3.18.19 \(\int (a+b x) (A+B x) (d+e x)^{5/2} \, dx\) [1719]

Optimal. Leaf size=83 \[ \frac {2 (b d-a e) (B d-A e) (d+e x)^{7/2}}{7 e^3}-\frac {2 (2 b B d-A b e-a B e) (d+e x)^{9/2}}{9 e^3}+\frac {2 b B (d+e x)^{11/2}}{11 e^3} \]

[Out]

2/7*(-a*e+b*d)*(-A*e+B*d)*(e*x+d)^(7/2)/e^3-2/9*(-A*b*e-B*a*e+2*B*b*d)*(e*x+d)^(9/2)/e^3+2/11*b*B*(e*x+d)^(11/
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)^{9/2} (-a B e-A b e+2 b B d)}{9 e^3}+\frac {2 (d+e x)^{7/2} (b d-a e) (B d-A e)}{7 e^3}+\frac {2 b B (d+e x)^{11/2}}{11 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(A + B*x)*(d + e*x)^(5/2),x]

[Out]

(2*(b*d - a*e)*(B*d - A*e)*(d + e*x)^(7/2))/(7*e^3) - (2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(9/2))/(9*e^3) +
(2*b*B*(d + e*x)^(11/2))/(11*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)^{5/2} \, dx &=\int \left (\frac {(-b d+a e) (-B d+A e) (d+e x)^{5/2}}{e^2}+\frac {(-2 b B d+A b e+a B e) (d+e x)^{7/2}}{e^2}+\frac {b B (d+e x)^{9/2}}{e^2}\right ) \, dx\\ &=\frac {2 (b d-a e) (B d-A e) (d+e x)^{7/2}}{7 e^3}-\frac {2 (2 b B d-A b e-a B e) (d+e x)^{9/2}}{9 e^3}+\frac {2 b B (d+e x)^{11/2}}{11 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)^{7/2} \left (11 A b e (-2 d+7 e x)+11 a e (-2 B d+9 A e+7 B e x)+b B \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[(a + b*x)*(A + B*x)*(d + e*x)^(5/2),x]

[Out]

(2*(d + e*x)^(7/2)*(11*A*b*e*(-2*d + 7*e*x) + 11*a*e*(-2*B*d + 9*A*e + 7*B*e*x) + b*B*(8*d^2 - 28*d*e*x + 63*e
^2*x^2)))/(693*e^3)

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

method	result	size
gosper	\(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (63 b B \,x^{2} e^{2}+77 A b \,e^{2} x +77 B a \,e^{2} x -28 B b d e x +99 A a \,e^{2}-22 A b d e -22 B a d e +8 B b \,d^{2}\right )}{693 e^{3}}\)	\(73\)
derivativedivides	\(\frac {\frac {2 B b \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -b d \right ) B +b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a e -b d \right ) \left (A e -B d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{3}}\)	\(73\)
default	\(\frac {\frac {2 B b \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -b d \right ) B +b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a e -b d \right ) \left (A e -B d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{3}}\)	\(73\)
trager	\(\frac {2 \left (63 B b \,e^{5} x^{5}+77 A b \,e^{5} x^{4}+77 B a \,e^{5} x^{4}+161 B b d \,e^{4} x^{4}+99 A a \,e^{5} x^{3}+209 A b d \,e^{4} x^{3}+209 B a d \,e^{4} x^{3}+113 B b \,d^{2} e^{3} x^{3}+297 A a d \,e^{4} x^{2}+165 A b \,d^{2} e^{3} x^{2}+165 B a \,d^{2} e^{3} x^{2}+3 B b \,d^{3} e^{2} x^{2}+297 A a \,d^{2} e^{3} x +11 A b \,d^{3} e^{2} x +11 B a \,d^{3} e^{2} x -4 B b \,d^{4} e x +99 A a \,d^{3} e^{2}-22 A b \,d^{4} e -22 B a \,d^{4} e +8 B b \,d^{5}\right ) \sqrt {e x +d}}{693 e^{3}}\)	\(225\)
risch	\(\frac {2 \left (63 B b \,e^{5} x^{5}+77 A b \,e^{5} x^{4}+77 B a \,e^{5} x^{4}+161 B b d \,e^{4} x^{4}+99 A a \,e^{5} x^{3}+209 A b d \,e^{4} x^{3}+209 B a d \,e^{4} x^{3}+113 B b \,d^{2} e^{3} x^{3}+297 A a d \,e^{4} x^{2}+165 A b \,d^{2} e^{3} x^{2}+165 B a \,d^{2} e^{3} x^{2}+3 B b \,d^{3} e^{2} x^{2}+297 A a \,d^{2} e^{3} x +11 A b \,d^{3} e^{2} x +11 B a \,d^{3} e^{2} x -4 B b \,d^{4} e x +99 A a \,d^{3} e^{2}-22 A b \,d^{4} e -22 B a \,d^{4} e +8 B b \,d^{5}\right ) \sqrt {e x +d}}{693 e^{3}}\)	\(225\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.42, size = 81, normalized size = 0.98 \begin {gather*} \frac {2}{693} \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} B b - 77 \, {\left (2 \, B b d - B a e - A b e\right )} {\left (x e + d\right )}^{\frac {9}{2}} + 99 \, {\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 {7}{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)^(5/2),x, algorithm="maxima")

[Out]

2/693*(63*(x*e + d)^(11/2)*B*b - 77*(2*B*b*d - B*a*e - A*b*e)*(x*e + d)^(9/2) + 99*(B*b*d^2 + A*a*e^2 - (B*a*e
+ A*b*e)*d)*(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. 179 vs. \(2 (75) = 150\).
time = 1.17, size = 179, normalized size = 2.16 \begin {gather*} \frac {2}{693} \, {\left (8 \, B b d^{5} + {\left (63 \, B b x^{5} + 99 \, A a x^{3} + 77 \, {\left (B a + A b\right )} x^{4}\right )} e^{5} + {\left (161 \, B b d x^{4} + 297 \, A a d x^{2} + 209 \, {\left (B a + A b\right )} d x^{3}\right )} e^{4} + {\left (113 \, B b d^{2} x^{3} + 297 \, A a d^{2} x + 165 \, {\left (B a + A b\right )} d^{2} x^{2}\right )} e^{3} + {\left (3 \, B b d^{3} x^{2} + 99 \, A a d^{3} + 11 \, {\left (B a + A b\right )} d^{3} x\right )} e^{2} - 2 \, {\left (2 \, B b d^{4} x + 11 \, {\left (B a + A b\right )} 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((b*x+a)*(B*x+A)*(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/693*(8*B*b*d^5 + (63*B*b*x^5 + 99*A*a*x^3 + 77*(B*a + A*b)*x^4)*e^5 + (161*B*b*d*x^4 + 297*A*a*d*x^2 + 209*(
B*a + A*b)*d*x^3)*e^4 + (113*B*b*d^2*x^3 + 297*A*a*d^2*x + 165*(B*a + A*b)*d^2*x^2)*e^3 + (3*B*b*d^3*x^2 + 99*
A*a*d^3 + 11*(B*a + A*b)*d^3*x)*e^2 - 2*(2*B*b*d^4*x + 11*(B*a + 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. 476 vs. \(2 (87) = 174\).
time = 0.36, size = 476, normalized size = 5.73 \begin {gather*} \begin {cases} \frac {2 A a d^{3} \sqrt {d + e x}}{7 e} + \frac {6 A a d^{2} x \sqrt {d + e x}}{7} + \frac {6 A a d e x^{2} \sqrt {d + e x}}{7} + \frac {2 A a e^{2} x^{3} \sqrt {d + e x}}{7} - \frac {4 A b d^{4} \sqrt {d + e x}}{63 e^{2}} + \frac {2 A b d^{3} x \sqrt {d + e x}}{63 e} + \frac {10 A b d^{2} x^{2} \sqrt {d + e x}}{21} + \frac {38 A b d e x^{3} \sqrt {d + e x}}{63} + \frac {2 A b e^{2} x^{4} \sqrt {d + e x}}{9} - \frac {4 B a d^{4} \sqrt {d + e x}}{63 e^{2}} + \frac {2 B a d^{3} x \sqrt {d + e x}}{63 e} + \frac {10 B a d^{2} x^{2} \sqrt {d + e x}}{21} + \frac {38 B a d e x^{3} \sqrt {d + e x}}{63} + \frac {2 B a e^{2} x^{4} \sqrt {d + e x}}{9} + \frac {16 B b d^{5} \sqrt {d + e x}}{693 e^{3}} - \frac {8 B b d^{4} x \sqrt {d + e x}}{693 e^{2}} + \frac {2 B b d^{3} x^{2} \sqrt {d + e x}}{231 e} + \frac {226 B b d^{2} x^{3} \sqrt {d + e x}}{693} + \frac {46 B b d e x^{4} \sqrt {d + e x}}{99} + \frac {2 B b e^{2} x^{5} \sqrt {d + e x}}{11} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (A a x + \frac {A b x^{2}}{2} + \frac {B a x^{2}}{2} + \frac {B b x^{3}}{3}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)*(e*x+d)**(5/2),x)

[Out]

Piecewise((2*A*a*d**3*sqrt(d + e*x)/(7*e) + 6*A*a*d**2*x*sqrt(d + e*x)/7 + 6*A*a*d*e*x**2*sqrt(d + e*x)/7 + 2*
A*a*e**2*x**3*sqrt(d + e*x)/7 - 4*A*b*d**4*sqrt(d + e*x)/(63*e**2) + 2*A*b*d**3*x*sqrt(d + e*x)/(63*e) + 10*A*
b*d**2*x**2*sqrt(d + e*x)/21 + 38*A*b*d*e*x**3*sqrt(d + e*x)/63 + 2*A*b*e**2*x**4*sqrt(d + e*x)/9 - 4*B*a*d**4
*sqrt(d + e*x)/(63*e**2) + 2*B*a*d**3*x*sqrt(d + e*x)/(63*e) + 10*B*a*d**2*x**2*sqrt(d + e*x)/21 + 38*B*a*d*e*
x**3*sqrt(d + e*x)/63 + 2*B*a*e**2*x**4*sqrt(d + e*x)/9 + 16*B*b*d**5*sqrt(d + e*x)/(693*e**3) - 8*B*b*d**4*x*
sqrt(d + e*x)/(693*e**2) + 2*B*b*d**3*x**2*sqrt(d + e*x)/(231*e) + 226*B*b*d**2*x**3*sqrt(d + e*x)/693 + 46*B*
b*d*e*x**4*sqrt(d + e*x)/99 + 2*B*b*e**2*x**5*sqrt(d + e*x)/11, Ne(e, 0)), (d**(5/2)*(A*a*x + A*b*x**2/2 + B*a
*x**2/2 + B*b*x**3/3), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 778 vs. \(2 (75) = 150\).
time = 0.63, size = 778, normalized size = 9.37 \begin {gather*} \frac {2}{3465} \, {\left (1155 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a d^{3} e^{\left (-1\right )} + 1155 \, {\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 b d^{3} 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 )} B a d^{2} e^{\left (-1\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 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 b d^{2} e^{\left (-2\right )} + 3465 \, \sqrt {x e + d} A a d^{3} + 3465 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a d^{2} + 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 a d 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 )} 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 b 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 a d + 11 \, {\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 a e^{\left (-1\right )} + 11 \, {\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 b 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 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)^(5/2),x, algorithm="giac")

[Out]

2/3465*(1155*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a*d^3*e^(-1) + 1155*((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*b*d^3*e^(-2) + 693*(
3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a*d^2*e^(-1) + 693*(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*b*d^2*e^(-2) + 3465*sqrt(x*e + d)*A*a*d^3 + 3465*((x*e + d)^(3/2
) - 3*sqrt(x*e + d)*d)*A*a*d^2 + 297*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*s
qrt(x*e + d)*d^3)*B*a*d*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*s
qrt(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 - 42
0*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*b*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*a*d + 11*(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*a*e^(-1) + 11*(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) - 3
85*(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*sqr
t(x*e + d)*d^5)*B*b*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*a)*e^(-1)

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Mupad [B]
time = 1.20, size = 80, normalized size = 0.96 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{7/2}\,\left (63\,B\,b\,{\left (d+e\,x\right )}^2+99\,A\,a\,e^2+99\,B\,b\,d^2+77\,A\,b\,e\,\left (d+e\,x\right )+77\,B\,a\,e\,\left (d+e\,x\right )-154\,B\,b\,d\,\left (d+e\,x\right )-99\,A\,b\,d\,e-99\,B\,a\,d\,e\right )}{693\,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)^(5/2),x)

[Out]

(2*(d + e*x)^(7/2)*(63*B*b*(d + e*x)^2 + 99*A*a*e^2 + 99*B*b*d^2 + 77*A*b*e*(d + e*x) + 77*B*a*e*(d + e*x) - 1
54*B*b*d*(d + e*x) - 99*A*b*d*e - 99*B*a*d*e))/(693*e^3)

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