3.1218 \(\int \frac{(A+B x) (b x+c x^2)}{(d+e x)^{3/2}} \, dx\)

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

[Out]

(2*d*(B*d - A*e)*(c*d - b*e))/(e^4*Sqrt[d + e*x]) + (2*(B*d*(3*c*d - 2*b*e) - A*e*(2*c*d - b*e))*Sqrt[d + e*x]
)/e^4 - (2*(3*B*c*d - b*B*e - A*c*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 = 0.0759785, 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} \[ -\frac{2 (d+e x)^{3/2} (-A c e-b B e+3 B c d)}{3 e^4}+\frac{2 \sqrt{d+e x} (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^4}+\frac{2 d (B d-A e) (c d-b e)}{e^4 \sqrt{d+e x}}+\frac{2 B c (d+e x)^{5/2}}{5 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d*(B*d - A*e)*(c*d - b*e))/(e^4*Sqrt[d + e*x]) + (2*(B*d*(3*c*d - 2*b*e) - A*e*(2*c*d - b*e))*Sqrt[d + e*x]
)/e^4 - (2*(3*B*c*d - b*B*e - A*c*e)*(d + e*x)^(3/2))/(3*e^4) + (2*B*c*(d + e*x)^(5/2))/(5*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)^{3/2}} \, dx &=\int \left (-\frac{d (B d-A e) (c d-b e)}{e^3 (d+e x)^{3/2}}+\frac{B d (3 c d-2 b e)-A e (2 c d-b e)}{e^3 \sqrt{d+e x}}+\frac{(-3 B c d+b B e+A c e) \sqrt{d+e x}}{e^3}+\frac{B c (d+e x)^{3/2}}{e^3}\right ) \, dx\\ &=\frac{2 d (B d-A e) (c d-b e)}{e^4 \sqrt{d+e x}}+\frac{2 (B d (3 c d-2 b e)-A e (2 c d-b e)) \sqrt{d+e x}}{e^4}-\frac{2 (3 B c d-b B e-A c 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 = 0.0470064, size = 110, normalized size = 0.9 \[ \frac{2 \left (5 A e \left (3 b e (2 d+e x)+c \left (-8 d^2-4 d e x+e^2 x^2\right )\right )+B \left (5 b e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 c \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )\right )\right )}{15 e^4 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(5*A*e*(3*b*e*(2*d + e*x) + c*(-8*d^2 - 4*d*e*x + e^2*x^2)) + B*(5*b*e*(-8*d^2 - 4*d*e*x + e^2*x^2) + 3*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 = 0.003, size = 121, normalized size = 1. \begin{align*}{\frac{6\,Bc{x}^{3}{e}^{3}+10\,Ac{e}^{3}{x}^{2}+10\,Bb{e}^{3}{x}^{2}-12\,Bcd{e}^{2}{x}^{2}+30\,Ab{e}^{3}x-40\,Acd{e}^{2}x-40\,Bbd{e}^{2}x+48\,Bc{d}^{2}ex+60\,Abd{e}^{2}-80\,Ac{d}^{2}e-80\,Bb{d}^{2}e+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+b*x)/(e*x+d)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.06913, size = 162, normalized size = 1.33 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} B c - 5 \,{\left (3 \, B c d -{\left (B b + A c\right )} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15 \,{\left (3 \, B c d^{2} + A b e^{2} - 2 \,{\left (B b + A c\right )} d e\right )} \sqrt{e x + d}}{e^{3}} + \frac{15 \,{\left (B c d^{3} + A b d e^{2} -{\left (B b + A c\right )} d^{2} e\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+b*x)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.74158, size = 267, normalized size = 2.19 \begin{align*} \frac{2 \,{\left (3 \, B c e^{3} x^{3} + 48 \, B c d^{3} + 30 \, A b d e^{2} - 40 \,{\left (B b + A c\right )} d^{2} e -{\left (6 \, B c d e^{2} - 5 \,{\left (B b + A c\right )} e^{3}\right )} x^{2} +{\left (24 \, B c d^{2} e + 15 \, A b e^{3} - 20 \,{\left (B b + A c\right )} d e^{2}\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+b*x)/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 20.3608, size = 126, normalized size = 1.03 \begin{align*} \frac{2 B c \left (d + e x\right )^{\frac{5}{2}}}{5 e^{4}} - \frac{2 d \left (- A e + B d\right ) \left (b e - c d\right )}{e^{4} \sqrt{d + e x}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (2 A c e + 2 B b e - 6 B c d\right )}{3 e^{4}} + \frac{\sqrt{d + e x} \left (2 A b e^{2} - 4 A c d e - 4 B b d e + 6 B c d^{2}\right )}{e^{4}} \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)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.25082, size = 225, normalized size = 1.84 \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}} B b e^{17} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A c e^{17} - 30 \, \sqrt{x e + d} B b d e^{17} - 30 \, \sqrt{x e + d} A c d e^{17} + 15 \, \sqrt{x e + d} A b e^{18}\right )} e^{\left (-20\right )} + \frac{2 \,{\left (B c d^{3} - B b d^{2} e - A c d^{2} e + A b d e^{2}\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+b*x)/(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)*B*b*e^17 + 5*(x*e + d)^(3/2)*A*c*e^17 - 30*sqrt(x*e + d)*B*b*d*e^17 - 30*sqrt(x*e + d)*A*c*d*e^17 + 15*
sqrt(x*e + d)*A*b*e^18)*e^(-20) + 2*(B*c*d^3 - B*b*d^2*e - A*c*d^2*e + A*b*d*e^2)*e^(-4)/sqrt(x*e + d)