3.1760 \(\int \frac{1}{(a+b x)^{5/2} (c+d x)^{5/6}} \, dx\)

Optimal. Leaf size=410 \[ \frac{16 d \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{9 \sqrt [4]{3} \sqrt{a+b x} (b c-a d)^{7/3} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{16 d \sqrt [6]{c+d x}}{9 \sqrt{a+b x} (b c-a d)^2}-\frac{2 \sqrt [6]{c+d x}}{3 (a+b x)^{3/2} (b c-a d)} \]

[Out]

(-2*(c + d*x)^(1/6))/(3*(b*c - a*d)*(a + b*x)^(3/2)) + (16*d*(c + d*x)^(1/6))/(9
*(b*c - a*d)^2*Sqrt[a + b*x]) + (16*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*c - a*d)^(7/3)*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 = 0.621258, antiderivative size = 410, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{16 d \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{9 \sqrt [4]{3} \sqrt{a+b x} (b c-a d)^{7/3} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{16 d \sqrt [6]{c+d x}}{9 \sqrt{a+b x} (b c-a d)^2}-\frac{2 \sqrt [6]{c+d x}}{3 (a+b x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(5/2)*(c + d*x)^(5/6)),x]

[Out]

(-2*(c + d*x)^(1/6))/(3*(b*c - a*d)*(a + b*x)^(3/2)) + (16*d*(c + d*x)^(1/6))/(9
*(b*c - a*d)^2*Sqrt[a + b*x]) + (16*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*c - a*d)^(7/3)*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 = 29.6919, size = 357, normalized size = 0.87 \[ \frac{16 \cdot 3^{\frac{3}{4}} d \sqrt{\frac{b^{\frac{2}{3}} \left (c + d x\right )^{\frac{2}{3}} - \sqrt [3]{b} \sqrt [3]{c + d x} \sqrt [3]{a d - b c} + \left (a d - b c\right )^{\frac{2}{3}}}{\left (\sqrt [3]{b} \left (1 + \sqrt{3}\right ) \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )^{2}}} \sqrt [6]{c + d x} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right ) F\left (\operatorname{acos}{\left (\frac{\sqrt [3]{b} \left (- \sqrt{3} + 1\right ) \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}}{\sqrt [3]{b} \left (1 + \sqrt{3}\right ) \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{27 \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c + d x} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )}{\left (\sqrt [3]{b} \left (1 + \sqrt{3}\right ) \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )^{2}}} \left (a d - b c\right )^{\frac{7}{3}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{16 d \sqrt [6]{c + d x}}{9 \sqrt{a + b x} \left (a d - b c\right )^{2}} + \frac{2 \sqrt [6]{c + d x}}{3 \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(5/2)/(d*x+c)**(5/6),x)

[Out]

16*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)*(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*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)**(7/3)*sqrt(a - b*c/d + b*(c + d*x)/d)) + 16*d*(c + d*x)**(1/6)/(9*
sqrt(a + b*x)*(a*d - b*c)**2) + 2*(c + d*x)**(1/6)/(3*(a + b*x)**(3/2)*(a*d - b*
c))

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Mathematica [C]  time = 0.203622, size = 102, normalized size = 0.25 \[ \frac{2 \sqrt [6]{c+d x} \left (16 d (a+b x) \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\frac{b (c+d x)}{b c-a d}\right )+11 a d-3 b c+8 b d x\right )}{9 (a+b x)^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(5/2)*(c + d*x)^(5/6)),x]

[Out]

(2*(c + d*x)^(1/6)*(-3*b*c + 11*a*d + 8*b*d*x + 16*d*(a + b*x)*Sqrt[(d*(a + b*x)
)/(-(b*c) + a*d)]*Hypergeometric2F1[1/6, 1/2, 7/6, (b*(c + d*x))/(b*c - a*d)]))/
(9*(b*c - a*d)^2*(a + b*x)^(3/2))

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{5}{2}}} \left ( dx+c \right ) ^{-{\frac{5}{6}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(5/2)/(d*x+c)^(5/6),x)

[Out]

int(1/(b*x+a)^(5/2)/(d*x+c)^(5/6),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{2}}{\left (d x + c\right )}^{\frac{5}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(5/6)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(5/2)*(d*x + c)^(5/6)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b x + a}{\left (d x + c\right )}^{\frac{5}{6}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(5/6)),x, algorithm="fricas")

[Out]

integral(1/((b^2*x^2 + 2*a*b*x + a^2)*sqrt(b*x + a)*(d*x + c)^(5/6)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{5}{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(5/2)/(d*x+c)**(5/6),x)

[Out]

Integral(1/((a + b*x)**(5/2)*(c + d*x)**(5/6)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{2}}{\left (d x + c\right )}^{\frac{5}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(5/6)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(5/2)*(d*x + c)^(5/6)), x)