3.23.69 \(\int (d+e x)^{3/2} (a+b x+c x^2) \, dx\) [2269]

Optimal. Leaf size=75 \[ \frac {2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}{5 e^3}-\frac {2 (2 c d-b e) (d+e x)^{7/2}}{7 e^3}+\frac {2 c (d+e x)^{9/2}}{9 e^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712} \begin {gather*} \frac {2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}{5 e^3}-\frac {2 (d+e x)^{7/2} (2 c d-b e)}{7 e^3}+\frac {2 c (d+e x)^{9/2}}{9 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 712

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.05, size = 55, normalized size = 0.73 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (9 e (-2 b d+7 a e+5 b e x)+c \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )}{315 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(9*e*(-2*b*d + 7*a*e + 5*b*e*x) + c*(8*d^2 - 20*d*e*x + 35*e^2*x^2)))/(315*e^3)

________________________________________________________________________________________

Maple [A]
time = 0.48, size = 59, normalized size = 0.79

method	result	size
gosper	\(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (35 x^{2} c \,e^{2}+45 b \,e^{2} x -20 c d e x +63 e^{2} a -18 b d e +8 c \,d^{2}\right )}{315 e^{3}}\)	\(53\)
derivativedivides	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a -b d e +c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{3}}\)	\(59\)
default	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a -b d e +c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{3}}\)	\(59\)
trager	\(\frac {2 \left (35 c \,e^{4} x^{4}+45 b \,e^{4} x^{3}+50 d \,e^{3} c \,x^{3}+63 a \,e^{4} x^{2}+72 b d \,e^{3} x^{2}+3 c \,d^{2} e^{2} x^{2}+126 a d \,e^{3} x +9 b \,d^{2} e^{2} x -4 c \,d^{3} e x +63 a \,d^{2} e^{2}-18 b \,d^{3} e +8 c \,d^{4}\right ) \sqrt {e x +d}}{315 e^{3}}\)	\(121\)
risch	\(\frac {2 \left (35 c \,e^{4} x^{4}+45 b \,e^{4} x^{3}+50 d \,e^{3} c \,x^{3}+63 a \,e^{4} x^{2}+72 b d \,e^{3} x^{2}+3 c \,d^{2} e^{2} x^{2}+126 a d \,e^{3} x +9 b \,d^{2} e^{2} x -4 c \,d^{3} e x +63 a \,d^{2} e^{2}-18 b \,d^{3} e +8 c \,d^{4}\right ) \sqrt {e x +d}}{315 e^{3}}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^3*(1/9*c*(e*x+d)^(9/2)+1/7*(b*e-2*c*d)*(e*x+d)^(7/2)+1/5*(a*e^2-b*d*e+c*d^2)*(e*x+d)^(5/2))

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 62, normalized size = 0.83 \begin {gather*} \frac {2}{315} \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} c - 45 \, {\left (2 \, c d - b e\right )} {\left (x e + d\right )}^{\frac {7}{2}} + 63 \, {\left (c d^{2} - b d e + a e^{2}\right )} {\left (x e + d\right )}^{\frac {5}{2}}\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*(x*e + d)^(9/2)*c - 45*(2*c*d - b*e)*(x*e + d)^(7/2) + 63*(c*d^2 - b*d*e + a*e^2)*(x*e + d)^(5/2))*e
^(-3)

________________________________________________________________________________________

Fricas [A]
time = 3.58, size = 109, normalized size = 1.45 \begin {gather*} \frac {2}{315} \, {\left (8 \, c d^{4} + {\left (35 \, c x^{4} + 45 \, b x^{3} + 63 \, a x^{2}\right )} e^{4} + 2 \, {\left (25 \, c d x^{3} + 36 \, b d x^{2} + 63 \, a d x\right )} e^{3} + 3 \, {\left (c d^{2} x^{2} + 3 \, b d^{2} x + 21 \, a d^{2}\right )} e^{2} - 2 \, {\left (2 \, c d^{3} x + 9 \, b d^{3}\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)^(3/2)*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

2/315*(8*c*d^4 + (35*c*x^4 + 45*b*x^3 + 63*a*x^2)*e^4 + 2*(25*c*d*x^3 + 36*b*d*x^2 + 63*a*d*x)*e^3 + 3*(c*d^2*
x^2 + 3*b*d^2*x + 21*a*d^2)*e^2 - 2*(2*c*d^3*x + 9*b*d^3)*e)*sqrt(x*e + d)*e^(-3)

________________________________________________________________________________________

Sympy [A]
time = 5.72, size = 230, normalized size = 3.07 \begin {gather*} a 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 \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 {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 {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 {2 c 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 c \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (64) = 128\).
time = 1.41, size = 365, normalized size = 4.87 \begin {gather*} \frac {2}{315} \, {\left (105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} 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 )} c d^{2} e^{\left (-2\right )} + 42 \, {\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 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 )} c d e^{\left (-2\right )} + 315 \, \sqrt {x e + d} a d^{2} + 210 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a d + 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 )} 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 )} c 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\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(105*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*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)*c*d^2*e^(-2) + 42*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*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)*c*d*e^
(-2) + 315*sqrt(x*e + d)*a*d^2 + 210*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*d + 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)*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)*c*e^(-2) + 21*(3*(x*e +
d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a)*e^(-1)

________________________________________________________________________________________

Mupad [B]
time = 0.81, size = 58, normalized size = 0.77 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{5/2}\,\left (35\,c\,{\left (d+e\,x\right )}^2+63\,a\,e^2+63\,c\,d^2+45\,b\,e\,\left (d+e\,x\right )-90\,c\,d\,\left (d+e\,x\right )-63\,b\,d\,e\right )}{315\,e^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(35*c*(d + e*x)^2 + 63*a*e^2 + 63*c*d^2 + 45*b*e*(d + e*x) - 90*c*d*(d + e*x) - 63*b*d*e))/
(315*e^3)

________________________________________________________________________________________