3.18.23 \(\int \frac {(a+b x) (A+B x)}{(d+e x)^{3/2}} \, dx\) [1723]

Optimal. Leaf size=79 \[ -\frac {2 (b d-a e) (B d-A e)}{e^3 \sqrt {d+e x}}-\frac {2 (2 b B d-A b e-a B e) \sqrt {d+e x}}{e^3}+\frac {2 b B (d+e x)^{3/2}}{3 e^3} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 79, 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 \sqrt {d+e x} (-a B e-A b e+2 b B d)}{e^3}-\frac {2 (b d-a e) (B d-A e)}{e^3 \sqrt {d+e x}}+\frac {2 b B (d+e x)^{3/2}}{3 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))/(e^3*Sqrt[d + e*x]) - (2*(2*b*B*d - A*b*e - a*B*e)*Sqrt[d + e*x])/e^3 + (2*b*B*(d
+ e*x)^(3/2))/(3*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 \frac {(a+b x) (A+B x)}{(d+e x)^{3/2}} \, dx &=\int \left (\frac {(-b d+a e) (-B d+A e)}{e^2 (d+e x)^{3/2}}+\frac {-2 b B d+A b e+a B e}{e^2 \sqrt {d+e x}}+\frac {b B \sqrt {d+e x}}{e^2}\right ) \, dx\\ &=-\frac {2 (b d-a e) (B d-A e)}{e^3 \sqrt {d+e x}}-\frac {2 (2 b B d-A b e-a B e) \sqrt {d+e x}}{e^3}+\frac {2 b B (d+e x)^{3/2}}{3 e^3}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 68, normalized size = 0.86 \begin {gather*} \frac {6 A b e (2 d+e x)+6 a e (2 B d-A e+B e x)+2 b B \left (-8 d^2-4 d e x+e^2 x^2\right )}{3 e^3 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*A*b*e*(2*d + e*x) + 6*a*e*(2*B*d - A*e + B*e*x) + 2*b*B*(-8*d^2 - 4*d*e*x + e^2*x^2))/(3*e^3*Sqrt[d + e*x])

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Maple [A]
time = 0.08, size = 86, normalized size = 1.09

method	result	size
risch	\(\frac {2 \left (b B x e +3 A b e +3 B a e -5 B b d \right ) \sqrt {e x +d}}{3 e^{3}}-\frac {2 \left (A a \,e^{2}-A b d e -B a d e +B b \,d^{2}\right )}{e^{3} \sqrt {e x +d}}\)	\(72\)
gosper	\(-\frac {2 \left (-b B \,x^{2} e^{2}-3 A b \,e^{2} x -3 B a \,e^{2} x +4 B b d e x +3 A a \,e^{2}-6 A b d e -6 B a d e +8 B b \,d^{2}\right )}{3 \sqrt {e x +d}\, e^{3}}\)	\(73\)
trager	\(-\frac {2 \left (-b B \,x^{2} e^{2}-3 A b \,e^{2} x -3 B a \,e^{2} x +4 B b d e x +3 A a \,e^{2}-6 A b d e -6 B a d e +8 B b \,d^{2}\right )}{3 \sqrt {e x +d}\, e^{3}}\)	\(73\)
derivativedivides	\(\frac {\frac {2 B b \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A b e \sqrt {e x +d}+2 a e B \sqrt {e x +d}-4 B b d \sqrt {e x +d}-\frac {2 \left (A a \,e^{2}-A b d e -B a d e +B b \,d^{2}\right )}{\sqrt {e x +d}}}{e^{3}}\)	\(86\)
default	\(\frac {\frac {2 B b \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A b e \sqrt {e x +d}+2 a e B \sqrt {e x +d}-4 B b d \sqrt {e x +d}-\frac {2 \left (A a \,e^{2}-A b d e -B a d e +B b \,d^{2}\right )}{\sqrt {e x +d}}}{e^{3}}\)	\(86\)

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

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Maxima [A]
time = 0.32, size = 86, normalized size = 1.09 \begin {gather*} \frac {2}{3} \, {\left ({\left ({\left (x e + d\right )}^{\frac {3}{2}} B b - 3 \, {\left (2 \, B b d - B a e - A b e\right )} \sqrt {x e + d}\right )} e^{\left (-2\right )} - \frac {3 \, {\left (B b d^{2} + A a e^{2} - {\left (B a e + A b e\right )} d\right )} e^{\left (-2\right )}}{\sqrt {x e + d}}\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="maxima")

[Out]

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

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Fricas [A]
time = 1.44, size = 75, normalized size = 0.95 \begin {gather*} -\frac {2 \, {\left (8 \, B b d^{2} - {\left (B b x^{2} - 3 \, A a + 3 \, {\left (B a + A b\right )} x\right )} e^{2} + 2 \, {\left (2 \, B b d x - 3 \, {\left (B a + A b\right )} d\right )} e\right )} \sqrt {x e + d}}{3 \, {\left (x e^{4} + d e^{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]

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

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Sympy [A]
time = 9.00, size = 76, normalized size = 0.96 \begin {gather*} \frac {2 B b \left (d + e x\right )^{\frac {3}{2}}}{3 e^{3}} + \frac {\sqrt {d + e x} \left (2 A b e + 2 B a e - 4 B b d\right )}{e^{3}} + \frac {2 \left (- A e + B d\right ) \left (a e - b d\right )}{e^{3} \sqrt {d + e x}} \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]

2*B*b*(d + e*x)**(3/2)/(3*e**3) + sqrt(d + e*x)*(2*A*b*e + 2*B*a*e - 4*B*b*d)/e**3 + 2*(-A*e + B*d)*(a*e - b*d
)/(e**3*sqrt(d + e*x))

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Giac [A]
time = 0.57, size = 100, normalized size = 1.27 \begin {gather*} \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B b e^{6} - 6 \, \sqrt {x e + d} B b d e^{6} + 3 \, \sqrt {x e + d} B a e^{7} + 3 \, \sqrt {x e + d} A b e^{7}\right )} e^{\left (-9\right )} - \frac {2 \, {\left (B b d^{2} - B a d e - A b d e + A a e^{2}\right )} e^{\left (-3\right )}}{\sqrt {x e + d}} \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]

2/3*((x*e + d)^(3/2)*B*b*e^6 - 6*sqrt(x*e + d)*B*b*d*e^6 + 3*sqrt(x*e + d)*B*a*e^7 + 3*sqrt(x*e + d)*A*b*e^7)*
e^(-9) - 2*(B*b*d^2 - B*a*d*e - A*b*d*e + A*a*e^2)*e^(-3)/sqrt(x*e + d)

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Mupad [B]
time = 0.06, size = 79, normalized size = 1.00 \begin {gather*} \frac {\frac {2\,B\,b\,{\left (d+e\,x\right )}^2}{3}-2\,A\,a\,e^2-2\,B\,b\,d^2+2\,A\,b\,e\,\left (d+e\,x\right )+2\,B\,a\,e\,\left (d+e\,x\right )-4\,B\,b\,d\,\left (d+e\,x\right )+2\,A\,b\,d\,e+2\,B\,a\,d\,e}{e^3\,\sqrt {d+e\,x}} \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*B*b*(d + e*x)^2)/3 - 2*A*a*e^2 - 2*B*b*d^2 + 2*A*b*e*(d + e*x) + 2*B*a*e*(d + e*x) - 4*B*b*d*(d + e*x) + 2
*A*b*d*e + 2*B*a*d*e)/(e^3*(d + e*x)^(1/2))

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