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

Optimal. Leaf size=71 \[ \frac {2 (b d-a e)^2 (d+e x)^{7/2}}{7 e^3}-\frac {4 b (b d-a e) (d+e x)^{9/2}}{9 e^3}+\frac {2 b^2 (d+e x)^{11/2}}{11 e^3} \]

[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

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Rubi [A]
time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\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*}

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Mathematica [A]
time = 0.05, size = 61, normalized size = 0.86 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [A]
time = 0.46, size = 70, normalized size = 0.99

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^3*(1/11*b^2*(e*x+d)^(11/2)+1/9*(2*a*b*e-2*b^2*d)*(e*x+d)^(9/2)+1/7*(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(7/
2))

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Maxima [A]
time = 0.28, size = 71, normalized size = 1.00 \begin {gather*} \frac {2}{693} \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} b^{2} - 154 \, {\left (b^{2} d - a b e\right )} {\left (x e + d\right )}^{\frac {9}{2}} + 99 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {7}{2}}\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (61) = 122\).
time = 3.01, size = 163, normalized size = 2.30 \begin {gather*} \frac {2}{693} \, {\left (8 \, b^{2} d^{5} + {\left (63 \, b^{2} x^{5} + 154 \, a b x^{4} + 99 \, a^{2} x^{3}\right )} e^{5} + {\left (161 \, b^{2} d x^{4} + 418 \, a b d x^{3} + 297 \, a^{2} d x^{2}\right )} e^{4} + {\left (113 \, b^{2} d^{2} x^{3} + 330 \, a b d^{2} x^{2} + 297 \, a^{2} d^{2} x\right )} e^{3} + {\left (3 \, b^{2} d^{3} x^{2} + 22 \, a b d^{3} x + 99 \, a^{2} d^{3}\right )} e^{2} - 4 \, {\left (b^{2} d^{4} x + 11 \, a b d^{4}\right )} e\right )} \sqrt {x e + d} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(8*b^2*d^5 + (63*b^2*x^5 + 154*a*b*x^4 + 99*a^2*x^3)*e^5 + (161*b^2*d*x^4 + 418*a*b*d*x^3 + 297*a^2*d*x^
2)*e^4 + (113*b^2*d^2*x^3 + 330*a*b*d^2*x^2 + 297*a^2*d^2*x)*e^3 + (3*b^2*d^3*x^2 + 22*a*b*d^3*x + 99*a^2*d^3)
*e^2 - 4*(b^2*d^4*x + 11*a*b*d^4)*e)*sqrt(x*e + d)*e^(-3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (65) = 130\).
time = 0.34, size = 355, normalized size = 5.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 591 vs. \(2 (61) = 122\).
time = 1.23, size = 591, normalized size = 8.32 \begin {gather*} \frac {2}{3465} \, {\left (2310 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a b d^{3} e^{\left (-1\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 )} b^{2} d^{3} e^{\left (-2\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 b d^{2} e^{\left (-1\right )} + 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 )} b^{2} d^{2} e^{\left (-2\right )} + 3465 \, \sqrt {x e + d} a^{2} d^{3} + 3465 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} d^{2} + 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 d 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 )} b^{2} d e^{\left (-2\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^{2} d + 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 )} a b e^{\left (-1\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^{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 )} a^{2}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(2310*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*b*d^3*e^(-1) + 231*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2
)*d + 15*sqrt(x*e + d)*d^2)*b^2*d^3*e^(-2) + 1386*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)
*d^2)*a*b*d^2*e^(-1) + 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)*b^2*d^2*e^(-2) + 3465*sqrt(x*e + d)*a^2*d^3 + 3465*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*d^2 + 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*d*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)*b^2*d*e^(-2) + 693*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*d + 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)*a*b*e^(-1) + 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^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)*a^2)*e^(-1)

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Mupad [B]
time = 0.06, size = 68, normalized size = 0.96 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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