3.775 \(\int (d+e x)^m (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx\)

Optimal. Leaf size=105 \[ \frac{2 (f+g x)^{5/2} (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \left (-\frac{g (a e+c d x)}{c d f-a e g}\right )^m \, _2F_1\left (\frac{5}{2},m;\frac{7}{2};\frac{c d (f+g x)}{c d f-a e g}\right )}{5 g} \]

[Out]

(2*(-((g*(a*e + c*d*x))/(c*d*f - a*e*g)))^m*(d + e*x)^m*(f + g*x)^(5/2)*Hypergeo
metric2F1[5/2, m, 7/2, (c*d*(f + g*x))/(c*d*f - a*e*g)])/(5*g*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^m)

_______________________________________________________________________________________

Rubi [A]  time = 0.21392, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{2 (f+g x)^{5/2} (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \left (-\frac{g (a e+c d x)}{c d f-a e g}\right )^m \, _2F_1\left (\frac{5}{2},m;\frac{7}{2};\frac{c d (f+g x)}{c d f-a e g}\right )}{5 g} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^m*(f + g*x)^(3/2))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m,x]

[Out]

(2*(-((g*(a*e + c*d*x))/(c*d*f - a*e*g)))^m*(d + e*x)^m*(f + g*x)^(5/2)*Hypergeo
metric2F1[5/2, m, 7/2, (c*d*(f + g*x))/(c*d*f - a*e*g)])/(5*g*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^m)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 46.4216, size = 94, normalized size = 0.9 \[ \frac{2 \left (\frac{g \left (a e + c d x\right )}{a e g - c d f}\right )^{m} \left (d + e x\right )^{m} \left (f + g x\right )^{\frac{5}{2}} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} m, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{c d \left (- f - g x\right )}{a e g - c d f}} \right )}}{5 g} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m*(g*x+f)**(3/2)/((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**m),x)

[Out]

2*(g*(a*e + c*d*x)/(a*e*g - c*d*f))**m*(d + e*x)**m*(f + g*x)**(5/2)*(a*d*e + c*
d*e*x**2 + x*(a*e**2 + c*d**2))**(-m)*hyper((m, 5/2), (7/2,), c*d*(-f - g*x)/(a*
e*g - c*d*f))/(5*g)

_______________________________________________________________________________________

Mathematica [C]  time = 0.62115, size = 234, normalized size = 2.23 \[ \frac{f \sqrt{f+g x} (d+e x)^m ((d+e x) (a e+c d x))^{-m} \left (\frac{9 a e g^2 x^2 F_1\left (2;m,-\frac{1}{2};3;-\frac{c d x}{a e},-\frac{g x}{f}\right )}{6 a e f F_1\left (2;m,-\frac{1}{2};3;-\frac{c d x}{a e},-\frac{g x}{f}\right )+a e g x F_1\left (3;m,\frac{1}{2};4;-\frac{c d x}{a e},-\frac{g x}{f}\right )-2 c d f m x F_1\left (3;m+1,-\frac{1}{2};4;-\frac{c d x}{a e},-\frac{g x}{f}\right )}+2 (f+g x) \left (\frac{g (a e+c d x)}{a e g-c d f}\right )^m \, _2F_1\left (\frac{3}{2},m;\frac{5}{2};\frac{c d (f+g x)}{c d f-a e g}\right )\right )}{3 g} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((d + e*x)^m*(f + g*x)^(3/2))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m,x]

[Out]

(f*(d + e*x)^m*Sqrt[f + g*x]*((9*a*e*g^2*x^2*AppellF1[2, m, -1/2, 3, -((c*d*x)/(
a*e)), -((g*x)/f)])/(6*a*e*f*AppellF1[2, m, -1/2, 3, -((c*d*x)/(a*e)), -((g*x)/f
)] + a*e*g*x*AppellF1[3, m, 1/2, 4, -((c*d*x)/(a*e)), -((g*x)/f)] - 2*c*d*f*m*x*
AppellF1[3, 1 + m, -1/2, 4, -((c*d*x)/(a*e)), -((g*x)/f)]) + 2*((g*(a*e + c*d*x)
)/(-(c*d*f) + a*e*g))^m*(f + g*x)*Hypergeometric2F1[3/2, m, 5/2, (c*d*(f + g*x))
/(c*d*f - a*e*g)]))/(3*g*((a*e + c*d*x)*(d + e*x))^m)

_______________________________________________________________________________________

Maple [F]  time = 0.112, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{m}} \left ( gx+f \right ) ^{{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m*(g*x+f)^(3/2)/((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^m),x)

[Out]

int((e*x+d)^m*(g*x+f)^(3/2)/((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^m),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (g x + f\right )}^{\frac{3}{2}}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{-m}{\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)^(3/2)*(e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^m,x, algorithm="maxima")

[Out]

integrate((g*x + f)^(3/2)*(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(-m)*(e*x + d)
^m, x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)^(3/2)*(e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^m,x, algorithm="fricas")

[Out]

integral((g*x + f)^(3/2)*(e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^m,
x)

_______________________________________________________________________________________

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((e*x+d)**m*(g*x+f)**(3/2)/((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**m),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)^(3/2)*(e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^m,x, algorithm="giac")

[Out]

integrate((g*x + f)^(3/2)*(e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^m,
x)