3.1219 \(\int \frac{(A+B x) (b x+c x^2)}{(d+e x)^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0744253, 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} \[ -\frac{2 \sqrt{d+e x} (-A c e-b B e+3 B c d)}{e^4}-\frac{2 (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^4 \sqrt{d+e x}}+\frac{2 d (B d-A e) (c d-b e)}{3 e^4 (d+e x)^{3/2}}+\frac{2 B c (d+e x)^{3/2}}{3 e^4} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2))/(d + e*x)^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.0517101, size = 110, normalized size = 0.9 \[ \frac{2 \left (A e \left (c \left (8 d^2+12 d e x+3 e^2 x^2\right )-b e (2 d+3 e x)\right )+B \left (b e \left (8 d^2+12 d e x+3 e^2 x^2\right )+c \left (-24 d^2 e x-16 d^3-6 d e^2 x^2+e^3 x^3\right )\right )\right )}{3 e^4 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2))/(d + e*x)^(5/2),x]

[Out]

(2*(A*e*(-(b*e*(2*d + 3*e*x)) + c*(8*d^2 + 12*d*e*x + 3*e^2*x^2)) + B*(b*e*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + c*
(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3))))/(3*e^4*(d + e*x)^(3/2))

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Maple [A]  time = 0.006, size = 121, normalized size = 1. \begin{align*} -{\frac{-2\,Bc{x}^{3}{e}^{3}-6\,Ac{e}^{3}{x}^{2}-6\,Bb{e}^{3}{x}^{2}+12\,Bcd{e}^{2}{x}^{2}+6\,Ab{e}^{3}x-24\,Acd{e}^{2}x-24\,Bbd{e}^{2}x+48\,Bc{d}^{2}ex+4\,Abd{e}^{2}-16\,Ac{d}^{2}e-16\,Bb{d}^{2}e+32\,Bc{d}^{3}}{3\,{e}^{4}} \left ( ex+d \right ) ^{-{\frac{3}{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)^(5/2),x)

[Out]

-2/3*(-B*c*e^3*x^3-3*A*c*e^3*x^2-3*B*b*e^3*x^2+6*B*c*d*e^2*x^2+3*A*b*e^3*x-12*A*c*d*e^2*x-12*B*b*d*e^2*x+24*B*
c*d^2*e*x+2*A*b*d*e^2-8*A*c*d^2*e-8*B*b*d^2*e+16*B*c*d^3)/(e*x+d)^(3/2)/e^4

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Maxima [A]  time = 1.05466, size = 157, normalized size = 1.29 \begin{align*} \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} B c - 3 \,{\left (3 \, B c d -{\left (B b + A c\right )} e\right )} \sqrt{e x + d}}{e^{3}} + \frac{B c d^{3} + A b d e^{2} -{\left (B b + A c\right )} d^{2} e - 3 \,{\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{3}{2}} e^{3}}\right )}}{3 \, 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)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.69689, size = 278, normalized size = 2.28 \begin{align*} \frac{2 \,{\left (B c e^{3} x^{3} - 16 \, B c d^{3} - 2 \, A b d e^{2} + 8 \,{\left (B b + A c\right )} d^{2} e - 3 \,{\left (2 \, B c d e^{2} -{\left (B b + A c\right )} e^{3}\right )} x^{2} - 3 \,{\left (8 \, 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}}{3 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} 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)^(5/2),x, algorithm="fricas")

[Out]

2/3*(B*c*e^3*x^3 - 16*B*c*d^3 - 2*A*b*d*e^2 + 8*(B*b + A*c)*d^2*e - 3*(2*B*c*d*e^2 - (B*b + A*c)*e^3)*x^2 - 3*
(8*B*c*d^2*e + A*b*e^3 - 4*(B*b + A*c)*d*e^2)*x)*sqrt(e*x + d)/(e^6*x^2 + 2*d*e^5*x + d^2*e^4)

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Sympy [A]  time = 1.81655, size = 539, normalized size = 4.42 \begin{align*} \begin{cases} - \frac{4 A b d e^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{6 A b e^{3} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{16 A c d^{2} e}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{24 A c d e^{2} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{6 A c e^{3} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{16 B b d^{2} e}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{24 B b d e^{2} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{6 B b e^{3} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{32 B c d^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{48 B c d^{2} e x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{12 B c d e^{2} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{2 B c e^{3} x^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \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{5}{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)**(5/2),x)

[Out]

Piecewise((-4*A*b*d*e**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 6*A*b*e**3*x/(3*d*e**4*sqrt(d + e
*x) + 3*e**5*x*sqrt(d + e*x)) + 16*A*c*d**2*e/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 24*A*c*d*e**
2*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 6*A*c*e**3*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqr
t(d + e*x)) + 16*B*b*d**2*e/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 24*B*b*d*e**2*x/(3*d*e**4*sqrt
(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 6*B*b*e**3*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 32*B
*c*d**3/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 48*B*c*d**2*e*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x
*sqrt(d + e*x)) - 12*B*c*d*e**2*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 2*B*c*e**3*x**3/(3*d*
e**4*sqrt(d + e*x) + 3*e**5*x*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**(5/2), True))

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Giac [A]  time = 1.23992, size = 211, normalized size = 1.73 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B c e^{8} - 9 \, \sqrt{x e + d} B c d e^{8} + 3 \, \sqrt{x e + d} B b e^{9} + 3 \, \sqrt{x e + d} A c e^{9}\right )} e^{\left (-12\right )} - \frac{2 \,{\left (9 \,{\left (x e + d\right )} B c d^{2} - B c d^{3} - 6 \,{\left (x e + d\right )} B b d e - 6 \,{\left (x e + d\right )} A c d e + B b d^{2} e + A c d^{2} e + 3 \,{\left (x e + d\right )} A b e^{2} - A b d e^{2}\right )} e^{\left (-4\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{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)^(5/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*B*c*e^8 - 9*sqrt(x*e + d)*B*c*d*e^8 + 3*sqrt(x*e + d)*B*b*e^9 + 3*sqrt(x*e + d)*A*c*e^9)*
e^(-12) - 2/3*(9*(x*e + d)*B*c*d^2 - B*c*d^3 - 6*(x*e + d)*B*b*d*e - 6*(x*e + d)*A*c*d*e + B*b*d^2*e + A*c*d^2
*e + 3*(x*e + d)*A*b*e^2 - A*b*d*e^2)*e^(-4)/(x*e + d)^(3/2)