\(\int (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2) \, dx\) [1622]

Optimal result

Integrand size = 26, antiderivative size = 71 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2 (b d-a e)^2 (d+e x)^{7/2}}{7 e^3}-\frac {4 b (b d-a e) (d+e x)^{9/2}}{9 e^3}+\frac {2 b^2 (d+e x)^{11/2}}{11 e^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45} \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=-\frac {4 b (d+e x)^{9/2} (b d-a e)}{9 e^3}+\frac {2 (d+e x)^{7/2} (b d-a e)^2}{7 e^3}+\frac {2 b^2 (d+e x)^{11/2}}{11 e^3} \]

[In]

Int[(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

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

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 45

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.86 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2 (d+e x)^{7/2} \left (99 a^2 e^2+22 a b e (-2 d+7 e x)+b^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3} \]

[In]

Integrate[(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(2*(d + e*x)^(7/2)*(99*a^2*e^2 + 22*a*b*e*(-2*d + 7*e*x) + b^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2)))/(693*e^3)

Maple [A] (verified)

Time = 2.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.76

method	result	size
pseudoelliptic	\(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (\left (\frac {7}{11} b^{2} x^{2}+\frac {14}{9} a b x +a^{2}\right ) e^{2}-\frac {4 b d \left (\frac {7 b x}{11}+a \right ) e}{9}+\frac {8 b^{2} d^{2}}{99}\right )}{7 e^{3}}\)	\(54\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (63 x^{2} b^{2} e^{2}+154 x a b \,e^{2}-28 b^{2} d e x +99 a^{2} e^{2}-44 a b d e +8 b^{2} d^{2}\right )}{693 e^{3}}\)	\(63\)
derivativedivides	\(\frac {\frac {2 b^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{3}}\)	\(70\)
default	\(\frac {\frac {2 b^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{3}}\)	\(70\)
trager	\(\frac {2 \left (63 b^{2} e^{5} x^{5}+154 a b \,e^{5} x^{4}+161 b^{2} d \,e^{4} x^{4}+99 a^{2} e^{5} x^{3}+418 a b d \,e^{4} x^{3}+113 b^{2} d^{2} e^{3} x^{3}+297 a^{2} d \,e^{4} x^{2}+330 a b \,d^{2} e^{3} x^{2}+3 b^{2} d^{3} e^{2} x^{2}+297 a^{2} d^{2} e^{3} x +22 a b \,d^{3} e^{2} x -4 b^{2} d^{4} e x +99 a^{2} d^{3} e^{2}-44 a b \,d^{4} e +8 b^{2} d^{5}\right ) \sqrt {e x +d}}{693 e^{3}}\)	\(182\)
risch	\(\frac {2 \left (63 b^{2} e^{5} x^{5}+154 a b \,e^{5} x^{4}+161 b^{2} d \,e^{4} x^{4}+99 a^{2} e^{5} x^{3}+418 a b d \,e^{4} x^{3}+113 b^{2} d^{2} e^{3} x^{3}+297 a^{2} d \,e^{4} x^{2}+330 a b \,d^{2} e^{3} x^{2}+3 b^{2} d^{3} e^{2} x^{2}+297 a^{2} d^{2} e^{3} x +22 a b \,d^{3} e^{2} x -4 b^{2} d^{4} e x +99 a^{2} d^{3} e^{2}-44 a b \,d^{4} e +8 b^{2} d^{5}\right ) \sqrt {e x +d}}{693 e^{3}}\)	\(182\)

[In]

int((e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2),x,method=_RETURNVERBOSE)

[Out]

2/7*(e*x+d)^(7/2)*((7/11*b^2*x^2+14/9*a*b*x+a^2)*e^2-4/9*b*d*(7/11*b*x+a)*e+8/99*b^2*d^2)/e^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (59) = 118\).

Time = 0.29 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.45 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2 \, {\left (63 \, b^{2} e^{5} x^{5} + 8 \, b^{2} d^{5} - 44 \, a b d^{4} e + 99 \, a^{2} d^{3} e^{2} + 7 \, {\left (23 \, b^{2} d e^{4} + 22 \, a b e^{5}\right )} x^{4} + {\left (113 \, b^{2} d^{2} e^{3} + 418 \, a b d e^{4} + 99 \, a^{2} e^{5}\right )} x^{3} + 3 \, {\left (b^{2} d^{3} e^{2} + 110 \, a b d^{2} e^{3} + 99 \, a^{2} d e^{4}\right )} x^{2} - {\left (4 \, b^{2} d^{4} e - 22 \, a b d^{3} e^{2} - 297 \, a^{2} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{693 \, e^{3}} \]

[In]

integrate((e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

2/693*(63*b^2*e^5*x^5 + 8*b^2*d^5 - 44*a*b*d^4*e + 99*a^2*d^3*e^2 + 7*(23*b^2*d*e^4 + 22*a*b*e^5)*x^4 + (113*b
^2*d^2*e^3 + 418*a*b*d*e^4 + 99*a^2*e^5)*x^3 + 3*(b^2*d^3*e^2 + 110*a*b*d^2*e^3 + 99*a^2*d*e^4)*x^2 - (4*b^2*d
^4*e - 22*a*b*d^3*e^2 - 297*a^2*d^2*e^3)*x)*sqrt(e*x + d)/e^3

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (65) = 130\).

Time = 0.37 (sec) , antiderivative size = 355, normalized size of antiderivative = 5.00 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\begin {cases} \frac {2 a^{2} d^{3} \sqrt {d + e x}}{7 e} + \frac {6 a^{2} d^{2} x \sqrt {d + e x}}{7} + \frac {6 a^{2} d e x^{2} \sqrt {d + e x}}{7} + \frac {2 a^{2} e^{2} x^{3} \sqrt {d + e x}}{7} - \frac {8 a b d^{4} \sqrt {d + e x}}{63 e^{2}} + \frac {4 a b d^{3} x \sqrt {d + e x}}{63 e} + \frac {20 a b d^{2} x^{2} \sqrt {d + e x}}{21} + \frac {76 a b d e x^{3} \sqrt {d + e x}}{63} + \frac {4 a b e^{2} x^{4} \sqrt {d + e x}}{9} + \frac {16 b^{2} d^{5} \sqrt {d + e x}}{693 e^{3}} - \frac {8 b^{2} d^{4} x \sqrt {d + e x}}{693 e^{2}} + \frac {2 b^{2} d^{3} x^{2} \sqrt {d + e x}}{231 e} + \frac {226 b^{2} d^{2} x^{3} \sqrt {d + e x}}{693} + \frac {46 b^{2} d e x^{4} \sqrt {d + e x}}{99} + \frac {2 b^{2} e^{2} x^{5} \sqrt {d + e x}}{11} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate((e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

Piecewise((2*a**2*d**3*sqrt(d + e*x)/(7*e) + 6*a**2*d**2*x*sqrt(d + e*x)/7 + 6*a**2*d*e*x**2*sqrt(d + e*x)/7 +
2*a**2*e**2*x**3*sqrt(d + e*x)/7 - 8*a*b*d**4*sqrt(d + e*x)/(63*e**2) + 4*a*b*d**3*x*sqrt(d + e*x)/(63*e) + 2
0*a*b*d**2*x**2*sqrt(d + e*x)/21 + 76*a*b*d*e*x**3*sqrt(d + e*x)/63 + 4*a*b*e**2*x**4*sqrt(d + e*x)/9 + 16*b**
2*d**5*sqrt(d + e*x)/(693*e**3) - 8*b**2*d**4*x*sqrt(d + e*x)/(693*e**2) + 2*b**2*d**3*x**2*sqrt(d + e*x)/(231
*e) + 226*b**2*d**2*x**3*sqrt(d + e*x)/693 + 46*b**2*d*e*x**4*sqrt(d + e*x)/99 + 2*b**2*e**2*x**5*sqrt(d + e*x
)/11, Ne(e, 0)), (d**(5/2)*(a**2*x + a*b*x**2 + b**2*x**3/3), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.96 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} b^{2} - 154 \, {\left (b^{2} d - a b e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 99 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{693 \, e^{3}} \]

[In]

integrate((e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

2/693*(63*(e*x + d)^(11/2)*b^2 - 154*(b^2*d - a*b*e)*(e*x + d)^(9/2) + 99*(b^2*d^2 - 2*a*b*d*e + a^2*e^2)*(e*x
+ d)^(7/2))/e^3

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (59) = 118\).

Time = 0.29 (sec) , antiderivative size = 558, normalized size of antiderivative = 7.86 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2 \, {\left (3465 \, \sqrt {e x + d} a^{2} d^{3} + 3465 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{2} d^{2} + \frac {2310 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a b d^{3}}{e} + 693 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{2} d + \frac {231 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} b^{2} d^{3}}{e^{2}} + \frac {1386 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a b d^{2}}{e} + 99 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a^{2} + \frac {297 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} b^{2} d^{2}}{e^{2}} + \frac {594 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a b d}{e} + \frac {33 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} b^{2} d}{e^{2}} + \frac {22 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} a b}{e} + \frac {5 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} b^{2}}{e^{2}}\right )}}{3465 \, e} \]

[In]

integrate((e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")

[Out]

2/3465*(3465*sqrt(e*x + d)*a^2*d^3 + 3465*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^2*d^2 + 2310*((e*x + d)^(3/2
) - 3*sqrt(e*x + d)*d)*a*b*d^3/e + 693*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*d
+ 231*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*b^2*d^3/e^2 + 1386*(3*(e*x + d)^(5/2)
- 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a*b*d^2/e + 99*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*
(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^2 + 297*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)
^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*b^2*d^2/e^2 + 594*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^
(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a*b*d/e + 33*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/
2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*b^2*d/e^2 + 22*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(
7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a*b/e + 5*(63*(e*x + d)^(1
1/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 -
693*sqrt(e*x + d)*d^5)*b^2/e^2)/e

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.96 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2\,{\left (d+e\,x\right )}^{7/2}\,\left (63\,b^2\,{\left (d+e\,x\right )}^2+99\,a^2\,e^2+99\,b^2\,d^2-154\,b^2\,d\,\left (d+e\,x\right )+154\,a\,b\,e\,\left (d+e\,x\right )-198\,a\,b\,d\,e\right )}{693\,e^3} \]

[In]

int((d + e*x)^(5/2)*(a^2 + b^2*x^2 + 2*a*b*x),x)

[Out]

(2*(d + e*x)^(7/2)*(63*b^2*(d + e*x)^2 + 99*a^2*e^2 + 99*b^2*d^2 - 154*b^2*d*(d + e*x) + 154*a*b*e*(d + e*x) -
198*a*b*d*e))/(693*e^3)