3.1764 \(\int \frac{1}{\sqrt{a+b x} (c+d x)^{7/6}} \, dx\)

Optimal. Leaf size=817 \[ \frac{6 \sqrt [4]{3} \sqrt [3]{b} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{d (b c-a d)^{2/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{3^{3/4} \left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{d (b c-a d)^{2/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{6 \sqrt{a+b x}}{(b c-a d) \sqrt [6]{c+d x}}+\frac{6 \left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt{a+b x} \sqrt [6]{c+d x}}{(b c-a d) \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )} \]

[Out]

(6*Sqrt[a + b*x])/((b*c - a*d)*(c + d*x)^(1/6)) + (6*(1 + Sqrt[3])*b^(1/3)*Sqrt[
a + b*x]*(c + d*x)^(1/6))/((b*c - a*d)*((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3
)*(c + d*x)^(1/3))) + (6*3^(1/4)*b^(1/3)*(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])/(d*(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 + Sq
rt[3])*b^(1/3)*(c + d*x)^(1/3))^2)]) + (3^(3/4)*(1 - Sqrt[3])*b^(1/3)*(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])/(d*(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 = 1.42454, antiderivative size = 817, 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 \[ \frac{6 \sqrt [4]{3} \sqrt [3]{b} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{d (b c-a d)^{2/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{3^{3/4} \left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{d (b c-a d)^{2/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{6 \sqrt{a+b x}}{(b c-a d) \sqrt [6]{c+d x}}+\frac{6 \left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt{a+b x} \sqrt [6]{c+d x}}{(b c-a d) \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b*x]*(c + d*x)^(7/6)),x]

[Out]

(6*Sqrt[a + b*x])/((b*c - a*d)*(c + d*x)^(1/6)) + (6*(1 + Sqrt[3])*b^(1/3)*Sqrt[
a + b*x]*(c + d*x)^(1/6))/((b*c - a*d)*((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3
)*(c + d*x)^(1/3))) + (6*3^(1/4)*b^(1/3)*(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])/(d*(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 + Sq
rt[3])*b^(1/3)*(c + d*x)^(1/3))^2)]) + (3^(3/4)*(1 - Sqrt[3])*b^(1/3)*(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])/(d*(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 = 66.9928, size = 716, normalized size = 0.88 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(1/2)/(d*x+c)**(7/6),x)

[Out]

6*b**(1/3)*(1 + sqrt(3))*(c + d*x)**(1/6)*sqrt(a - b*c/d + b*(c + d*x)/d)/((a*d
- b*c)*(b**(1/3)*(1 + sqrt(3))*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))) - 6*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)*(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)/(d*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)) - 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)*(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
)/(d*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)) - 6*sqrt(a + b*x)/((c + d*x)**(1/6)*
(a*d - b*c))

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Mathematica [C]  time = 0.18217, size = 100, normalized size = 0.12 \[ \frac{6 \left (5 d (a+b x)-2 b (c+d x) \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\frac{b (c+d x)}{b c-a d}\right )\right )}{5 d \sqrt{a+b x} \sqrt [6]{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b*x]*(c + d*x)^(7/6)),x]

[Out]

(6*(5*d*(a + b*x) - 2*b*Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]*(c + d*x)*Hypergeomet
ric2F1[1/2, 5/6, 11/6, (b*(c + d*x))/(b*c - a*d)]))/(5*d*(b*c - a*d)*Sqrt[a + b*
x]*(c + d*x)^(1/6))

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt{bx+a}}} \left ( dx+c \right ) ^{-{\frac{7}{6}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(1/2)/(d*x+c)^(7/6),x)

[Out]

int(1/(b*x+a)^(1/2)/(d*x+c)^(7/6),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{7}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*(d*x + c)^(7/6)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x + a)*(d*x + c)^(7/6)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{7}{6}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*(d*x + c)^(7/6)),x, algorithm="fricas")

[Out]

integral(1/(sqrt(b*x + a)*(d*x + c)^(7/6)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + b x} \left (c + d x\right )^{\frac{7}{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(1/2)/(d*x+c)**(7/6),x)

[Out]

Integral(1/(sqrt(a + b*x)*(c + d*x)**(7/6)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*(d*x + c)^(7/6)),x, algorithm="giac")

[Out]

Timed out