3.18.20 \(\int (a+b x) (A+B x) (d+e x)^{3/2} \, dx\) [1720]

Optimal. Leaf size=83 \[ \frac {2 (b d-a e) (B d-A e) (d+e x)^{5/2}}{5 e^3}-\frac {2 (2 b B d-A b e-a B e) (d+e x)^{7/2}}{7 e^3}+\frac {2 b B (d+e x)^{9/2}}{9 e^3} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 83, 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 = {78} \begin {gather*} -\frac {2 (d+e x)^{7/2} (-a B e-A b e+2 b B d)}{7 e^3}+\frac {2 (d+e x)^{5/2} (b d-a e) (B d-A e)}{5 e^3}+\frac {2 b B (d+e x)^{9/2}}{9 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

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Mathematica [A]
time = 0.07, size = 70, normalized size = 0.84 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (9 A b e (-2 d+5 e x)+9 a e (-2 B d+7 A e+5 B e x)+b B \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[(a + b*x)*(A + B*x)*(d + e*x)^(3/2),x]

[Out]

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

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Maple [A]
time = 0.08, size = 73, normalized size = 0.88

method	result	size
gosper	\(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (35 b B \,x^{2} e^{2}+45 A b \,e^{2} x +45 B a \,e^{2} x -20 B b d e x +63 A a \,e^{2}-18 A b d e -18 B a d e +8 B b \,d^{2}\right )}{315 e^{3}}\)	\(73\)
derivativedivides	\(\frac {\frac {2 B b \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a e -b d \right ) B +b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a e -b d \right ) \left (A e -B d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{3}}\)	\(73\)
default	\(\frac {\frac {2 B b \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a e -b d \right ) B +b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a e -b d \right ) \left (A e -B d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{3}}\)	\(73\)
trager	\(\frac {2 \left (35 B b \,e^{4} x^{4}+45 A b \,e^{4} x^{3}+45 B a \,e^{4} x^{3}+50 B b d \,e^{3} x^{3}+63 A a \,e^{4} x^{2}+72 A b d \,e^{3} x^{2}+72 B a d \,e^{3} x^{2}+3 B b \,d^{2} e^{2} x^{2}+126 A a d \,e^{3} x +9 A b \,d^{2} e^{2} x +9 B a \,d^{2} e^{2} x -4 B b \,d^{3} e x +63 A a \,d^{2} e^{2}-18 A b \,d^{3} e -18 B a \,d^{3} e +8 B b \,d^{4}\right ) \sqrt {e x +d}}{315 e^{3}}\)	\(173\)
risch	\(\frac {2 \left (35 B b \,e^{4} x^{4}+45 A b \,e^{4} x^{3}+45 B a \,e^{4} x^{3}+50 B b d \,e^{3} x^{3}+63 A a \,e^{4} x^{2}+72 A b d \,e^{3} x^{2}+72 B a d \,e^{3} x^{2}+3 B b \,d^{2} e^{2} x^{2}+126 A a d \,e^{3} x +9 A b \,d^{2} e^{2} x +9 B a \,d^{2} e^{2} x -4 B b \,d^{3} e x +63 A a \,d^{2} e^{2}-18 A b \,d^{3} e -18 B a \,d^{3} e +8 B b \,d^{4}\right ) \sqrt {e x +d}}{315 e^{3}}\)	\(173\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 81, normalized size = 0.98 \begin {gather*} \frac {2}{315} \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} B b - 45 \, {\left (2 \, B b d - B a e - A b e\right )} {\left (x e + d\right )}^{\frac {7}{2}} + 63 \, {\left (B b d^{2} + A a e^{2} - {\left (B a e + A b e\right )} d\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((b*x+a)*(B*x+A)*(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.01, size = 141, normalized size = 1.70 \begin {gather*} \frac {2}{315} \, {\left (8 \, B b d^{4} + {\left (35 \, B b x^{4} + 63 \, A a x^{2} + 45 \, {\left (B a + A b\right )} x^{3}\right )} e^{4} + 2 \, {\left (25 \, B b d x^{3} + 63 \, A a d x + 36 \, {\left (B a + A b\right )} d x^{2}\right )} e^{3} + 3 \, {\left (B b d^{2} x^{2} + 21 \, A a d^{2} + 3 \, {\left (B a + A b\right )} d^{2} x\right )} e^{2} - 2 \, {\left (2 \, B b d^{3} x + 9 \, {\left (B a + A b\right )} 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((b*x+a)*(B*x+A)*(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 8.39, size = 318, normalized size = 3.83 \begin {gather*} A 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 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 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 {2 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 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 )}{e^{2}} + \frac {2 B 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 )}{e^{2}} + \frac {2 B b 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 b \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((b*x+a)*(B*x+A)*(e*x+d)**(3/2),x)

[Out]

A*a*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*A*a*(-d*(d + e*x)**(3/2)/3 + (d +
e*x)**(5/2)/5)/e + 2*A*b*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 2*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*a*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e*
*2 + 2*B*a*(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*b*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*b*(-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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 499 vs. \(2 (75) = 150\).
time = 0.55, size = 499, normalized size = 6.01 \begin {gather*} \frac {2}{315} \, {\left (105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a d^{2} e^{\left (-1\right )} + 105 \, {\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 b 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 a d e^{\left (-1\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 )} 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 b d e^{\left (-2\right )} + 315 \, \sqrt {x e + d} A a d^{2} + 210 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A 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 a 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 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 b 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 a\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.07, size = 80, normalized size = 0.96 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{5/2}\,\left (35\,B\,b\,{\left (d+e\,x\right )}^2+63\,A\,a\,e^2+63\,B\,b\,d^2+45\,A\,b\,e\,\left (d+e\,x\right )+45\,B\,a\,e\,\left (d+e\,x\right )-90\,B\,b\,d\,\left (d+e\,x\right )-63\,A\,b\,d\,e-63\,B\,a\,d\,e\right )}{315\,e^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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