3.1747 \(\int \frac{(c+d x)^{5/6}}{(a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=854 \[ -\frac{10 \left (1+\sqrt{3}\right ) \sqrt{a+b x} \sqrt [6]{c+d x} d^2}{9 b^{5/3} (b c-a d) \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}-\frac{10 \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) d}{3\ 3^{3/4} b^{5/3} (b c-a d)^{2/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{5 \left (1-\sqrt{3}\right ) \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) d}{9 \sqrt [4]{3} b^{5/3} (b c-a d)^{2/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{10 (c+d x)^{5/6} d}{9 b (b c-a d) \sqrt{a+b x}}-\frac{2 (c+d x)^{5/6}}{3 b (a+b x)^{3/2}} \]

[Out]

(-2*(c + d*x)^(5/6))/(3*b*(a + b*x)^(3/2)) - (10*d*(c + d*x)^(5/6))/(9*b*(b*c -
a*d)*Sqrt[a + b*x]) - (10*(1 + Sqrt[3])*d^2*Sqrt[a + b*x]*(c + d*x)^(1/6))/(9*b^
(5/3)*(b*c - a*d)*((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))) -
(10*d*(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]*EllipticE[Arc
Cos[((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])/(3*3^(3/4)*b^(5/
3)*(b*c - a*d)^(2/3)*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)]) - (5*(1 - Sqrt[3])*d*(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])/(9*3^(1/4)*b^(5/3)*(b*c - a*d)^(2/3)*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 = 1.63174, antiderivative size = 854, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{10 \left (1+\sqrt{3}\right ) \sqrt{a+b x} \sqrt [6]{c+d x} d^2}{9 b^{5/3} (b c-a d) \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}-\frac{10 \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) d}{3\ 3^{3/4} b^{5/3} (b c-a d)^{2/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{5 \left (1-\sqrt{3}\right ) \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) d}{9 \sqrt [4]{3} b^{5/3} (b c-a d)^{2/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{10 (c+d x)^{5/6} d}{9 b (b c-a d) \sqrt{a+b x}}-\frac{2 (c+d x)^{5/6}}{3 b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/6)/(a + b*x)^(5/2),x]

[Out]

(-2*(c + d*x)^(5/6))/(3*b*(a + b*x)^(3/2)) - (10*d*(c + d*x)^(5/6))/(9*b*(b*c -
a*d)*Sqrt[a + b*x]) - (10*(1 + Sqrt[3])*d^2*Sqrt[a + b*x]*(c + d*x)^(1/6))/(9*b^
(5/3)*(b*c - a*d)*((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))) -
(10*d*(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]*EllipticE[Arc
Cos[((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])/(3*3^(3/4)*b^(5/
3)*(b*c - a*d)^(2/3)*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)]) - (5*(1 - Sqrt[3])*d*(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])/(9*3^(1/4)*b^(5/3)*(b*c - a*d)^(2/3)*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 = 82.4518, size = 755, normalized size = 0.88 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/6)/(b*x+a)**(5/2),x)

[Out]

10*d*(c + d*x)**(5/6)/(9*b*sqrt(a + b*x)*(a*d - b*c)) - 2*(c + d*x)**(5/6)/(3*b*
(a + b*x)**(3/2)) - d**2*(10/9 + 10*sqrt(3)/9)*(c + d*x)**(1/6)*sqrt(a - b*c/d +
b*(c + d*x)/d)/(b**(5/3)*(a*d - b*c)*(b**(1/3)*(1 + sqrt(3))*(c + d*x)**(1/3) +
(a*d - b*c)**(1/3))) + 10*3**(1/4)*d*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)*(b**(1/3)*(c + d*
x)**(1/3) + (a*d - b*c)**(1/3))*elliptic_e(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)/(9*b**(5/3)*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)*(a*d - b*c)**(2/3)*sqrt(a - b*c/d + b*(c + d*x)
/d)) + 5*3**(3/4)*d*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)*(-sqrt(3) + 1)*(c + d*x)**(1/6)*(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)/(27*b**(5/3)*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)*(a*d - b*c)**(2/3)*sqrt(a - b*c/d + b*(c + d*x)/d
))

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Mathematica [C]  time = 0.239232, size = 105, normalized size = 0.12 \[ -\frac{2 (c+d x)^{5/6} \left (-2 d (a+b x) \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\frac{b (c+d x)}{b c-a d}\right )+2 a d+3 b c+5 b d x\right )}{9 b (a+b x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/6)/(a + b*x)^(5/2),x]

[Out]

(-2*(c + d*x)^(5/6)*(3*b*c + 2*a*d + 5*b*d*x - 2*d*(a + b*x)*Sqrt[(d*(a + b*x))/
(-(b*c) + a*d)]*Hypergeometric2F1[1/2, 5/6, 11/6, (b*(c + d*x))/(b*c - a*d)]))/(
9*b*(b*c - a*d)*(a + b*x)^(3/2))

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{5}{6}}} \left ( bx+a \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/6)/(b*x+a)^(5/2),x)

[Out]

int((d*x+c)^(5/6)/(b*x+a)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{5}{6}}}{{\left (b x + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/6)/(b*x + a)^(5/2),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(5/6)/(b*x + a)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x + c\right )}^{\frac{5}{6}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/6)/(b*x + a)^(5/2),x, algorithm="fricas")

[Out]

integral((d*x + c)^(5/6)/((b^2*x^2 + 2*a*b*x + a^2)*sqrt(b*x + a)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**(5/6)/(b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{5}{6}}}{{\left (b x + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/6)/(b*x + a)^(5/2),x, algorithm="giac")

[Out]

integrate((d*x + c)^(5/6)/(b*x + a)^(5/2), x)