3.1220 \(\int \frac{(A+B x) (b x+c x^2)}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=122 \[ \frac{2 (-A c e-b B e+3 B c d)}{e^4 \sqrt{d+e x}}-\frac{2 (B d (3 c d-2 b e)-A e (2 c d-b e))}{3 e^4 (d+e x)^{3/2}}+\frac{2 d (B d-A e) (c d-b e)}{5 e^4 (d+e x)^{5/2}}+\frac{2 B c \sqrt{d+e x}}{e^4} \]

[Out]

(2*d*(B*d - A*e)*(c*d - b*e))/(5*e^4*(d + e*x)^(5/2)) - (2*(B*d*(3*c*d - 2*b*e) - A*e*(2*c*d - b*e)))/(3*e^4*(
d + e*x)^(3/2)) + (2*(3*B*c*d - b*B*e - A*c*e))/(e^4*Sqrt[d + e*x]) + (2*B*c*Sqrt[d + e*x])/e^4

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Rubi [A]  time = 0.074826, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ \frac{2 (-A c e-b B e+3 B c d)}{e^4 \sqrt{d+e x}}-\frac{2 (B d (3 c d-2 b e)-A e (2 c d-b e))}{3 e^4 (d+e x)^{3/2}}+\frac{2 d (B d-A e) (c d-b e)}{5 e^4 (d+e x)^{5/2}}+\frac{2 B c \sqrt{d+e x}}{e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d*(B*d - A*e)*(c*d - b*e))/(5*e^4*(d + e*x)^(5/2)) - (2*(B*d*(3*c*d - 2*b*e) - A*e*(2*c*d - b*e)))/(3*e^4*(
d + e*x)^(3/2)) + (2*(3*B*c*d - b*B*e - A*c*e))/(e^4*Sqrt[d + e*x]) + (2*B*c*Sqrt[d + e*x])/e^4

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )}{(d+e x)^{7/2}} \, dx &=\int \left (-\frac{d (B d-A e) (c d-b e)}{e^3 (d+e x)^{7/2}}+\frac{B d (3 c d-2 b e)-A e (2 c d-b e)}{e^3 (d+e x)^{5/2}}+\frac{-3 B c d+b B e+A c e}{e^3 (d+e x)^{3/2}}+\frac{B c}{e^3 \sqrt{d+e x}}\right ) \, dx\\ &=\frac{2 d (B d-A e) (c d-b e)}{5 e^4 (d+e x)^{5/2}}-\frac{2 (B d (3 c d-2 b e)-A e (2 c d-b e))}{3 e^4 (d+e x)^{3/2}}+\frac{2 (3 B c d-b B e-A c e)}{e^4 \sqrt{d+e x}}+\frac{2 B c \sqrt{d+e x}}{e^4}\\ \end{align*}

Mathematica [A]  time = 0.0503867, size = 111, normalized size = 0.91 \[ -\frac{2 \left (A e \left (b e (2 d+5 e x)+c \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )+B \left (b e \left (8 d^2+20 d e x+15 e^2 x^2\right )-3 c \left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )\right )\right )}{15 e^4 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(A*e*(b*e*(2*d + 5*e*x) + c*(8*d^2 + 20*d*e*x + 15*e^2*x^2)) + B*(b*e*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - 3*
c*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3))))/(15*e^4*(d + e*x)^(5/2))

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Maple [A]  time = 0.006, size = 121, normalized size = 1. \begin{align*} -{\frac{-30\,Bc{x}^{3}{e}^{3}+30\,Ac{e}^{3}{x}^{2}+30\,Bb{e}^{3}{x}^{2}-180\,Bcd{e}^{2}{x}^{2}+10\,Ab{e}^{3}x+40\,Acd{e}^{2}x+40\,Bbd{e}^{2}x-240\,Bc{d}^{2}ex+4\,Abd{e}^{2}+16\,Ac{d}^{2}e+16\,Bb{d}^{2}e-96\,Bc{d}^{3}}{15\,{e}^{4}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)/(e*x+d)^(7/2),x)

[Out]

-2/15*(-15*B*c*e^3*x^3+15*A*c*e^3*x^2+15*B*b*e^3*x^2-90*B*c*d*e^2*x^2+5*A*b*e^3*x+20*A*c*d*e^2*x+20*B*b*d*e^2*
x-120*B*c*d^2*e*x+2*A*b*d*e^2+8*A*c*d^2*e+8*B*b*d^2*e-48*B*c*d^3)/(e*x+d)^(5/2)/e^4

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Maxima [A]  time = 1.05007, size = 158, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (\frac{15 \, \sqrt{e x + d} B c}{e^{3}} + \frac{3 \, B c d^{3} + 3 \, A b d e^{2} - 3 \,{\left (B b + A c\right )} d^{2} e + 15 \,{\left (3 \, B c d -{\left (B b + A c\right )} e\right )}{\left (e x + d\right )}^{2} - 5 \,{\left (3 \, B c d^{2} + A b e^{2} - 2 \,{\left (B b + A c\right )} d e\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{3}}\right )}}{15 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.75838, size = 308, normalized size = 2.52 \begin{align*} \frac{2 \,{\left (15 \, B c e^{3} x^{3} + 48 \, B c d^{3} - 2 \, A b d e^{2} - 8 \,{\left (B b + A c\right )} d^{2} e + 15 \,{\left (6 \, B c d e^{2} -{\left (B b + A c\right )} e^{3}\right )} x^{2} + 5 \,{\left (24 \, B c d^{2} e - A b e^{3} - 4 \,{\left (B b + A c\right )} d e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

2/15*(15*B*c*e^3*x^3 + 48*B*c*d^3 - 2*A*b*d*e^2 - 8*(B*b + A*c)*d^2*e + 15*(6*B*c*d*e^2 - (B*b + A*c)*e^3)*x^2
+ 5*(24*B*c*d^2*e - A*b*e^3 - 4*(B*b + A*c)*d*e^2)*x)*sqrt(e*x + d)/(e^7*x^3 + 3*d*e^6*x^2 + 3*d^2*e^5*x + d^
3*e^4)

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Sympy [A]  time = 4.34298, size = 784, normalized size = 6.43 \begin{align*} \begin{cases} - \frac{4 A b d e^{2}}{15 d^{2} e^{4} \sqrt{d + e x} + 30 d e^{5} x \sqrt{d + e x} + 15 e^{6} x^{2} \sqrt{d + e x}} - \frac{10 A b e^{3} x}{15 d^{2} e^{4} \sqrt{d + e x} + 30 d e^{5} x \sqrt{d + e x} + 15 e^{6} x^{2} \sqrt{d + e x}} - \frac{16 A c d^{2} e}{15 d^{2} e^{4} \sqrt{d + e x} + 30 d e^{5} x \sqrt{d + e x} + 15 e^{6} x^{2} \sqrt{d + e x}} - \frac{40 A c d e^{2} x}{15 d^{2} e^{4} \sqrt{d + e x} + 30 d e^{5} x \sqrt{d + e x} + 15 e^{6} x^{2} \sqrt{d + e x}} - \frac{30 A c e^{3} x^{2}}{15 d^{2} e^{4} \sqrt{d + e x} + 30 d e^{5} x \sqrt{d + e x} + 15 e^{6} x^{2} \sqrt{d + e x}} - \frac{16 B b d^{2} e}{15 d^{2} e^{4} \sqrt{d + e x} + 30 d e^{5} x \sqrt{d + e x} + 15 e^{6} x^{2} \sqrt{d + e x}} - \frac{40 B b d e^{2} x}{15 d^{2} e^{4} \sqrt{d + e x} + 30 d e^{5} x \sqrt{d + e x} + 15 e^{6} x^{2} \sqrt{d + e x}} - \frac{30 B b e^{3} x^{2}}{15 d^{2} e^{4} \sqrt{d + e x} + 30 d e^{5} x \sqrt{d + e x} + 15 e^{6} x^{2} \sqrt{d + e x}} + \frac{96 B c d^{3}}{15 d^{2} e^{4} \sqrt{d + e x} + 30 d e^{5} x \sqrt{d + e x} + 15 e^{6} x^{2} \sqrt{d + e x}} + \frac{240 B c d^{2} e x}{15 d^{2} e^{4} \sqrt{d + e x} + 30 d e^{5} x \sqrt{d + e x} + 15 e^{6} x^{2} \sqrt{d + e x}} + \frac{180 B c d e^{2} x^{2}}{15 d^{2} e^{4} \sqrt{d + e x} + 30 d e^{5} x \sqrt{d + e x} + 15 e^{6} x^{2} \sqrt{d + e x}} + \frac{30 B c e^{3} x^{3}}{15 d^{2} e^{4} \sqrt{d + e x} + 30 d e^{5} x \sqrt{d + e x} + 15 e^{6} x^{2} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{\frac{A b x^{2}}{2} + \frac{A c x^{3}}{3} + \frac{B b x^{3}}{3} + \frac{B c x^{4}}{4}}{d^{\frac{7}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)/(e*x+d)**(7/2),x)

[Out]

Piecewise((-4*A*b*d*e**2/(15*d**2*e**4*sqrt(d + e*x) + 30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x))
- 10*A*b*e**3*x/(15*d**2*e**4*sqrt(d + e*x) + 30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) - 16*A*
c*d**2*e/(15*d**2*e**4*sqrt(d + e*x) + 30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) - 40*A*c*d*e**2
*x/(15*d**2*e**4*sqrt(d + e*x) + 30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) - 30*A*c*e**3*x**2/(1
5*d**2*e**4*sqrt(d + e*x) + 30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) - 16*B*b*d**2*e/(15*d**2*e
**4*sqrt(d + e*x) + 30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) - 40*B*b*d*e**2*x/(15*d**2*e**4*sq
rt(d + e*x) + 30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) - 30*B*b*e**3*x**2/(15*d**2*e**4*sqrt(d
+ e*x) + 30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) + 96*B*c*d**3/(15*d**2*e**4*sqrt(d + e*x) + 3
0*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) + 240*B*c*d**2*e*x/(15*d**2*e**4*sqrt(d + e*x) + 30*d*e
**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) + 180*B*c*d*e**2*x**2/(15*d**2*e**4*sqrt(d + e*x) + 30*d*e**
5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) + 30*B*c*e**3*x**3/(15*d**2*e**4*sqrt(d + e*x) + 30*d*e**5*x*s
qrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)), Ne(e, 0)), ((A*b*x**2/2 + A*c*x**3/3 + B*b*x**3/3 + B*c*x**4/4)/d*
*(7/2), True))

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Giac [A]  time = 1.25376, size = 205, normalized size = 1.68 \begin{align*} 2 \, \sqrt{x e + d} B c e^{\left (-4\right )} + \frac{2 \,{\left (45 \,{\left (x e + d\right )}^{2} B c d - 15 \,{\left (x e + d\right )} B c d^{2} + 3 \, B c d^{3} - 15 \,{\left (x e + d\right )}^{2} B b e - 15 \,{\left (x e + d\right )}^{2} A c e + 10 \,{\left (x e + d\right )} B b d e + 10 \,{\left (x e + d\right )} A c d e - 3 \, B b d^{2} e - 3 \, A c d^{2} e - 5 \,{\left (x e + d\right )} A b e^{2} + 3 \, A b d e^{2}\right )} e^{\left (-4\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*B*c*e^(-4) + 2/15*(45*(x*e + d)^2*B*c*d - 15*(x*e + d)*B*c*d^2 + 3*B*c*d^3 - 15*(x*e + d)^2*B*
b*e - 15*(x*e + d)^2*A*c*e + 10*(x*e + d)*B*b*d*e + 10*(x*e + d)*A*c*d*e - 3*B*b*d^2*e - 3*A*c*d^2*e - 5*(x*e
+ d)*A*b*e^2 + 3*A*b*d*e^2)*e^(-4)/(x*e + d)^(5/2)