3.20.76 \(\int (d+e x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2) \, dx\) [1976]

Optimal. Leaf size=43 \[ \frac {2}{7} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^{7/2}+\frac {2 c d (d+e x)^{9/2}}{9 e^2} \]

[Out]

2/7*(a-c*d^2/e^2)*(e*x+d)^(7/2)+2/9*c*d*(e*x+d)^(9/2)/e^2

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45} \begin {gather*} \frac {2}{7} (d+e x)^{7/2} \left (a-\frac {c d^2}{e^2}\right )+\frac {2 c d (d+e x)^{9/2}}{9 e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a - (c*d^2)/e^2)*(d + e*x)^(7/2))/7 + (2*c*d*(d + e*x)^(9/2))/(9*e^2)

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])

Rule 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.04, size = 34, normalized size = 0.79 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (9 a e^2+c d (-2 d+7 e x)\right )}{63 e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(9*a*e^2 + c*d*(-2*d + 7*e*x)))/(63*e^2)

________________________________________________________________________________________

Maple [A]
time = 0.48, size = 39, normalized size = 0.91

method	result	size
gosper	\(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (7 c d e x +9 e^{2} a -2 c \,d^{2}\right )}{63 e^{2}}\)	\(32\)
derivativedivides	\(\frac {\frac {2 c d \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{2}}\)	\(39\)
default	\(\frac {\frac {2 c d \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{2}}\)	\(39\)
trager	\(\frac {2 \left (7 c d \,e^{4} x^{4}+9 a \,e^{5} x^{3}+19 c \,d^{2} e^{3} x^{3}+27 a d \,e^{4} x^{2}+15 c \,d^{3} e^{2} x^{2}+27 a \,e^{3} d^{2} x +c \,d^{4} e x +9 a \,e^{2} d^{3}-2 c \,d^{5}\right ) \sqrt {e x +d}}{63 e^{2}}\)	\(99\)
risch	\(\frac {2 \left (7 c d \,e^{4} x^{4}+9 a \,e^{5} x^{3}+19 c \,d^{2} e^{3} x^{3}+27 a d \,e^{4} x^{2}+15 c \,d^{3} e^{2} x^{2}+27 a \,e^{3} d^{2} x +c \,d^{4} e x +9 a \,e^{2} d^{3}-2 c \,d^{5}\right ) \sqrt {e x +d}}{63 e^{2}}\)	\(99\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x,method=_RETURNVERBOSE)

[Out]

2/e^2*(1/9*c*d*(e*x+d)^(9/2)+1/7*(a*e^2-c*d^2)*(e*x+d)^(7/2))

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 38, normalized size = 0.88 \begin {gather*} \frac {2}{63} \, {\left (7 \, {\left (x e + d\right )}^{\frac {9}{2}} c d - 9 \, {\left (c d^{2} - a e^{2}\right )} {\left (x e + d\right )}^{\frac {7}{2}}\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/63*(7*(x*e + d)^(9/2)*c*d - 9*(c*d^2 - a*e^2)*(x*e + d)^(7/2))*e^(-2)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (36) = 72\).
time = 3.66, size = 93, normalized size = 2.16 \begin {gather*} \frac {2}{63} \, {\left (c d^{4} x e - 2 \, c d^{5} + 9 \, a x^{3} e^{5} + {\left (7 \, c d x^{4} + 27 \, a d x^{2}\right )} e^{4} + {\left (19 \, c d^{2} x^{3} + 27 \, a d^{2} x\right )} e^{3} + 3 \, {\left (5 \, c d^{3} x^{2} + 3 \, a d^{3}\right )} e^{2}\right )} \sqrt {x e + d} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/63*(c*d^4*x*e - 2*c*d^5 + 9*a*x^3*e^5 + (7*c*d*x^4 + 27*a*d*x^2)*e^4 + (19*c*d^2*x^3 + 27*a*d^2*x)*e^3 + 3*(
5*c*d^3*x^2 + 3*a*d^3)*e^2)*sqrt(x*e + d)*e^(-2)

________________________________________________________________________________________

Sympy [A]
time = 5.38, size = 235, normalized size = 5.47 \begin {gather*} a d^{2} e \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 ) + 4 a d \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right ) + 2 a \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 ) + \frac {2 c d^{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^{2}} + \frac {4 c d^{2} \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 c 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^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (36) = 72\).
time = 1.46, size = 336, normalized size = 7.81 \begin {gather*} \frac {2}{315} \, {\left (105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} c d^{4} e^{\left (-1\right )} + 63 \, {\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 )} c d^{3} e^{\left (-1\right )} + 315 \, \sqrt {x e + d} a d^{3} e + 315 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a d^{2} e + 27 \, {\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 )} c d^{2} e^{\left (-1\right )} + 63 \, {\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 d e + {\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 )} c d e^{\left (-1\right )} + 9 \, {\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 e\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(105*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*c*d^4*e^(-1) + 63*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d +
15*sqrt(x*e + d)*d^2)*c*d^3*e^(-1) + 315*sqrt(x*e + d)*a*d^3*e + 315*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*
d^2*e + 27*(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)*c*d^2*e^
(-1) + 63*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*d*e + (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)*c*d*e^(-1) + 9*
(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*e)*e^(-1)

________________________________________________________________________________________

Mupad [B]
time = 0.61, size = 34, normalized size = 0.79 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{7/2}\,\left (9\,a\,e^2-9\,c\,d^2+7\,c\,d\,\left (d+e\,x\right )\right )}{63\,e^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(9*a*e^2 - 9*c*d^2 + 7*c*d*(d + e*x)))/(63*e^2)

________________________________________________________________________________________