3.344 \(\int \frac{b x+c x^2}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=66 \[ \frac{2 (2 c d-b e)}{3 e^3 (d+e x)^{3/2}}-\frac{2 d (c d-b e)}{5 e^3 (d+e x)^{5/2}}-\frac{2 c}{e^3 \sqrt{d+e x}} \]

[Out]

(-2*d*(c*d - b*e))/(5*e^3*(d + e*x)^(5/2)) + (2*(2*c*d - b*e))/(3*e^3*(d + e*x)^
(3/2)) - (2*c)/(e^3*Sqrt[d + e*x])

_______________________________________________________________________________________

Rubi [A]  time = 0.0934363, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 (2 c d-b e)}{3 e^3 (d+e x)^{3/2}}-\frac{2 d (c d-b e)}{5 e^3 (d+e x)^{5/2}}-\frac{2 c}{e^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*d*(c*d - b*e))/(5*e^3*(d + e*x)^(5/2)) + (2*(2*c*d - b*e))/(3*e^3*(d + e*x)^
(3/2)) - (2*c)/(e^3*Sqrt[d + e*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.5544, size = 61, normalized size = 0.92 \[ - \frac{2 c}{e^{3} \sqrt{d + e x}} + \frac{2 d \left (b e - c d\right )}{5 e^{3} \left (d + e x\right )^{\frac{5}{2}}} - \frac{2 \left (b e - 2 c d\right )}{3 e^{3} \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*c/(e**3*sqrt(d + e*x)) + 2*d*(b*e - c*d)/(5*e**3*(d + e*x)**(5/2)) - 2*(b*e -
2*c*d)/(3*e**3*(d + e*x)**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0515512, size = 49, normalized size = 0.74 \[ -\frac{2 \left (b e (2 d+5 e x)+c \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )}{15 e^3 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*e*(2*d + 5*e*x) + c*(8*d^2 + 20*d*e*x + 15*e^2*x^2)))/(15*e^3*(d + e*x)^(
5/2))

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 47, normalized size = 0.7 \[ -{\frac{30\,c{e}^{2}{x}^{2}+10\,b{e}^{2}x+40\,cdex+4\,bde+16\,c{d}^{2}}{15\,{e}^{3}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(15*c*e^2*x^2+5*b*e^2*x+20*c*d*e*x+2*b*d*e+8*c*d^2)/(e*x+d)^(5/2)/e^3

_______________________________________________________________________________________

Maxima [A]  time = 0.6865, size = 68, normalized size = 1.03 \[ -\frac{2 \,{\left (15 \,{\left (e x + d\right )}^{2} c + 3 \, c d^{2} - 3 \, b d e - 5 \,{\left (2 \, c d - b e\right )}{\left (e x + d\right )}\right )}}{15 \,{\left (e x + d\right )}^{\frac{5}{2}} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(15*(e*x + d)^2*c + 3*c*d^2 - 3*b*d*e - 5*(2*c*d - b*e)*(e*x + d))/((e*x +
d)^(5/2)*e^3)

_______________________________________________________________________________________

Fricas [A]  time = 0.2165, size = 92, normalized size = 1.39 \[ -\frac{2 \,{\left (15 \, c e^{2} x^{2} + 8 \, c d^{2} + 2 \, b d e + 5 \,{\left (4 \, c d e + b e^{2}\right )} x\right )}}{15 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(15*c*e^2*x^2 + 8*c*d^2 + 2*b*d*e + 5*(4*c*d*e + b*e^2)*x)/((e^5*x^2 + 2*d
*e^4*x + d^2*e^3)*sqrt(e*x + d))

_______________________________________________________________________________________

Sympy [A]  time = 8.66775, size = 314, normalized size = 4.76 \[ \begin{cases} - \frac{4 b d e}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{10 b e^{2} x}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{16 c d^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{40 c d e x}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{30 c e^{2} x^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{\frac{b x^{2}}{2} + \frac{c x^{3}}{3}}{d^{\frac{7}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-4*b*d*e/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15
*e**5*x**2*sqrt(d + e*x)) - 10*b*e**2*x/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*
x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 16*c*d**2/(15*d**2*e**3*sqrt(d +
e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 40*c*d*e*x/(15
*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x
)) - 30*c*e**2*x**2/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15
*e**5*x**2*sqrt(d + e*x)), Ne(e, 0)), ((b*x**2/2 + c*x**3/3)/d**(7/2), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.206421, size = 77, normalized size = 1.17 \[ -\frac{2 \,{\left (15 \,{\left (x e + d\right )}^{2} c - 10 \,{\left (x e + d\right )} c d + 3 \, c d^{2} + 5 \,{\left (x e + d\right )} b e - 3 \, b d e\right )} e^{\left (-3\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(15*(x*e + d)^2*c - 10*(x*e + d)*c*d + 3*c*d^2 + 5*(x*e + d)*b*e - 3*b*d*e
)*e^(-3)/(x*e + d)^(5/2)