3.18.21 \(\int (a+b x) (A+B x) \sqrt {d+e x} \, dx\) [1721]

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

[Out]

2/3*(-a*e+b*d)*(-A*e+B*d)*(e*x+d)^(3/2)/e^3-2/5*(-A*b*e-B*a*e+2*B*b*d)*(e*x+d)^(5/2)/e^3+2/7*b*B*(e*x+d)^(7/2)
/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)^{5/2} (-a B e-A b e+2 b B d)}{5 e^3}+\frac {2 (d+e x)^{3/2} (b d-a e) (B d-A e)}{3 e^3}+\frac {2 b B (d+e x)^{7/2}}{7 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(A + B*x)*Sqrt[d + e*x],x]

[Out]

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

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Mathematica [A]
time = 0.06, size = 70, normalized size = 0.84 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (7 A b e (-2 d+3 e x)+7 a e (-2 B d+5 A e+3 B e x)+b B \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(A + B*x)*Sqrt[d + e*x],x]

[Out]

(2*(d + e*x)^(3/2)*(7*A*b*e*(-2*d + 3*e*x) + 7*a*e*(-2*B*d + 5*A*e + 3*B*e*x) + b*B*(8*d^2 - 12*d*e*x + 15*e^2
*x^2)))/(105*e^3)

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

method	result	size
gosper	\(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (15 b B \,x^{2} e^{2}+21 A b \,e^{2} x +21 B a \,e^{2} x -12 B b d e x +35 A a \,e^{2}-14 A b d e -14 B a d e +8 B b \,d^{2}\right )}{105 e^{3}}\)	\(73\)
derivativedivides	\(\frac {\frac {2 B b \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a e -b d \right ) B +b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a e -b d \right ) \left (A e -B d \right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{3}}\)	\(73\)
default	\(\frac {\frac {2 B b \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a e -b d \right ) B +b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a e -b d \right ) \left (A e -B d \right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{3}}\)	\(73\)
trager	\(\frac {2 \left (15 B b \,e^{3} x^{3}+21 A b \,e^{3} x^{2}+21 B a \,e^{3} x^{2}+3 B b d \,e^{2} x^{2}+35 A a \,e^{3} x +7 A b d \,e^{2} x +7 B a d \,e^{2} x -4 B b \,d^{2} e x +35 A a d \,e^{2}-14 A b \,d^{2} e -14 B a \,d^{2} e +8 B b \,d^{3}\right ) \sqrt {e x +d}}{105 e^{3}}\)	\(121\)
risch	\(\frac {2 \left (15 B b \,e^{3} x^{3}+21 A b \,e^{3} x^{2}+21 B a \,e^{3} x^{2}+3 B b d \,e^{2} x^{2}+35 A a \,e^{3} x +7 A b d \,e^{2} x +7 B a d \,e^{2} x -4 B b \,d^{2} e x +35 A a d \,e^{2}-14 A b \,d^{2} e -14 B a \,d^{2} e +8 B b \,d^{3}\right ) \sqrt {e x +d}}{105 e^{3}}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(B*x+A)*(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 81, normalized size = 0.98 \begin {gather*} \frac {2}{105} \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} B b - 21 \, {\left (2 \, B b d - B a e - A b e\right )} {\left (x e + d\right )}^{\frac {5}{2}} + 35 \, {\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 {3}{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)^(1/2),x, algorithm="maxima")

[Out]

2/105*(15*(x*e + d)^(7/2)*B*b - 21*(2*B*b*d - B*a*e - A*b*e)*(x*e + d)^(5/2) + 35*(B*b*d^2 + A*a*e^2 - (B*a*e
+ A*b*e)*d)*(x*e + d)^(3/2))*e^(-3)

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Fricas [A]
time = 1.31, size = 101, normalized size = 1.22 \begin {gather*} \frac {2}{105} \, {\left (8 \, B b d^{3} + {\left (15 \, B b x^{3} + 35 \, A a x + 21 \, {\left (B a + A b\right )} x^{2}\right )} e^{3} + {\left (3 \, B b d x^{2} + 35 \, A a d + 7 \, {\left (B a + A b\right )} d x\right )} e^{2} - 2 \, {\left (2 \, B b d^{2} x + 7 \, {\left (B a + A b\right )} d^{2}\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)^(1/2),x, algorithm="fricas")

[Out]

2/105*(8*B*b*d^3 + (15*B*b*x^3 + 35*A*a*x + 21*(B*a + A*b)*x^2)*e^3 + (3*B*b*d*x^2 + 35*A*a*d + 7*(B*a + A*b)*
d*x)*e^2 - 2*(2*B*b*d^2*x + 7*(B*a + A*b)*d^2)*e)*sqrt(x*e + d)*e^(-3)

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Sympy [A]
time = 1.74, size = 94, normalized size = 1.13 \begin {gather*} \frac {2 \left (\frac {B b \left (d + e x\right )^{\frac {7}{2}}}{7 e^{2}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (A b e + B a e - 2 B b d\right )}{5 e^{2}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A a e^{2} - A b d e - B a d e + B b d^{2}\right )}{3 e^{2}}\right )}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)*(e*x+d)**(1/2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (75) = 150\).
time = 0.65, size = 274, normalized size = 3.30 \begin {gather*} \frac {2}{105} \, {\left (35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a d e^{\left (-1\right )} + 35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A b d e^{\left (-1\right )} + 7 \, {\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 e^{\left (-2\right )} + 7 \, {\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 e^{\left (-1\right )} + 7 \, {\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 e^{\left (-1\right )} + 3 \, {\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 e^{\left (-2\right )} + 105 \, \sqrt {x e + d} A a d + 35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\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)^(1/2),x, algorithm="giac")

[Out]

2/105*(35*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a*d*e^(-1) + 35*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*b*d*
e^(-1) + 7*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*b*d*e^(-2) + 7*(3*(x*e + d)^(5/
2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a*e^(-1) + 7*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 1
5*sqrt(x*e + d)*d^2)*A*b*e^(-1) + 3*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sq
rt(x*e + d)*d^3)*B*b*e^(-2) + 105*sqrt(x*e + d)*A*a*d + 35*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*a)*e^(-1)

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Mupad [B]
time = 1.21, size = 80, normalized size = 0.96 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{3/2}\,\left (15\,B\,b\,{\left (d+e\,x\right )}^2+35\,A\,a\,e^2+35\,B\,b\,d^2+21\,A\,b\,e\,\left (d+e\,x\right )+21\,B\,a\,e\,\left (d+e\,x\right )-42\,B\,b\,d\,\left (d+e\,x\right )-35\,A\,b\,d\,e-35\,B\,a\,d\,e\right )}{105\,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)^(1/2),x)

[Out]

(2*(d + e*x)^(3/2)*(15*B*b*(d + e*x)^2 + 35*A*a*e^2 + 35*B*b*d^2 + 21*A*b*e*(d + e*x) + 21*B*a*e*(d + e*x) - 4
2*B*b*d*(d + e*x) - 35*A*b*d*e - 35*B*a*d*e))/(105*e^3)

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