3.1567 \(\int \frac{\sqrt{a+b x}}{\sqrt [3]{c+d x}} \, dx\)

Optimal. Leaf size=804 \[ -\frac{9 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right ) (b c-a d)^{4/3}}{7 b^{2/3} d^2 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{6 \sqrt{2} 3^{3/4} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right ) (b c-a d)^{4/3}}{7 b^{2/3} d^2 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{18 \sqrt{a+b x} (b c-a d)}{7 b^{2/3} d \left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac{6 \sqrt{a+b x} (c+d x)^{2/3}}{7 d} \]

[Out]

(6*Sqrt[a + b*x]*(c + d*x)^(2/3))/(7*d) + (18*(b*c - a*d)*Sqrt[a + b*x])/(7*b^(2
/3)*d*((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))) - (9*3^(1/4)*
Sqrt[2 + Sqrt[3]]*(b*c - a*d)^(4/3)*((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]*
EllipticE[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]])/(7*b
^(2/3)*d^2*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)
]) + (6*Sqrt[2]*3^(3/4)*(b*c - a*d)^(4/3)*((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]]
)/(7*b^(2/3)*d^2*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)])

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Rubi [A]  time = 1.54371, antiderivative size = 804, 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 \[ -\frac{9 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right ) (b c-a d)^{4/3}}{7 b^{2/3} d^2 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{6 \sqrt{2} 3^{3/4} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right ) (b c-a d)^{4/3}}{7 b^{2/3} d^2 \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{18 \sqrt{a+b x} (b c-a d)}{7 b^{2/3} d \left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac{6 \sqrt{a+b x} (c+d x)^{2/3}}{7 d} \]

Antiderivative was successfully verified.

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

[Out]

(6*Sqrt[a + b*x]*(c + d*x)^(2/3))/(7*d) + (18*(b*c - a*d)*Sqrt[a + b*x])/(7*b^(2
/3)*d*((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))) - (9*3^(1/4)*
Sqrt[2 + Sqrt[3]]*(b*c - a*d)^(4/3)*((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]*
EllipticE[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]])/(7*b
^(2/3)*d^2*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)
]) + (6*Sqrt[2]*3^(3/4)*(b*c - a*d)^(4/3)*((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]]
)/(7*b^(2/3)*d^2*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)])

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Rubi in Sympy [A]  time = 86.4593, size = 694, normalized size = 0.86 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*sqrt(a + b*x)*(c + d*x)**(2/3)/(7*d) + 18*(a*d - b*c)*sqrt(a - b*c/d + b*(c +
d*x)/d)/(7*b**(2/3)*d*(b**(1/3)*(c + d*x)**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(1
/3))) - 9*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)*(c + d*x)**(1/3) + (1 + sqrt(3
))*(a*d - b*c)**(1/3))**2)*sqrt(-sqrt(3) + 2)*(a*d - b*c)**(4/3)*(b**(1/3)*(c +
d*x)**(1/3) + (a*d - b*c)**(1/3))*elliptic_e(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))/(7*b**(2/3)*d**2*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)) + 6*sqrt(2)*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 + sqrt(3))*(a*d - b*c
)**(1/3))**2)*(a*d - b*c)**(4/3)*(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
))/(7*b**(2/3)*d**2*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 + sqrt(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 = 0.185775, size = 77, normalized size = 0.1 \[ \frac{3 \sqrt{a+b x} (c+d x)^{2/3} \left (\frac{3 \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\frac{b (c+d x)}{b c-a d}\right )}{\sqrt{\frac{d (a+b x)}{a d-b c}}}+4\right )}{14 d} \]

Antiderivative was successfully verified.

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

[Out]

(3*Sqrt[a + b*x]*(c + d*x)^(2/3)*(4 + (3*Hypergeometric2F1[1/2, 2/3, 5/3, (b*(c
+ d*x))/(b*c - a*d)])/Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]))/(14*d)

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{1\sqrt{bx+a}{\frac{1}{\sqrt [3]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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