3.1748 \(\int \frac{(c+d x)^{5/6}}{(a+b x)^{7/2}} \, dx\)

Optimal. Leaf size=896 \[ \frac{8 \left (1+\sqrt{3}\right ) \sqrt{a+b x} \sqrt [6]{c+d x} d^3}{27 b^{5/3} (b c-a d)^2 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac{8 \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) d^2}{9\ 3^{3/4} b^{5/3} (b c-a d)^{5/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{4 \left (1-\sqrt{3}\right ) \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) d^2}{27 \sqrt [4]{3} b^{5/3} (b c-a d)^{5/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{8 (c+d x)^{5/6} d^2}{27 b (b c-a d)^2 \sqrt{a+b x}}-\frac{2 (c+d x)^{5/6} d}{9 b (b c-a d) (a+b x)^{3/2}}-\frac{2 (c+d x)^{5/6}}{5 b (a+b x)^{5/2}} \]

[Out]

(-2*(c + d*x)^(5/6))/(5*b*(a + b*x)^(5/2)) - (2*d*(c + d*x)^(5/6))/(9*b*(b*c - a
*d)*(a + b*x)^(3/2)) + (8*d^2*(c + d*x)^(5/6))/(27*b*(b*c - a*d)^2*Sqrt[a + b*x]
) + (8*(1 + Sqrt[3])*d^3*Sqrt[a + b*x]*(c + d*x)^(1/6))/(27*b^(5/3)*(b*c - a*d)^
2*((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))) + (8*d^2*(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[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^(3/4)*b^(5/3)*(b*c - a*d)^
(5/3)*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
)]) + (4*(1 - Sqrt[3])*d^2*(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])/(27*3^(1/4)*b^(5/3)*(b*c - a*d)^(5/3)*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 = 1.87069, antiderivative size = 896, 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 \[ \frac{8 \left (1+\sqrt{3}\right ) \sqrt{a+b x} \sqrt [6]{c+d x} d^3}{27 b^{5/3} (b c-a d)^2 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac{8 \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) d^2}{9\ 3^{3/4} b^{5/3} (b c-a d)^{5/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{4 \left (1-\sqrt{3}\right ) \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) d^2}{27 \sqrt [4]{3} b^{5/3} (b c-a d)^{5/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{8 (c+d x)^{5/6} d^2}{27 b (b c-a d)^2 \sqrt{a+b x}}-\frac{2 (c+d x)^{5/6} d}{9 b (b c-a d) (a+b x)^{3/2}}-\frac{2 (c+d x)^{5/6}}{5 b (a+b x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/6)/(a + b*x)^(7/2),x]

[Out]

(-2*(c + d*x)^(5/6))/(5*b*(a + b*x)^(5/2)) - (2*d*(c + d*x)^(5/6))/(9*b*(b*c - a
*d)*(a + b*x)^(3/2)) + (8*d^2*(c + d*x)^(5/6))/(27*b*(b*c - a*d)^2*Sqrt[a + b*x]
) + (8*(1 + Sqrt[3])*d^3*Sqrt[a + b*x]*(c + d*x)^(1/6))/(27*b^(5/3)*(b*c - a*d)^
2*((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))) + (8*d^2*(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[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^(3/4)*b^(5/3)*(b*c - a*d)^
(5/3)*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
)]) + (4*(1 - Sqrt[3])*d^2*(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])/(27*3^(1/4)*b^(5/3)*(b*c - a*d)^(5/3)*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 = 104.021, size = 794, normalized size = 0.89 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/6)/(b*x+a)**(7/2),x)

[Out]

8*d**2*(c + d*x)**(5/6)/(27*b*sqrt(a + b*x)*(a*d - b*c)**2) + 2*d*(c + d*x)**(5/
6)/(9*b*(a + b*x)**(3/2)*(a*d - b*c)) - 2*(c + d*x)**(5/6)/(5*b*(a + b*x)**(5/2)
) - d**3*(8/27 + 8*sqrt(3)/27)*(c + d*x)**(1/6)*sqrt(a - b*c/d + b*(c + d*x)/d)/
(b**(5/3)*(a*d - b*c)**2*(b**(1/3)*(1 + sqrt(3))*(c + d*x)**(1/3) + (a*d - b*c)*
*(1/3))) + 8*3**(1/4)*d**2*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)/(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)**(5/3)*sqrt(a - b*c/d + b*(c + d*x)/d)) + 4*3
**(3/4)*d**2*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)/(81*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)**(5/3)*sqrt(a - b*c/d + b*(c + d*x)/d))

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Mathematica [C]  time = 0.326907, size = 140, normalized size = 0.16 \[ -\frac{2 (c+d x)^{5/6} \left (-8 a^2 d^2+8 d^2 (a+b x)^2 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\frac{b (c+d x)}{b c-a d}\right )-a b d (39 c+55 d x)+b^2 \left (27 c^2+15 c d x-20 d^2 x^2\right )\right )}{135 b (a+b x)^{5/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/6)/(a + b*x)^(7/2),x]

[Out]

(-2*(c + d*x)^(5/6)*(-8*a^2*d^2 - a*b*d*(39*c + 55*d*x) + b^2*(27*c^2 + 15*c*d*x
- 20*d^2*x^2) + 8*d^2*(a + b*x)^2*Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]*Hypergeome
tric2F1[1/2, 5/6, 11/6, (b*(c + d*x))/(b*c - a*d)]))/(135*b*(b*c - a*d)^2*(a + b
*x)^(5/2))

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{5}{6}}} \left ( bx+a \right ) ^{-{\frac{7}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/6)/(b*x+a)^(7/2),x)

[Out]

int((d*x+c)^(5/6)/(b*x+a)^(7/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{5}{6}}}{{\left (b x + a\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/6)/(b*x + a)^(7/2),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(5/6)/(b*x + a)^(7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x + c\right )}^{\frac{5}{6}}}{{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt{b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/6)/(b*x + a)^(7/2),x, algorithm="fricas")

[Out]

integral((d*x + c)^(5/6)/((b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*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)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.837715, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/6)/(b*x + a)^(7/2),x, algorithm="giac")

[Out]

Done