3.1433 \(\int \frac{(A+B x) (a+c x^2)}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=112 \[ -\frac{2 \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4 \sqrt{d+e x}}+\frac{2 \left (a e^2+c d^2\right ) (B d-A e)}{3 e^4 (d+e x)^{3/2}}-\frac{2 c \sqrt{d+e x} (3 B d-A e)}{e^4}+\frac{2 B c (d+e x)^{3/2}}{3 e^4} \]

[Out]

(2*(B*d - A*e)*(c*d^2 + a*e^2))/(3*e^4*(d + e*x)^(3/2)) - (2*(3*B*c*d^2 - 2*A*c*d*e + a*B*e^2))/(e^4*Sqrt[d +
e*x]) - (2*c*(3*B*d - A*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 = 0.0496668, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {772} \[ -\frac{2 \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4 \sqrt{d+e x}}+\frac{2 \left (a e^2+c d^2\right ) (B d-A e)}{3 e^4 (d+e x)^{3/2}}-\frac{2 c \sqrt{d+e x} (3 B d-A e)}{e^4}+\frac{2 B c (d+e x)^{3/2}}{3 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0779698, size = 94, normalized size = 0.84 \[ -\frac{2 \left (a A e^3+a B e^2 (2 d+3 e x)-A c e \left (8 d^2+12 d e x+3 e^2 x^2\right )+B c \left (24 d^2 e x+16 d^3+6 d e^2 x^2-e^3 x^3\right )\right )}{3 e^4 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a*A*e^3 + a*B*e^2*(2*d + 3*e*x) - A*c*e*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + B*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 = 0.003, size = 100, normalized size = 0.9 \begin{align*} -{\frac{-2\,Bc{x}^{3}{e}^{3}-6\,Ac{e}^{3}{x}^{2}+12\,Bcd{e}^{2}{x}^{2}-24\,Acd{e}^{2}x+6\,Ba{e}^{3}x+48\,Bc{d}^{2}ex+2\,aA{e}^{3}-16\,Ac{d}^{2}e+4\,aBd{e}^{2}+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+a)/(e*x+d)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.02886, size = 146, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} B c - 3 \,{\left (3 \, B c d - A c e\right )} \sqrt{e x + d}}{e^{3}} + \frac{B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3} - 3 \,{\left (3 \, B c d^{2} - 2 \, A c d e + B a e^{2}\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+a)/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.47714, size = 259, normalized size = 2.31 \begin{align*} \frac{2 \,{\left (B c e^{3} x^{3} - 16 \, B c d^{3} + 8 \, A c d^{2} e - 2 \, B a d e^{2} - A a e^{3} - 3 \,{\left (2 \, B c d e^{2} - A c e^{3}\right )} x^{2} - 3 \,{\left (8 \, B c d^{2} e - 4 \, A c d e^{2} + B a e^{3}\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+a)/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/3*(B*c*e^3*x^3 - 16*B*c*d^3 + 8*A*c*d^2*e - 2*B*a*d*e^2 - A*a*e^3 - 3*(2*B*c*d*e^2 - A*c*e^3)*x^2 - 3*(8*B*c
*d^2*e - 4*A*c*d*e^2 + B*a*e^3)*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 = 1.51504, size = 449, normalized size = 4.01 \begin{align*} \begin{cases} - \frac{2 A a e^{3}}{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{4 B a d e^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{6 B a e^{3} x}{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{A a x + \frac{A c x^{3}}{3} + \frac{B a x^{2}}{2} + \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+a)/(e*x+d)**(5/2),x)

[Out]

Piecewise((-2*A*a*e**3/(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*sqrt(d + e*x)) - 4*B*a*d*e**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(
d + e*x)) - 6*B*a*e**3*x/(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*a*x + A*c*x**3/3 + B*a*x**2/2 + B*c*x**4/4)/d**(5/2), True))

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Giac [A]  time = 1.16531, size = 170, normalized size = 1.52 \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} 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 )} A c d e + A c d^{2} e + 3 \,{\left (x e + d\right )} B a e^{2} - B a d e^{2} + A a e^{3}\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+a)/(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)*A*c*e^9)*e^(-12) - 2/3*(9*(x*e + d)
*B*c*d^2 - B*c*d^3 - 6*(x*e + d)*A*c*d*e + A*c*d^2*e + 3*(x*e + d)*B*a*e^2 - B*a*d*e^2 + A*a*e^3)*e^(-4)/(x*e
+ d)^(3/2)