3.1575 \(\int \frac{1}{(a+b x)^{5/2} (c+d x)^{2/3}} \, dx\)

Optimal. Leaf size=421 \[ -\frac{14 \sqrt{2-\sqrt{3}} d \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right )}{9 \sqrt [4]{3} \sqrt [3]{b} \sqrt{a+b x} (b c-a d)^2 \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{14 d \sqrt [3]{c+d x}}{9 \sqrt{a+b x} (b c-a d)^2}-\frac{2 \sqrt [3]{c+d x}}{3 (a+b x)^{3/2} (b c-a d)} \]

[Out]

(-2*(c + d*x)^(1/3))/(3*(b*c - a*d)*(a + b*x)^(3/2)) + (14*d*(c + d*x)^(1/3))/(9
*(b*c - a*d)^2*Sqrt[a + b*x]) - (14*Sqrt[2 - Sqrt[3]]*d*((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))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*
(c + d*x)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*
(c + d*x)^(1/3))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))], -
7 + 4*Sqrt[3]])/(9*3^(1/4)*b^(1/3)*(b*c - a*d)^2*Sqrt[a + b*x]*Sqrt[-(((b*c - a*
d)^(1/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((1 - Sqrt[3])*(b*c - a*
d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))^2)])

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Rubi [A]  time = 0.719291, antiderivative size = 421, 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 \[ -\frac{14 \sqrt{2-\sqrt{3}} d \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right )}{9 \sqrt [4]{3} \sqrt [3]{b} \sqrt{a+b x} (b c-a d)^2 \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{14 d \sqrt [3]{c+d x}}{9 \sqrt{a+b x} (b c-a d)^2}-\frac{2 \sqrt [3]{c+d x}}{3 (a+b x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(5/2)*(c + d*x)^(2/3)),x]

[Out]

(-2*(c + d*x)^(1/3))/(3*(b*c - a*d)*(a + b*x)^(3/2)) + (14*d*(c + d*x)^(1/3))/(9
*(b*c - a*d)^2*Sqrt[a + b*x]) - (14*Sqrt[2 - Sqrt[3]]*d*((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))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*
(c + d*x)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*
(c + d*x)^(1/3))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))], -
7 + 4*Sqrt[3]])/(9*3^(1/4)*b^(1/3)*(b*c - a*d)^2*Sqrt[a + b*x]*Sqrt[-(((b*c - a*
d)^(1/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((1 - Sqrt[3])*(b*c - a*
d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))^2)])

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Rubi in Sympy [A]  time = 36.9857, size = 359, normalized size = 0.85 \[ \frac{14 d \sqrt [3]{c + d x}}{9 \sqrt{a + b x} \left (a d - b c\right )^{2}} + \frac{2 \sqrt [3]{c + d x}}{3 \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{14 \cdot 3^{\frac{3}{4}} d \sqrt{\frac{b^{\frac{2}{3}} \left (c + d x\right )^{\frac{2}{3}} - \sqrt [3]{b} \sqrt [3]{c + d x} \sqrt [3]{a d - b c} + \left (a d - b c\right )^{\frac{2}{3}}}{\left (\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x} - \left (-1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}}{\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{27 \sqrt [3]{b} \sqrt{\frac{\sqrt [3]{a d - b c} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )}{\left (\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}\right )^{2}}} \left (a d - b c\right )^{2} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(5/2)/(d*x+c)**(2/3),x)

[Out]

14*d*(c + d*x)**(1/3)/(9*sqrt(a + b*x)*(a*d - b*c)**2) + 2*(c + d*x)**(1/3)/(3*(
a + b*x)**(3/2)*(a*d - b*c)) + 14*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)*(c +
d*x)**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(1/3))**2)*sqrt(sqrt(3) + 2)*(b**(1/3)
*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))*elliptic_f(asin((b**(1/3)*(c + d*x)**(1/
3) - (-1 + sqrt(3))*(a*d - b*c)**(1/3))/(b**(1/3)*(c + d*x)**(1/3) + (1 + sqrt(3
))*(a*d - b*c)**(1/3))), -7 - 4*sqrt(3))/(27*b**(1/3)*sqrt((a*d - b*c)**(1/3)*(b
**(1/3)*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))/(b**(1/3)*(c + d*x)**(1/3) + (1 +
sqrt(3))*(a*d - b*c)**(1/3))**2)*(a*d - b*c)**2*sqrt(a - b*c/d + b*(c + d*x)/d)
)

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Mathematica [C]  time = 0.2031, size = 102, normalized size = 0.24 \[ \frac{\sqrt [3]{c+d x} \left (7 d (a+b x) \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\frac{b (c+d x)}{b c-a d}\right )+20 a d-6 b c+14 b d x\right )}{9 (a+b x)^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(5/2)*(c + d*x)^(2/3)),x]

[Out]

((c + d*x)^(1/3)*(-6*b*c + 20*a*d + 14*b*d*x + 7*d*(a + b*x)*Sqrt[(d*(a + b*x))/
(-(b*c) + a*d)]*Hypergeometric2F1[1/3, 1/2, 4/3, (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 = 0.089, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{5}{2}}} \left ( dx+c \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(5/2)/(d*x+c)^(2/3),x)

[Out]

int(1/(b*x+a)^(5/2)/(d*x+c)^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{2}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(2/3)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(5/2)*(d*x + c)^(2/3)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b x + a}{\left (d x + c\right )}^{\frac{2}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(2/3)),x, algorithm="fricas")

[Out]

integral(1/((b^2*x^2 + 2*a*b*x + a^2)*sqrt(b*x + a)*(d*x + c)^(2/3)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(5/2)/(d*x+c)**(2/3),x)

[Out]

Integral(1/((a + b*x)**(5/2)*(c + d*x)**(2/3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{2}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(2/3)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(5/2)*(d*x + c)^(2/3)), x)