3.1434 \(\int \frac{(A+B x) (a+c x^2)}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=112 \[ -\frac{2 \left (a B e^2-2 A c d e+3 B c d^2\right )}{3 e^4 (d+e x)^{3/2}}+\frac{2 \left (a e^2+c d^2\right ) (B d-A e)}{5 e^4 (d+e x)^{5/2}}+\frac{2 c (3 B d-A e)}{e^4 \sqrt{d+e x}}+\frac{2 B c \sqrt{d+e x}}{e^4} \]

[Out]

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

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Rubi [A]  time = 0.05017, 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} \[ -\frac{2 \left (a B e^2-2 A c d e+3 B c d^2\right )}{3 e^4 (d+e x)^{3/2}}+\frac{2 \left (a e^2+c d^2\right ) (B d-A e)}{5 e^4 (d+e x)^{5/2}}+\frac{2 c (3 B d-A e)}{e^4 \sqrt{d+e x}}+\frac{2 B c \sqrt{d+e x}}{e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0801244, size = 95, normalized size = 0.85 \[ -\frac{2 \left (3 a A e^3+a B e^2 (2 d+5 e x)+A c e \left (8 d^2+20 d e x+15 e^2 x^2\right )-3 B c \left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )\right )}{15 e^4 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(3*a*A*e^3 + a*B*e^2*(2*d + 5*e*x) + A*c*e*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - 3*B*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 = 101, normalized size = 0.9 \begin{align*} -{\frac{-30\,Bc{x}^{3}{e}^{3}+30\,Ac{e}^{3}{x}^{2}-180\,Bcd{e}^{2}{x}^{2}+40\,Acd{e}^{2}x+10\,Ba{e}^{3}x-240\,Bc{d}^{2}ex+6\,aA{e}^{3}+16\,Ac{d}^{2}e+4\,aBd{e}^{2}-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+a)/(e*x+d)^(7/2),x)

[Out]

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

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Maxima [A]  time = 1.01926, size = 147, normalized size = 1.31 \begin{align*} \frac{2 \,{\left (\frac{15 \, \sqrt{e x + d} B c}{e^{3}} + \frac{3 \, B c d^{3} - 3 \, A c d^{2} e + 3 \, B a d e^{2} - 3 \, A a e^{3} + 15 \,{\left (3 \, B c d - A c e\right )}{\left (e x + d\right )}^{2} - 5 \,{\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{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+a)/(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*c*d^2*e + 3*B*a*d*e^2 - 3*A*a*e^3 + 15*(3*B*c*d - A*c*e)*(e*
x + d)^2 - 5*(3*B*c*d^2 - 2*A*c*d*e + B*a*e^2)*(e*x + d))/((e*x + d)^(5/2)*e^3))/e

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Fricas [A]  time = 1.46523, size = 292, normalized size = 2.61 \begin{align*} \frac{2 \,{\left (15 \, B c e^{3} x^{3} + 48 \, B c d^{3} - 8 \, A c d^{2} e - 2 \, B a d e^{2} - 3 \, A a e^{3} + 15 \,{\left (6 \, B c d e^{2} - A c e^{3}\right )} x^{2} + 5 \,{\left (24 \, B c d^{2} e - 4 \, A c d e^{2} - B a e^{3}\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+a)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

2/15*(15*B*c*e^3*x^3 + 48*B*c*d^3 - 8*A*c*d^2*e - 2*B*a*d*e^2 - 3*A*a*e^3 + 15*(6*B*c*d*e^2 - A*c*e^3)*x^2 + 5
*(24*B*c*d^2*e - 4*A*c*d*e^2 - B*a*e^3)*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 = 3.38702, size = 653, normalized size = 5.83 \begin{align*} \begin{cases} - \frac{6 A a e^{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{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{4 B a 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 B a 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{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{A a x + \frac{A c x^{3}}{3} + \frac{B a x^{2}}{2} + \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+a)/(e*x+d)**(7/2),x)

[Out]

Piecewise((-6*A*a*e**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)) -
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/(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)) - 4*B*a*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*B*a*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)) + 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)) + 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*sqrt(d + e*x) + 15*e**6*x**2*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**(7/2), True))

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Giac [A]  time = 1.17291, size = 165, normalized size = 1.47 \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} A c e + 10 \,{\left (x e + d\right )} A c d e - 3 \, A c d^{2} e - 5 \,{\left (x e + d\right )} B a e^{2} + 3 \, B a d e^{2} - 3 \, A a e^{3}\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+a)/(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*A*
c*e + 10*(x*e + d)*A*c*d*e - 3*A*c*d^2*e - 5*(x*e + d)*B*a*e^2 + 3*B*a*d*e^2 - 3*A*a*e^3)*e^(-4)/(x*e + d)^(5/
2)