3.1788 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=126 \[ -\frac{2 b (d+e x)^{5/2} (-2 a B e-A b e+3 b B d)}{5 e^4}+\frac{2 (d+e x)^{3/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{3 e^4}-\frac{2 \sqrt{d+e x} (b d-a e)^2 (B d-A e)}{e^4}+\frac{2 b^2 B (d+e x)^{7/2}}{7 e^4} \]

[Out]

(-2*(b*d - a*e)^2*(B*d - A*e)*Sqrt[d + e*x])/e^4 + (2*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^(3/2))
/(3*e^4) - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(5/2))/(5*e^4) + (2*b^2*B*(d + e*x)^(7/2))/(7*e^4)

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Rubi [A]  time = 0.0524879, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 77} \[ -\frac{2 b (d+e x)^{5/2} (-2 a B e-A b e+3 b B d)}{5 e^4}+\frac{2 (d+e x)^{3/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{3 e^4}-\frac{2 \sqrt{d+e x} (b d-a e)^2 (B d-A e)}{e^4}+\frac{2 b^2 B (d+e x)^{7/2}}{7 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^2*(B*d - A*e)*Sqrt[d + e*x])/e^4 + (2*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^(3/2))
/(3*e^4) - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(5/2))/(5*e^4) + (2*b^2*B*(d + e*x)^(7/2))/(7*e^4)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0924312, size = 107, normalized size = 0.85 \[ \frac{2 \sqrt{d+e x} \left (-21 b (d+e x)^2 (-2 a B e-A b e+3 b B d)+35 (d+e x) (b d-a e) (-a B e-2 A b e+3 b B d)-105 (b d-a e)^2 (B d-A e)+15 b^2 B (d+e x)^3\right )}{105 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(-105*(b*d - a*e)^2*(B*d - A*e) + 35*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x) - 21*b
*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^2 + 15*b^2*B*(d + e*x)^3))/(105*e^4)

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Maple [A]  time = 0.009, size = 169, normalized size = 1.3 \begin{align*}{\frac{30\,{b}^{2}B{x}^{3}{e}^{3}+42\,A{b}^{2}{e}^{3}{x}^{2}+84\,Bab{e}^{3}{x}^{2}-36\,B{b}^{2}d{e}^{2}{x}^{2}+140\,Axab{e}^{3}-56\,Ax{b}^{2}d{e}^{2}+70\,Bx{a}^{2}{e}^{3}-112\,Bxabd{e}^{2}+48\,B{b}^{2}{d}^{2}ex+210\,A{a}^{2}{e}^{3}-280\,Aabd{e}^{2}+112\,A{b}^{2}{d}^{2}e-140\,B{a}^{2}d{e}^{2}+224\,Bab{d}^{2}e-96\,{b}^{2}B{d}^{3}}{105\,{e}^{4}}\sqrt{ex+d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(1/2),x)

[Out]

2/105*(15*B*b^2*e^3*x^3+21*A*b^2*e^3*x^2+42*B*a*b*e^3*x^2-18*B*b^2*d*e^2*x^2+70*A*a*b*e^3*x-28*A*b^2*d*e^2*x+3
5*B*a^2*e^3*x-56*B*a*b*d*e^2*x+24*B*b^2*d^2*e*x+105*A*a^2*e^3-140*A*a*b*d*e^2+56*A*b^2*d^2*e-70*B*a^2*d*e^2+11
2*B*a*b*d^2*e-48*B*b^2*d^3)*(e*x+d)^(1/2)/e^4

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Maxima [A]  time = 0.988725, size = 215, normalized size = 1.71 \begin{align*} \frac{2 \,{\left (15 \,{\left (e x + d\right )}^{\frac{7}{2}} B b^{2} - 21 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (3 \, B b^{2} d^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 105 \,{\left (B b^{2} d^{3} - A a^{2} e^{3} -{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} \sqrt{e x + d}\right )}}{105 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/105*(15*(e*x + d)^(7/2)*B*b^2 - 21*(3*B*b^2*d - (2*B*a*b + A*b^2)*e)*(e*x + d)^(5/2) + 35*(3*B*b^2*d^2 - 2*(
2*B*a*b + A*b^2)*d*e + (B*a^2 + 2*A*a*b)*e^2)*(e*x + d)^(3/2) - 105*(B*b^2*d^3 - A*a^2*e^3 - (2*B*a*b + A*b^2)
*d^2*e + (B*a^2 + 2*A*a*b)*d*e^2)*sqrt(e*x + d))/e^4

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Fricas [A]  time = 1.36742, size = 351, normalized size = 2.79 \begin{align*} \frac{2 \,{\left (15 \, B b^{2} e^{3} x^{3} - 48 \, B b^{2} d^{3} + 105 \, A a^{2} e^{3} + 56 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e - 70 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} - 3 \,{\left (6 \, B b^{2} d e^{2} - 7 \,{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} +{\left (24 \, B b^{2} d^{2} e - 28 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 35 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/105*(15*B*b^2*e^3*x^3 - 48*B*b^2*d^3 + 105*A*a^2*e^3 + 56*(2*B*a*b + A*b^2)*d^2*e - 70*(B*a^2 + 2*A*a*b)*d*e
^2 - 3*(6*B*b^2*d*e^2 - 7*(2*B*a*b + A*b^2)*e^3)*x^2 + (24*B*b^2*d^2*e - 28*(2*B*a*b + A*b^2)*d*e^2 + 35*(B*a^
2 + 2*A*a*b)*e^3)*x)*sqrt(e*x + d)/e^4

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Sympy [A]  time = 44.7561, size = 583, normalized size = 4.63 \begin{align*} \begin{cases} - \frac{\frac{2 A a^{2} d}{\sqrt{d + e x}} + 2 A a^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{4 A a b d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{4 A a b \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{2 A b^{2} d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{2 A b^{2} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 B a^{2} d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{2 B a^{2} \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{4 B a b d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{4 B a b \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 B b^{2} d \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{3}} + \frac{2 B b^{2} \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}}}{e} & \text{for}\: e \neq 0 \\\frac{A a^{2} x + \frac{B b^{2} x^{4}}{4} + \frac{x^{3} \left (A b^{2} + 2 B a b\right )}{3} + \frac{x^{2} \left (2 A a b + B a^{2}\right )}{2}}{\sqrt{d}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)/(e*x+d)**(1/2),x)

[Out]

Piecewise((-(2*A*a**2*d/sqrt(d + e*x) + 2*A*a**2*(-d/sqrt(d + e*x) - sqrt(d + e*x)) + 4*A*a*b*d*(-d/sqrt(d + e
*x) - sqrt(d + e*x))/e + 4*A*a*b*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 2*A*b**2*d*
(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 2*A*b**2*(-d**3/sqrt(d + e*x) - 3*d**2*sq
rt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 + 2*B*a**2*d*(-d/sqrt(d + e*x) - sqrt(d + e*x))/e
+ 2*B*a**2*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 4*B*a*b*d*(d**2/sqrt(d + e*x) + 2
*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 4*B*a*b*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x
)**(3/2) - (d + e*x)**(5/2)/5)/e**2 + 2*B*b**2*d*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3
/2) - (d + e*x)**(5/2)/5)/e**3 + 2*B*b**2*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2)
+ 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3)/e, Ne(e, 0)), ((A*a**2*x + B*b**2*x**4/4 + x**3*(A*b**2
+ 2*B*a*b)/3 + x**2*(2*A*a*b + B*a**2)/2)/sqrt(d), True))

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Giac [A]  time = 1.13638, size = 290, normalized size = 2.3 \begin{align*} \frac{2}{105} \,{\left (35 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} B a^{2} e^{\left (-1\right )} + 70 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} A a b e^{\left (-1\right )} + 14 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} B a b e^{\left (-2\right )} + 7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} A b^{2} e^{\left (-2\right )} + 3 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{x e + d} d^{3}\right )} B b^{2} e^{\left (-3\right )} + 105 \, \sqrt{x e + d} A a^{2}\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/105*(35*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^2*e^(-1) + 70*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*a*b*
e^(-1) + 14*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a*b*e^(-2) + 7*(3*(x*e + d)^(5
/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*b^2*e^(-2) + 3*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d
+ 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*b^2*e^(-3) + 105*sqrt(x*e + d)*A*a^2)*e^(-1)