3.1596 \(\int \frac{(a+b x)^{5/3}}{\sqrt [3]{c+d x}} \, dx\)

Optimal. Leaf size=1330 \[ \text{result too large to display} \]

[Out]

(-15*(b*c - a*d)*(a + b*x)^(2/3)*(c + d*x)^(2/3))/(28*d^2) + (3*(a + b*x)^(5/3)*
(c + d*x)^(2/3))/(7*d) + (15*(b*c - a*d)^2*((a + b*x)*(c + d*x))^(1/3)*Sqrt[(b*c
+ a*d + 2*b*d*x)^2]*Sqrt[(a*d + b*(c + 2*d*x))^2])/(14*2^(1/3)*b^(2/3)*d^(8/3)*
(a + b*x)^(1/3)*(c + d*x)^(1/3)*(b*c + a*d + 2*b*d*x)*((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))) - (15*3^(1/4)*Sqr
t[2 - Sqrt[3]]*(b*c - a*d)^(8/3)*((a + b*x)*(c + d*x))^(1/3)*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]*EllipticE[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]])/(28*2^(1/3)*b^(2/3)
*d^(8/3)*(a + b*x)^(1/3)*(c + d*x)^(1/3)*(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]) + (5*3^(3/4)*(b*c - a*d)^(8/3)*((a +
b*x)*(c + d*x))^(1/3)*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]])/(7*2^(5/6)*b^(2/3)*d^(8/3)*(a + b*x)^(1/3)*(c + d*x)^(1/3
)*(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 = 4.40534, antiderivative size = 1330, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{15 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{(a+b x) (c+d x)} \sqrt{(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt{\frac{(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)} (b c-a d)^{2/3}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\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 )^2}} E\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 ) (b c-a d)^{8/3}}{28 \sqrt [3]{2} b^{2/3} d^{8/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\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 )^2}} \sqrt{(a d+b (c+2 d x))^2}}+\frac{5\ 3^{3/4} \sqrt [3]{(a+b x) (c+d x)} \sqrt{(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt{\frac{(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)} (b c-a d)^{2/3}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\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 )^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 ) (b c-a d)^{8/3}}{7\ 2^{5/6} b^{2/3} d^{8/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\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 )^2}} \sqrt{(a d+b (c+2 d x))^2}}+\frac{15 \sqrt [3]{(a+b x) (c+d x)} \sqrt{(b c+a d+2 b d x)^2} \sqrt{(a d+b (c+2 d x))^2} (b c-a d)^2}{14 \sqrt [3]{2} b^{2/3} d^{8/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \left (\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 )}-\frac{15 (a+b x)^{2/3} (c+d x)^{2/3} (b c-a d)}{28 d^2}+\frac{3 (a+b x)^{5/3} (c+d x)^{2/3}}{7 d} \]

Warning: Unable to verify antiderivative.

[In]  Int[(a + b*x)^(5/3)/(c + d*x)^(1/3),x]

[Out]

(-15*(b*c - a*d)*(a + b*x)^(2/3)*(c + d*x)^(2/3))/(28*d^2) + (3*(a + b*x)^(5/3)*
(c + d*x)^(2/3))/(7*d) + (15*(b*c - a*d)^2*((a + b*x)*(c + d*x))^(1/3)*Sqrt[(b*c
+ a*d + 2*b*d*x)^2]*Sqrt[(a*d + b*(c + 2*d*x))^2])/(14*2^(1/3)*b^(2/3)*d^(8/3)*
(a + b*x)^(1/3)*(c + d*x)^(1/3)*(b*c + a*d + 2*b*d*x)*((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))) - (15*3^(1/4)*Sqr
t[2 - Sqrt[3]]*(b*c - a*d)^(8/3)*((a + b*x)*(c + d*x))^(1/3)*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]*EllipticE[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]])/(28*2^(1/3)*b^(2/3)
*d^(8/3)*(a + b*x)^(1/3)*(c + d*x)^(1/3)*(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]) + (5*3^(3/4)*(b*c - a*d)^(8/3)*((a +
b*x)*(c + d*x))^(1/3)*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]])/(7*2^(5/6)*b^(2/3)*d^(8/3)*(a + b*x)^(1/3)*(c + d*x)^(1/3
)*(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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/3)/(d*x+c)**(1/3),x)

[Out]

Timed out

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Mathematica [C]  time = 0.192126, size = 107, normalized size = 0.08 \[ \frac{3 (c+d x)^{2/3} \left (5 (b c-a d)^2 \sqrt [3]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{b (c+d x)}{b c-a d}\right )+d (a+b x) (9 a d-5 b c+4 b d x)\right )}{28 d^3 \sqrt [3]{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(5/3)/(c + d*x)^(1/3),x]

[Out]

(3*(c + d*x)^(2/3)*(d*(a + b*x)*(-5*b*c + 9*a*d + 4*b*d*x) + 5*(b*c - a*d)^2*((d
*(a + b*x))/(-(b*c) + a*d))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, (b*(c + d*x))
/(b*c - a*d)]))/(28*d^3*(a + b*x)^(1/3))

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{5}{3}}}{\frac{1}{\sqrt [3]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/3)/(d*x+c)^(1/3),x)

[Out]

int((b*x+a)^(5/3)/(d*x+c)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{5}{3}}}{{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/3)/(d*x + c)^(1/3),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(5/3)/(d*x + c)^(1/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{\frac{5}{3}}}{{\left (d x + c\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/3)/(d*x + c)^(1/3),x, algorithm="fricas")

[Out]

integral((b*x + a)^(5/3)/(d*x + c)^(1/3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{\frac{5}{3}}}{\sqrt [3]{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(5/3)/(d*x+c)**(1/3),x)

[Out]

Integral((a + b*x)**(5/3)/(c + d*x)**(1/3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{5}{3}}}{{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/3)/(d*x + c)^(1/3),x, algorithm="giac")

[Out]

integrate((b*x + a)^(5/3)/(d*x + c)^(1/3), x)