3.2044 \(\int (a+b x) (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2) \, dx\)

Optimal. Leaf size=100 \[ -\frac{2 b^2 (d+e x)^{9/2} (b d-a e)}{3 e^4}+\frac{6 b (d+e x)^{7/2} (b d-a e)^2}{7 e^4}-\frac{2 (d+e x)^{5/2} (b d-a e)^3}{5 e^4}+\frac{2 b^3 (d+e x)^{11/2}}{11 e^4} \]

[Out]

(-2*(b*d - a*e)^3*(d + e*x)^(5/2))/(5*e^4) + (6*b*(b*d - a*e)^2*(d + e*x)^(7/2))/(7*e^4) - (2*b^2*(b*d - a*e)*
(d + e*x)^(9/2))/(3*e^4) + (2*b^3*(d + e*x)^(11/2))/(11*e^4)

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Rubi [A]  time = 0.033601, antiderivative size = 100, 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, 43} \[ -\frac{2 b^2 (d+e x)^{9/2} (b d-a e)}{3 e^4}+\frac{6 b (d+e x)^{7/2} (b d-a e)^2}{7 e^4}-\frac{2 (d+e x)^{5/2} (b d-a e)^3}{5 e^4}+\frac{2 b^3 (d+e x)^{11/2}}{11 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^3*(d + e*x)^(5/2))/(5*e^4) + (6*b*(b*d - a*e)^2*(d + e*x)^(7/2))/(7*e^4) - (2*b^2*(b*d - a*e)*
(d + e*x)^(9/2))/(3*e^4) + (2*b^3*(d + e*x)^(11/2))/(11*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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^3 (d+e x)^{3/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^3 (d+e x)^{3/2}}{e^3}+\frac{3 b (b d-a e)^2 (d+e x)^{5/2}}{e^3}-\frac{3 b^2 (b d-a e) (d+e x)^{7/2}}{e^3}+\frac{b^3 (d+e x)^{9/2}}{e^3}\right ) \, dx\\ &=-\frac{2 (b d-a e)^3 (d+e x)^{5/2}}{5 e^4}+\frac{6 b (b d-a e)^2 (d+e x)^{7/2}}{7 e^4}-\frac{2 b^2 (b d-a e) (d+e x)^{9/2}}{3 e^4}+\frac{2 b^3 (d+e x)^{11/2}}{11 e^4}\\ \end{align*}

Mathematica [A]  time = 0.0628938, size = 79, normalized size = 0.79 \[ \frac{2 (d+e x)^{5/2} \left (-385 b^2 (d+e x)^2 (b d-a e)+495 b (d+e x) (b d-a e)^2-231 (b d-a e)^3+105 b^3 (d+e x)^3\right )}{1155 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(-231*(b*d - a*e)^3 + 495*b*(b*d - a*e)^2*(d + e*x) - 385*b^2*(b*d - a*e)*(d + e*x)^2 + 105
*b^3*(d + e*x)^3))/(1155*e^4)

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Maple [A]  time = 0.007, size = 116, normalized size = 1.2 \begin{align*}{\frac{210\,{x}^{3}{b}^{3}{e}^{3}+770\,{x}^{2}a{b}^{2}{e}^{3}-140\,{x}^{2}{b}^{3}d{e}^{2}+990\,x{a}^{2}b{e}^{3}-440\,xa{b}^{2}d{e}^{2}+80\,x{b}^{3}{d}^{2}e+462\,{e}^{3}{a}^{3}-396\,d{e}^{2}{a}^{2}b+176\,a{d}^{2}e{b}^{2}-32\,{d}^{3}{b}^{3}}{1155\,{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)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2),x)

[Out]

2/1155*(e*x+d)^(5/2)*(105*b^3*e^3*x^3+385*a*b^2*e^3*x^2-70*b^3*d*e^2*x^2+495*a^2*b*e^3*x-220*a*b^2*d*e^2*x+40*
b^3*d^2*e*x+231*a^3*e^3-198*a^2*b*d*e^2+88*a*b^2*d^2*e-16*b^3*d^3)/e^4

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Maxima [A]  time = 0.965409, size = 159, normalized size = 1.59 \begin{align*} \frac{2 \,{\left (105 \,{\left (e x + d\right )}^{\frac{11}{2}} b^{3} - 385 \,{\left (b^{3} d - a b^{2} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 495 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 231 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{1155 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

2/1155*(105*(e*x + d)^(11/2)*b^3 - 385*(b^3*d - a*b^2*e)*(e*x + d)^(9/2) + 495*(b^3*d^2 - 2*a*b^2*d*e + a^2*b*
e^2)*(e*x + d)^(7/2) - 231*(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*(e*x + d)^(5/2))/e^4

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Fricas [B]  time = 1.28424, size = 474, normalized size = 4.74 \begin{align*} \frac{2 \,{\left (105 \, b^{3} e^{5} x^{5} - 16 \, b^{3} d^{5} + 88 \, a b^{2} d^{4} e - 198 \, a^{2} b d^{3} e^{2} + 231 \, a^{3} d^{2} e^{3} + 35 \,{\left (4 \, b^{3} d e^{4} + 11 \, a b^{2} e^{5}\right )} x^{4} + 5 \,{\left (b^{3} d^{2} e^{3} + 110 \, a b^{2} d e^{4} + 99 \, a^{2} b e^{5}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{3} e^{2} - 11 \, a b^{2} d^{2} e^{3} - 264 \, a^{2} b d e^{4} - 77 \, a^{3} e^{5}\right )} x^{2} +{\left (8 \, b^{3} d^{4} e - 44 \, a b^{2} d^{3} e^{2} + 99 \, a^{2} b d^{2} e^{3} + 462 \, a^{3} d e^{4}\right )} x\right )} \sqrt{e x + d}}{1155 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

2/1155*(105*b^3*e^5*x^5 - 16*b^3*d^5 + 88*a*b^2*d^4*e - 198*a^2*b*d^3*e^2 + 231*a^3*d^2*e^3 + 35*(4*b^3*d*e^4
+ 11*a*b^2*e^5)*x^4 + 5*(b^3*d^2*e^3 + 110*a*b^2*d*e^4 + 99*a^2*b*e^5)*x^3 - 3*(2*b^3*d^3*e^2 - 11*a*b^2*d^2*e
^3 - 264*a^2*b*d*e^4 - 77*a^3*e^5)*x^2 + (8*b^3*d^4*e - 44*a*b^2*d^3*e^2 + 99*a^2*b*d^2*e^3 + 462*a^3*d*e^4)*x
)*sqrt(e*x + d)/e^4

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Sympy [A]  time = 12.2111, size = 386, normalized size = 3.86 \begin{align*} a^{3} d \left (\begin{cases} \sqrt{d} x & \text{for}\: e = 0 \\\frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 e} & \text{otherwise} \end{cases}\right ) + \frac{2 a^{3} \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e} + \frac{6 a^{2} b d \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{6 a^{2} b \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{2}} + \frac{6 a b^{2} d \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{6 a b^{2} \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{3}} + \frac{2 b^{3} d \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{4}} + \frac{2 b^{3} \left (\frac{d^{4} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{4 d^{3} \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{6 d^{2} \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{4 d \left (d + e x\right )^{\frac{9}{2}}}{9} + \frac{\left (d + e x\right )^{\frac{11}{2}}}{11}\right )}{e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

a**3*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a**3*(-d*(d + e*x)**(3/2)/3 + (d
+ e*x)**(5/2)/5)/e + 6*a**2*b*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 6*a**2*b*(d**2*(d + e*x)*
*(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 6*a*b**2*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d +
e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 6*a*b**2*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*
d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 2*b**3*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)
/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 2*b**3*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(
5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4

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Giac [B]  time = 1.12133, size = 463, normalized size = 4.63 \begin{align*} \frac{2}{3465} \,{\left (693 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{2} b d e^{\left (-1\right )} + 99 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} a b^{2} d e^{\left (-2\right )} + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} b^{3} d e^{\left (-3\right )} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} d + 99 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} a^{2} b e^{\left (-1\right )} + 33 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} a b^{2} e^{\left (-2\right )} +{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4}\right )} b^{3} e^{\left (-3\right )} + 231 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{3}\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(693*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^2*b*d*e^(-1) + 99*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(
5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a*b^2*d*e^(-2) + 11*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e +
d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*b^3*d*e^(-3) + 1155*(x*e + d)^(3/2)*a^3*d + 99*(15*(x*e + d)^(7/2) - 4
2*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^2*b*e^(-1) + 33*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d +
189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a*b^2*e^(-2) + (315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)
*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*b^3*e^(-3) + 231*(3*(x*e
+ d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^3)*e^(-1)