3.1719 \(\int (a+b x) (A+B x) (d+e x)^{5/2} \, dx\)

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

[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 = 0.04, 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 = {77} \[ -\frac {2 (d+e x)^{9/2} (-a B e-A b e+2 b B d)}{9 e^3}+\frac {2 (d+e x)^{7/2} (b d-a e) (B d-A e)}{7 e^3}+\frac {2 b B (d+e x)^{11/2}}{11 e^3} \]

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 77

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 = 0.08, size = 70, normalized size = 0.84 \[ \frac {2 (d+e x)^{7/2} \left (11 a e (9 A e-2 B d+7 B e x)+11 A b e (7 e x-2 d)+b B \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3} \]

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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fricas [B]  time = 1.05, size = 189, normalized size = 2.28 \[ \frac {2 \, {\left (63 \, B b e^{5} x^{5} + 8 \, B b d^{5} + 99 \, A a d^{3} e^{2} - 22 \, {\left (B a + A b\right )} d^{4} e + 7 \, {\left (23 \, B b d e^{4} + 11 \, {\left (B a + A b\right )} e^{5}\right )} x^{4} + {\left (113 \, B b d^{2} e^{3} + 99 \, A a e^{5} + 209 \, {\left (B a + A b\right )} d e^{4}\right )} x^{3} + 3 \, {\left (B b d^{3} e^{2} + 99 \, A a d e^{4} + 55 \, {\left (B a + A b\right )} d^{2} e^{3}\right )} x^{2} - {\left (4 \, B b d^{4} e - 297 \, A a d^{2} e^{3} - 11 \, {\left (B a + A b\right )} d^{3} e^{2}\right )} x\right )} \sqrt {e x + d}}{693 \, e^{3}} \]

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*(63*B*b*e^5*x^5 + 8*B*b*d^5 + 99*A*a*d^3*e^2 - 22*(B*a + A*b)*d^4*e + 7*(23*B*b*d*e^4 + 11*(B*a + A*b)*e
^5)*x^4 + (113*B*b*d^2*e^3 + 99*A*a*e^5 + 209*(B*a + A*b)*d*e^4)*x^3 + 3*(B*b*d^3*e^2 + 99*A*a*d*e^4 + 55*(B*a
+ A*b)*d^2*e^3)*x^2 - (4*B*b*d^4*e - 297*A*a*d^2*e^3 - 11*(B*a + A*b)*d^3*e^2)*x)*sqrt(e*x + d)/e^3

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giac [B]  time = 1.35, size = 778, normalized size = 9.37 \[ \text {result too large to display} \]

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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maple [A]  time = 0.00, size = 73, normalized size = 0.88 \[ \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (63 B b \,x^{2} e^{2}+77 A b \,e^{2} x +77 B a \,e^{2} x -28 B b d e x +99 A a \,e^{2}-22 A b d e -22 B a d e +8 B b \,d^{2}\right )}{693 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(e*x+d)^(7/2)*(63*B*b*e^2*x^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)/e^3

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maxima [A]  time = 0.58, size = 75, normalized size = 0.90 \[ \frac {2 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} B b - 77 \, {\left (2 \, B b d - {\left (B a + A b\right )} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 99 \, {\left (B b d^{2} + A a e^{2} - {\left (B a + A b\right )} d e\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{693 \, e^{3}} \]

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*(e*x + d)^(11/2)*B*b - 77*(2*B*b*d - (B*a + A*b)*e)*(e*x + d)^(9/2) + 99*(B*b*d^2 + A*a*e^2 - (B*a +
A*b)*d*e)*(e*x + d)^(7/2))/e^3

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mupad [B]  time = 1.20, size = 80, normalized size = 0.96 \[ \frac {2\,{\left (d+e\,x\right )}^{7/2}\,\left (63\,B\,b\,{\left (d+e\,x\right )}^2+99\,A\,a\,e^2+99\,B\,b\,d^2+77\,A\,b\,e\,\left (d+e\,x\right )+77\,B\,a\,e\,\left (d+e\,x\right )-154\,B\,b\,d\,\left (d+e\,x\right )-99\,A\,b\,d\,e-99\,B\,a\,d\,e\right )}{693\,e^3} \]

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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sympy [A]  time = 3.66, size = 476, normalized size = 5.73 \[ \begin {cases} \frac {2 A a d^{3} \sqrt {d + e x}}{7 e} + \frac {6 A a d^{2} x \sqrt {d + e x}}{7} + \frac {6 A a d e x^{2} \sqrt {d + e x}}{7} + \frac {2 A a e^{2} x^{3} \sqrt {d + e x}}{7} - \frac {4 A b d^{4} \sqrt {d + e x}}{63 e^{2}} + \frac {2 A b d^{3} x \sqrt {d + e x}}{63 e} + \frac {10 A b d^{2} x^{2} \sqrt {d + e x}}{21} + \frac {38 A b d e x^{3} \sqrt {d + e x}}{63} + \frac {2 A b e^{2} x^{4} \sqrt {d + e x}}{9} - \frac {4 B a d^{4} \sqrt {d + e x}}{63 e^{2}} + \frac {2 B a d^{3} x \sqrt {d + e x}}{63 e} + \frac {10 B a d^{2} x^{2} \sqrt {d + e x}}{21} + \frac {38 B a d e x^{3} \sqrt {d + e x}}{63} + \frac {2 B a e^{2} x^{4} \sqrt {d + e x}}{9} + \frac {16 B b d^{5} \sqrt {d + e x}}{693 e^{3}} - \frac {8 B b d^{4} x \sqrt {d + e x}}{693 e^{2}} + \frac {2 B b d^{3} x^{2} \sqrt {d + e x}}{231 e} + \frac {226 B b d^{2} x^{3} \sqrt {d + e x}}{693} + \frac {46 B b d e x^{4} \sqrt {d + e x}}{99} + \frac {2 B b e^{2} x^{5} \sqrt {d + e x}}{11} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (A a x + \frac {A b x^{2}}{2} + \frac {B a x^{2}}{2} + \frac {B b x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]

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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