c)^(2/3),x, algorithm="giac")

[Out]

integrate((b*x + a)^(4/3)/(d*x + c)^(2/3), x)