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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 100, normalized size of antiderivative = 1.00, 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 = 0.06, 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)

________________________________________________________________________________________

fricas [B]  time = 0.78, size = 216, normalized size = 2.16 \[ \frac {2 \, {\left (105 \, b^{3} e^{5} x^{5} - 16 \, b^{3} d^{5} + 88 \, a b^{2} d^{4} e - 198 \, a^{2} b d^{3} e^{2} + 231 \, a^{3} d^{2} e^{3} + 35 \, {\left (4 \, b^{3} d e^{4} + 11 \, a b^{2} e^{5}\right )} x^{4} + 5 \, {\left (b^{3} d^{2} e^{3} + 110 \, a b^{2} d e^{4} + 99 \, a^{2} b e^{5}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{3} e^{2} - 11 \, a b^{2} d^{2} e^{3} - 264 \, a^{2} b d e^{4} - 77 \, a^{3} e^{5}\right )} x^{2} + {\left (8 \, b^{3} d^{4} e - 44 \, a b^{2} d^{3} e^{2} + 99 \, a^{2} b d^{2} e^{3} + 462 \, a^{3} d e^{4}\right )} x\right )} \sqrt {e x + d}}{1155 \, e^{4}} \]

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

________________________________________________________________________________________

giac [B]  time = 0.18, size = 598, normalized size = 5.98 \[ \frac {2}{3465} \, {\left (3465 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} b d^{2} e^{\left (-1\right )} + 693 \, {\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} d^{2} e^{\left (-2\right )} + 99 \, {\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^{3} d^{2} e^{\left (-3\right )} + 1386 \, {\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^{2} b d e^{\left (-1\right )} + 594 \, {\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 )} a b^{2} d e^{\left (-2\right )} + 22 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} b^{3} d e^{\left (-3\right )} + 3465 \, \sqrt {x e + d} a^{3} d^{2} + 2310 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{3} d + 297 \, {\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 )} a^{2} b e^{\left (-1\right )} + 33 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a b^{2} e^{\left (-2\right )} + 5 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} b^{3} e^{\left (-3\right )} + 231 \, {\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^{3}\right )} e^{\left (-1\right )} \]

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*(3465*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*b*d^2*e^(-1) + 693*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3
/2)*d + 15*sqrt(x*e + d)*d^2)*a*b^2*d^2*e^(-2) + 99*(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^3*d^2*e^(-3) + 1386*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e
+ d)*d^2)*a^2*b*d*e^(-1) + 594*(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)*a*b^2*d*e^(-2) + 22*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(
x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*b^3*d*e^(-3) + 3465*sqrt(x*e + d)*a^3*d^2 + 2310*((x*e + d)^(3/2)
- 3*sqrt(x*e + d)*d)*a^3*d + 297*(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)*a^2*b*e^(-1) + 33*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x
*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a*b^2*e^(-2) + 5*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990
*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*b^3*e^(-3)
+ 231*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^3)*e^(-1)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 116, normalized size = 1.16 \[ \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (105 b^{3} e^{3} x^{3}+385 a \,b^{2} e^{3} x^{2}-70 b^{3} d \,e^{2} x^{2}+495 a^{2} b \,e^{3} x -220 a \,b^{2} d \,e^{2} x +40 b^{3} d^{2} e x +231 a^{3} e^{3}-198 a^{2} b d \,e^{2}+88 a \,b^{2} d^{2} e -16 b^{3} d^{3}\right )}{1155 e^{4}} \]

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

________________________________________________________________________________________

maxima [A]  time = 0.55, size = 118, normalized size = 1.18 \[ \frac {2 \, {\left (105 \, {\left (e x + d\right )}^{\frac {11}{2}} b^{3} - 385 \, {\left (b^{3} d - a b^{2} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 231 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{1155 \, e^{4}} \]

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

________________________________________________________________________________________

mupad [B]  time = 0.06, size = 87, normalized size = 0.87 \[ \frac {2\,b^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4}-\frac {\left (6\,b^3\,d-6\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}+\frac {2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4}+\frac {6\,b\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 16.77, size = 386, normalized size = 3.86 \[ a^{3} d \left (\begin {cases} \sqrt {d} x & \text {for}\: e = 0 \\\frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 e} & \text {otherwise} \end {cases}\right ) + \frac {2 a^{3} \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e} + \frac {6 a^{2} b d \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} + \frac {6 a^{2} b \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{2}} + \frac {6 a b^{2} d \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} + \frac {6 a b^{2} \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} + \frac {2 b^{3} d \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}} + \frac {2 b^{3} \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{4}} \]

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

________________________________________________________________________________________