3.1722 \(\int \frac {(a+b x) (A+B x)}{\sqrt {d+e x}} \, dx\)

Optimal. Leaf size=81 \[ -\frac {2 (d+e x)^{3/2} (-a B e-A b e+2 b B d)}{3 e^3}+\frac {2 \sqrt {d+e x} (b d-a e) (B d-A e)}{e^3}+\frac {2 b B (d+e x)^{5/2}}{5 e^3} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 81, 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} \[ -\frac {2 (d+e x)^{3/2} (-a B e-A b e+2 b B d)}{3 e^3}+\frac {2 \sqrt {d+e x} (b d-a e) (B d-A e)}{e^3}+\frac {2 b B (d+e x)^{5/2}}{5 e^3} \]

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

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Mathematica [A]  time = 0.08, size = 68, normalized size = 0.84 \[ \frac {2 \sqrt {d+e x} \left (5 a e (3 A e-2 B d+B e x)+5 A b e (e x-2 d)+b B \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(5*A*b*e*(-2*d + e*x) + 5*a*e*(-2*B*d + 3*A*e + B*e*x) + b*B*(8*d^2 - 4*d*e*x + 3*e^2*x^2)))/
(15*e^3)

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fricas [A]  time = 0.84, size = 70, normalized size = 0.86 \[ \frac {2 \, {\left (3 \, B b e^{2} x^{2} + 8 \, B b d^{2} + 15 \, A a e^{2} - 10 \, {\left (B a + A b\right )} d e - {\left (4 \, B b d e - 5 \, {\left (B a + A b\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{15 \, e^{3}} \]

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]

2/15*(3*B*b*e^2*x^2 + 8*B*b*d^2 + 15*A*a*e^2 - 10*(B*a + A*b)*d*e - (4*B*b*d*e - 5*(B*a + A*b)*e^2)*x)*sqrt(e*
x + d)/e^3

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giac [A]  time = 1.20, size = 109, normalized size = 1.35 \[ \frac {2}{15} \, {\left (5 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a e^{\left (-1\right )} + 5 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A b e^{\left (-1\right )} + {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} B b e^{\left (-2\right )} + 15 \, \sqrt {x e + d} A a\right )} e^{\left (-1\right )} \]

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]

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

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maple [A]  time = 0.00, size = 73, normalized size = 0.90 \[ \frac {2 \sqrt {e x +d}\, \left (3 B b \,x^{2} e^{2}+5 A b \,e^{2} x +5 B a \,e^{2} x -4 B b d e x +15 A a \,e^{2}-10 A b d e -10 B a d e +8 B b \,d^{2}\right )}{15 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(e*x+d)^(1/2)*(3*B*b*e^2*x^2+5*A*b*e^2*x+5*B*a*e^2*x-4*B*b*d*e*x+15*A*a*e^2-10*A*b*d*e-10*B*a*d*e+8*B*b*d
^2)/e^3

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maxima [A]  time = 0.55, size = 75, normalized size = 0.93 \[ \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} B b - 5 \, {\left (2 \, B b d - {\left (B a + A b\right )} e\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 15 \, {\left (B b d^{2} + A a e^{2} - {\left (B a + A b\right )} d e\right )} \sqrt {e x + d}\right )}}{15 \, e^{3}} \]

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]

2/15*(3*(e*x + d)^(5/2)*B*b - 5*(2*B*b*d - (B*a + A*b)*e)*(e*x + d)^(3/2) + 15*(B*b*d^2 + A*a*e^2 - (B*a + A*b
)*d*e)*sqrt(e*x + d))/e^3

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mupad [B]  time = 1.21, size = 80, normalized size = 0.99 \[ \frac {2\,\sqrt {d+e\,x}\,\left (3\,B\,b\,{\left (d+e\,x\right )}^2+15\,A\,a\,e^2+15\,B\,b\,d^2+5\,A\,b\,e\,\left (d+e\,x\right )+5\,B\,a\,e\,\left (d+e\,x\right )-10\,B\,b\,d\,\left (d+e\,x\right )-15\,A\,b\,d\,e-15\,B\,a\,d\,e\right )}{15\,e^3} \]

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)^(1/2)*(3*B*b*(d + e*x)^2 + 15*A*a*e^2 + 15*B*b*d^2 + 5*A*b*e*(d + e*x) + 5*B*a*e*(d + e*x) - 10*B
*b*d*(d + e*x) - 15*A*b*d*e - 15*B*a*d*e))/(15*e^3)

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sympy [A]  time = 26.72, size = 311, normalized size = 3.84 \[ \begin {cases} \frac {- \frac {2 A a d}{\sqrt {d + e x}} - 2 A a \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - \frac {2 A b d \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {2 A b \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {2 B a d \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {2 B a \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {2 B b d \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e^{2}} - \frac {2 B b \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}}}{e} & \text {for}\: e \neq 0 \\\frac {A a x + \frac {B b x^{3}}{3} + \frac {x^{2} \left (A b + B a\right )}{2}}{\sqrt {d}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-2*A*a*d/sqrt(d + e*x) - 2*A*a*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 2*A*b*d*(-d/sqrt(d + e*x) - sq
rt(d + e*x))/e - 2*A*b*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 2*B*a*d*(-d/sqrt(d +
e*x) - sqrt(d + e*x))/e - 2*B*a*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 2*B*b*d*(d**
2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 2*B*b*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d +
e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2)/e, Ne(e, 0)), ((A*a*x + B*b*x**3/3 + x**2*(A*b + B*a)/2)
/sqrt(d), True))

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