3.1623 \(\int (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2) \, dx\)

Optimal. Leaf size=71 \[ -\frac {4 b (d+e x)^{7/2} (b d-a e)}{7 e^3}+\frac {2 (d+e x)^{5/2} (b d-a e)^2}{5 e^3}+\frac {2 b^2 (d+e x)^{9/2}}{9 e^3} \]

[Out]

2/5*(-a*e+b*d)^2*(e*x+d)^(5/2)/e^3-4/7*b*(-a*e+b*d)*(e*x+d)^(7/2)/e^3+2/9*b^2*(e*x+d)^(9/2)/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, 43} \[ -\frac {4 b (d+e x)^{7/2} (b d-a e)}{7 e^3}+\frac {2 (d+e x)^{5/2} (b d-a e)^2}{5 e^3}+\frac {2 b^2 (d+e x)^{9/2}}{9 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.04, size = 61, normalized size = 0.86 \[ \frac {2 (d+e x)^{5/2} \left (63 a^2 e^2+18 a b e (5 e x-2 d)+b^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )}{315 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(63*a^2*e^2 + 18*a*b*e*(-2*d + 5*e*x) + b^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2)))/(315*e^3)

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fricas [B]  time = 0.91, size = 137, normalized size = 1.93 \[ \frac {2 \, {\left (35 \, b^{2} e^{4} x^{4} + 8 \, b^{2} d^{4} - 36 \, a b d^{3} e + 63 \, a^{2} d^{2} e^{2} + 10 \, {\left (5 \, b^{2} d e^{3} + 9 \, a b e^{4}\right )} x^{3} + 3 \, {\left (b^{2} d^{2} e^{2} + 48 \, a b d e^{3} + 21 \, a^{2} e^{4}\right )} x^{2} - 2 \, {\left (2 \, b^{2} d^{3} e - 9 \, a b d^{2} e^{2} - 63 \, a^{2} d e^{3}\right )} x\right )} \sqrt {e x + d}}{315 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*b^2*e^4*x^4 + 8*b^2*d^4 - 36*a*b*d^3*e + 63*a^2*d^2*e^2 + 10*(5*b^2*d*e^3 + 9*a*b*e^4)*x^3 + 3*(b^2*
d^2*e^2 + 48*a*b*d*e^3 + 21*a^2*e^4)*x^2 - 2*(2*b^2*d^3*e - 9*a*b*d^2*e^2 - 63*a^2*d*e^3)*x)*sqrt(e*x + d)/e^3

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giac [B]  time = 0.20, size = 380, normalized size = 5.35 \[ \frac {2}{315} \, {\left (210 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a b d^{2} e^{\left (-1\right )} + 21 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} b^{2} d^{2} e^{\left (-2\right )} + 84 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a b d e^{\left (-1\right )} + 18 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} b^{2} d e^{\left (-2\right )} + 315 \, \sqrt {x e + d} a^{2} d^{2} + 210 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} d + 18 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a b e^{\left (-1\right )} + {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} b^{2} e^{\left (-2\right )} + 21 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{2}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(210*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*b*d^2*e^(-1) + 21*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d
+ 15*sqrt(x*e + d)*d^2)*b^2*d^2*e^(-2) + 84*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)
*a*b*d*e^(-1) + 18*(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*e^(-2) + 315*sqrt(x*e + d)*a^2*d^2 + 210*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*d + 18*(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*e^(-1) + (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*e^(-2
) + 21*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2)*e^(-1)

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maple [A]  time = 0.05, size = 63, normalized size = 0.89 \[ \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (35 b^{2} e^{2} x^{2}+90 a b \,e^{2} x -20 b^{2} d e x +63 a^{2} e^{2}-36 a b d e +8 b^{2} d^{2}\right )}{315 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(e*x+d)^(5/2)*(35*b^2*e^2*x^2+90*a*b*e^2*x-20*b^2*d*e*x+63*a^2*e^2-36*a*b*d*e+8*b^2*d^2)/e^3

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maxima [A]  time = 1.06, size = 68, normalized size = 0.96 \[ \frac {2 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{2} - 90 \, {\left (b^{2} d - a b e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 63 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{315 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*(e*x + d)^(9/2)*b^2 - 90*(b^2*d - a*b*e)*(e*x + d)^(7/2) + 63*(b^2*d^2 - 2*a*b*d*e + a^2*e^2)*(e*x +
d)^(5/2))/e^3

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mupad [B]  time = 0.55, size = 68, normalized size = 0.96 \[ \frac {2\,{\left (d+e\,x\right )}^{5/2}\,\left (35\,b^2\,{\left (d+e\,x\right )}^2+63\,a^2\,e^2+63\,b^2\,d^2-90\,b^2\,d\,\left (d+e\,x\right )+90\,a\,b\,e\,\left (d+e\,x\right )-126\,a\,b\,d\,e\right )}{315\,e^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(35*b^2*(d + e*x)^2 + 63*a^2*e^2 + 63*b^2*d^2 - 90*b^2*d*(d + e*x) + 90*a*b*e*(d + e*x) - 1
26*a*b*d*e))/(315*e^3)

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sympy [A]  time = 10.70, size = 240, normalized size = 3.38 \[ a^{2} d \left (\begin {cases} \sqrt {d} x & \text {for}\: e = 0 \\\frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 e} & \text {otherwise} \end {cases}\right ) + \frac {2 a^{2} \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e} + \frac {4 a b d \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} + \frac {4 a b \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{2}} + \frac {2 b^{2} d \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} + \frac {2 b^{2} \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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