3.1762 \(\int \frac{(a+b x)^{3/2}}{(c+d x)^{7/6}} \, dx\)

Optimal. Leaf size=844 \[ -\frac{6 (a+b x)^{3/2}}{d \sqrt [6]{c+d x}}+\frac{27 b (c+d x)^{5/6} \sqrt{a+b x}}{4 d^2}+\frac{81 \left (1+\sqrt{3}\right ) \sqrt [3]{b} (b c-a d) \sqrt [6]{c+d x} \sqrt{a+b x}}{8 d^2 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac{81 \sqrt [4]{3} \sqrt [3]{b} (b c-a d)^{4/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{8 d^3 \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \sqrt{a+b x}}+\frac{27\ 3^{3/4} \left (1-\sqrt{3}\right ) \sqrt [3]{b} (b c-a d)^{4/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{16 d^3 \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \sqrt{a+b x}} \]

[Out]

(-6*(a + b*x)^(3/2))/(d*(c + d*x)^(1/6)) + (27*b*Sqrt[a + b*x]*(c + d*x)^(5/6))/
(4*d^2) + (81*(1 + Sqrt[3])*b^(1/3)*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(1/6))/(
8*d^2*((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))) + (81*3^(1/4)
*b^(1/3)*(b*c - a*d)^(4/3)*(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])/(8*d^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)]) + (27*3^(3/4)*(1 - Sqrt[3])*b^(1/3)*(b*c - a*d)^(4/3)*(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])/(16*d^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 = 1.6579, antiderivative size = 844, 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 \[ -\frac{6 (a+b x)^{3/2}}{d \sqrt [6]{c+d x}}+\frac{27 b (c+d x)^{5/6} \sqrt{a+b x}}{4 d^2}+\frac{81 \left (1+\sqrt{3}\right ) \sqrt [3]{b} (b c-a d) \sqrt [6]{c+d x} \sqrt{a+b x}}{8 d^2 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac{81 \sqrt [4]{3} \sqrt [3]{b} (b c-a d)^{4/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{8 d^3 \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \sqrt{a+b x}}+\frac{27\ 3^{3/4} \left (1-\sqrt{3}\right ) \sqrt [3]{b} (b c-a d)^{4/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{16 d^3 \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(3/2)/(c + d*x)^(7/6),x]

[Out]

(-6*(a + b*x)^(3/2))/(d*(c + d*x)^(1/6)) + (27*b*Sqrt[a + b*x]*(c + d*x)^(5/6))/
(4*d^2) + (81*(1 + Sqrt[3])*b^(1/3)*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(1/6))/(
8*d^2*((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))) + (81*3^(1/4)
*b^(1/3)*(b*c - a*d)^(4/3)*(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])/(8*d^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)]) + (27*3^(3/4)*(1 - Sqrt[3])*b^(1/3)*(b*c - a*d)^(4/3)*(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])/(16*d^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 = 83.6657, size = 751, normalized size = 0.89 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)/(d*x+c)**(7/6),x)

[Out]

b**(1/3)*(81/8 + 81*sqrt(3)/8)*(c + d*x)**(1/6)*(a*d - b*c)*sqrt(a - b*c/d + b*(
c + d*x)/d)/(d**2*(b**(1/3)*(1 + sqrt(3))*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))
) - 81*3**(1/4)*b**(1/3)*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)*(a*d - b*c)**(4/3)*(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)/(8*d**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)*sqrt(a - b*c/d + b*(c + d*x)/d)) - 27*3**(3/4
)*b**(1/3)*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)*(a*d - b*c)**(4/3)*(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)/(16*d**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)*sqrt(a - b*c/d + b*(c + d*x)/d)) + 27*b*sqr
t(a + b*x)*(c + d*x)**(5/6)/(4*d**2) - 6*(a + b*x)**(3/2)/(d*(c + d*x)**(1/6))

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Mathematica [C]  time = 0.340406, size = 99, normalized size = 0.12 \[ \frac{3 \sqrt{a+b x} (c+d x)^{5/6} \left (\frac{27 b \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\frac{b (c+d x)}{b c-a d}\right )}{\sqrt{\frac{d (a+b x)}{a d-b c}}}+\frac{5 (-8 a d+9 b c+b d x)}{c+d x}\right )}{20 d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(3/2)/(c + d*x)^(7/6),x]

[Out]

(3*Sqrt[a + b*x]*(c + d*x)^(5/6)*((5*(9*b*c - 8*a*d + b*d*x))/(c + d*x) + (27*b*
Hypergeometric2F1[1/2, 5/6, 11/6, (b*(c + d*x))/(b*c - a*d)])/Sqrt[(d*(a + b*x))
/(-(b*c) + a*d)]))/(20*d^2)

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Maple [F]  time = 0.086, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( dx+c \right ) ^{-{\frac{7}{6}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)/(d*x+c)^(7/6),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{3}{2}}}{{\left (d x + c\right )}^{\frac{7}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)/(d*x + c)^(7/6),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{\frac{3}{2}}}{{\left (d x + c\right )}^{\frac{7}{6}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)/(d*x + c)^(7/6),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{\frac{3}{2}}}{\left (c + d x\right )^{\frac{7}{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(3/2)/(d*x+c)**(7/6),x)

[Out]

Integral((a + b*x)**(3/2)/(c + d*x)**(7/6), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{3}{2}}}{{\left (d x + c\right )}^{\frac{7}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)/(d*x + c)^(7/6),x, algorithm="giac")

[Out]

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