3.1432 \(\int \frac{(A+B x) (a+c x^2)}{(d+e x)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0501356, 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} \[ \frac{2 \sqrt{d+e x} \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4}+\frac{2 \left (a e^2+c d^2\right ) (B d-A e)}{e^4 \sqrt{d+e x}}-\frac{2 c (d+e x)^{3/2} (3 B d-A e)}{3 e^4}+\frac{2 B c (d+e x)^{5/2}}{5 e^4} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2))/(d + e*x)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.076224, size = 97, normalized size = 0.87 \[ \frac{6 B \left (5 a e^2 (2 d+e x)+c \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )\right )-10 A e \left (3 a e^2+c \left (8 d^2+4 d e x-e^2 x^2\right )\right )}{15 e^4 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2))/(d + e*x)^(3/2),x]

[Out]

(-10*A*e*(3*a*e^2 + c*(8*d^2 + 4*d*e*x - e^2*x^2)) + 6*B*(5*a*e^2*(2*d + e*x) + c*(16*d^3 + 8*d^2*e*x - 2*d*e^
2*x^2 + e^3*x^3)))/(15*e^4*Sqrt[d + e*x])

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Maple [A]  time = 0.004, size = 101, normalized size = 0.9 \begin{align*} -{\frac{-6\,Bc{x}^{3}{e}^{3}-10\,Ac{e}^{3}{x}^{2}+12\,Bcd{e}^{2}{x}^{2}+40\,Acd{e}^{2}x-30\,Ba{e}^{3}x-48\,Bc{d}^{2}ex+30\,aA{e}^{3}+80\,Ac{d}^{2}e-60\,aBd{e}^{2}-96\,Bc{d}^{3}}{15\,{e}^{4}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)/(e*x+d)^(3/2),x)

[Out]

-2/15/(e*x+d)^(1/2)*(-3*B*c*e^3*x^3-5*A*c*e^3*x^2+6*B*c*d*e^2*x^2+20*A*c*d*e^2*x-15*B*a*e^3*x-24*B*c*d^2*e*x+1
5*A*a*e^3+40*A*c*d^2*e-30*B*a*d*e^2-48*B*c*d^3)/e^4

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Maxima [A]  time = 1.02192, size = 151, normalized size = 1.35 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} B c - 5 \,{\left (3 \, B c d - A c e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15 \,{\left (3 \, B c d^{2} - 2 \, A c d e + B a e^{2}\right )} \sqrt{e x + d}}{e^{3}} + \frac{15 \,{\left (B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3}\right )}}{\sqrt{e x + d} 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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.651, size = 252, normalized size = 2.25 \begin{align*} \frac{2 \,{\left (3 \, B c e^{3} x^{3} + 48 \, B c d^{3} - 40 \, A c d^{2} e + 30 \, B a d e^{2} - 15 \, A a e^{3} -{\left (6 \, B c d e^{2} - 5 \, A c e^{3}\right )} x^{2} +{\left (24 \, B c d^{2} e - 20 \, A c d e^{2} + 15 \, B a e^{3}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{5} x + d 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)^(3/2),x, algorithm="fricas")

[Out]

2/15*(3*B*c*e^3*x^3 + 48*B*c*d^3 - 40*A*c*d^2*e + 30*B*a*d*e^2 - 15*A*a*e^3 - (6*B*c*d*e^2 - 5*A*c*e^3)*x^2 +
(24*B*c*d^2*e - 20*A*c*d*e^2 + 15*B*a*e^3)*x)*sqrt(e*x + d)/(e^5*x + d*e^4)

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Sympy [A]  time = 15.101, size = 112, normalized size = 1. \begin{align*} \frac{2 B c \left (d + e x\right )^{\frac{5}{2}}}{5 e^{4}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (2 A c e - 6 B c d\right )}{3 e^{4}} + \frac{\sqrt{d + e x} \left (- 4 A c d e + 2 B a e^{2} + 6 B c d^{2}\right )}{e^{4}} + \frac{2 \left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )}{e^{4} \sqrt{d + e x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)/(e*x+d)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.143, size = 182, normalized size = 1.62 \begin{align*} \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B c e^{16} - 15 \,{\left (x e + d\right )}^{\frac{3}{2}} B c d e^{16} + 45 \, \sqrt{x e + d} B c d^{2} e^{16} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A c e^{17} - 30 \, \sqrt{x e + d} A c d e^{17} + 15 \, \sqrt{x e + d} B a e^{18}\right )} e^{\left (-20\right )} + \frac{2 \,{\left (B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3}\right )} e^{\left (-4\right )}}{\sqrt{x e + d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

2/15*(3*(x*e + d)^(5/2)*B*c*e^16 - 15*(x*e + d)^(3/2)*B*c*d*e^16 + 45*sqrt(x*e + d)*B*c*d^2*e^16 + 5*(x*e + d)
^(3/2)*A*c*e^17 - 30*sqrt(x*e + d)*A*c*d*e^17 + 15*sqrt(x*e + d)*B*a*e^18)*e^(-20) + 2*(B*c*d^3 - A*c*d^2*e +
B*a*d*e^2 - A*a*e^3)*e^(-4)/sqrt(x*e + d)