3.4.32 \(\int \frac {d+e x}{(b x+c x^2)^{5/2}} \, dx\) [332]

Optimal. Leaf size=71 \[ -\frac {2 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {8 (2 c d-b e) (b+2 c x)}{3 b^4 \sqrt {b x+c x^2}} \]

[Out]

-2/3*(b*d+(-b*e+2*c*d)*x)/b^2/(c*x^2+b*x)^(3/2)+8/3*(-b*e+2*c*d)*(2*c*x+b)/b^4/(c*x^2+b*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {652, 627} \begin {gather*} \frac {8 (b+2 c x) (2 c d-b e)}{3 b^4 \sqrt {b x+c x^2}}-\frac {2 (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(b*x + c*x^2)^(5/2),x]

[Out]

(-2*(b*d + (2*c*d - b*e)*x))/(3*b^2*(b*x + c*x^2)^(3/2)) + (8*(2*c*d - b*e)*(b + 2*c*x))/(3*b^4*Sqrt[b*x + c*x
^2])

Rule 627

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 652

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), x] - Dist[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 -
4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rubi steps

\begin {align*} \int \frac {d+e x}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {(4 (2 c d-b e)) \int \frac {1}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2}\\ &=-\frac {2 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {8 (2 c d-b e) (b+2 c x)}{3 b^4 \sqrt {b x+c x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.14, size = 67, normalized size = 0.94 \begin {gather*} -\frac {2 \left (-16 c^3 d x^3-6 b^2 c x (d-2 e x)+8 b c^2 x^2 (-3 d+e x)+b^3 (d+3 e x)\right )}{3 b^4 (x (b+c x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(b*x + c*x^2)^(5/2),x]

[Out]

(-2*(-16*c^3*d*x^3 - 6*b^2*c*x*(d - 2*e*x) + 8*b*c^2*x^2*(-3*d + e*x) + b^3*(d + 3*e*x)))/(3*b^4*(x*(b + c*x))
^(3/2))

________________________________________________________________________________________

Maple [A]
time = 0.43, size = 121, normalized size = 1.70

method	result	size
gosper	\(-\frac {2 x \left (c x +b \right ) \left (8 b \,c^{2} x^{3} e -16 c^{3} d \,x^{3}+12 b^{2} c e \,x^{2}-24 b \,c^{2} d \,x^{2}+3 b^{3} e x -6 b^{2} c d x +b^{3} d \right )}{3 b^{4} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}\)	\(83\)
risch	\(-\frac {2 \left (c x +b \right ) \left (3 b e x -8 c d x +b d \right )}{3 b^{4} x \sqrt {x \left (c x +b \right )}}-\frac {2 c \left (5 b e x c -8 c^{2} d x +6 b^{2} e -9 b c d \right ) x}{3 \sqrt {x \left (c x +b \right )}\, \left (c x +b \right ) b^{4}}\)	\(86\)
trager	\(-\frac {2 \left (8 b \,c^{2} x^{3} e -16 c^{3} d \,x^{3}+12 b^{2} c e \,x^{2}-24 b \,c^{2} d \,x^{2}+3 b^{3} e x -6 b^{2} c d x +b^{3} d \right ) \sqrt {c \,x^{2}+b x}}{3 b^{4} x^{2} \left (c x +b \right )^{2}}\)	\(87\)
default	\(e \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )+d \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*(-1/3/c/(c*x^2+b*x)^(3/2)-1/2*b/c*(-2/3*(2*c*x+b)/b^2/(c*x^2+b*x)^(3/2)+16/3*c*(2*c*x+b)/b^4/(c*x^2+b*x)^(1/
2)))+d*(-2/3*(2*c*x+b)/b^2/(c*x^2+b*x)^(3/2)+16/3*c*(2*c*x+b)/b^4/(c*x^2+b*x)^(1/2))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (65) = 130\).
time = 0.28, size = 133, normalized size = 1.87 \begin {gather*} -\frac {4 \, c d x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, c^{2} d x}{3 \, \sqrt {c x^{2} + b x} b^{4}} + \frac {2 \, x e}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {16 \, c x e}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {2 \, d}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, c d}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {8 \, e}{3 \, \sqrt {c x^{2} + b x} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

-4/3*c*d*x/((c*x^2 + b*x)^(3/2)*b^2) + 32/3*c^2*d*x/(sqrt(c*x^2 + b*x)*b^4) + 2/3*x*e/((c*x^2 + b*x)^(3/2)*b)
- 16/3*c*x*e/(sqrt(c*x^2 + b*x)*b^3) - 2/3*d/((c*x^2 + b*x)^(3/2)*b) + 16/3*c*d/(sqrt(c*x^2 + b*x)*b^3) - 8/3*
e/(sqrt(c*x^2 + b*x)*b^2)

________________________________________________________________________________________

Fricas [A]
time = 1.62, size = 105, normalized size = 1.48 \begin {gather*} \frac {2 \, {\left (16 \, c^{3} d x^{3} + 24 \, b c^{2} d x^{2} + 6 \, b^{2} c d x - b^{3} d - {\left (8 \, b c^{2} x^{3} + 12 \, b^{2} c x^{2} + 3 \, b^{3} x\right )} e\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

2/3*(16*c^3*d*x^3 + 24*b*c^2*d*x^2 + 6*b^2*c*d*x - b^3*d - (8*b*c^2*x^3 + 12*b^2*c*x^2 + 3*b^3*x)*e)*sqrt(c*x^
2 + b*x)/(b^4*c^2*x^4 + 2*b^5*c*x^3 + b^6*x^2)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d + e x}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x**2+b*x)**(5/2),x)

[Out]

Integral((d + e*x)/(x*(b + c*x))**(5/2), x)

________________________________________________________________________________________

Giac [A]
time = 2.30, size = 89, normalized size = 1.25 \begin {gather*} \frac {2 \, {\left ({\left (4 \, x {\left (\frac {2 \, {\left (2 \, c^{3} d - b c^{2} e\right )} x}{b^{4}} + \frac {3 \, {\left (2 \, b c^{2} d - b^{2} c e\right )}}{b^{4}}\right )} + \frac {3 \, {\left (2 \, b^{2} c d - b^{3} e\right )}}{b^{4}}\right )} x - \frac {d}{b}\right )}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

2/3*((4*x*(2*(2*c^3*d - b*c^2*e)*x/b^4 + 3*(2*b*c^2*d - b^2*c*e)/b^4) + 3*(2*b^2*c*d - b^3*e)/b^4)*x - d/b)/(c
*x^2 + b*x)^(3/2)

________________________________________________________________________________________

Mupad [B]
time = 0.30, size = 76, normalized size = 1.07 \begin {gather*} -\frac {2\,\left (3\,e\,b^3\,x+d\,b^3+12\,e\,b^2\,c\,x^2-6\,d\,b^2\,c\,x+8\,e\,b\,c^2\,x^3-24\,d\,b\,c^2\,x^2-16\,d\,c^3\,x^3\right )}{3\,b^4\,{\left (c\,x^2+b\,x\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(b*x + c*x^2)^(5/2),x)

[Out]

-(2*(b^3*d - 16*c^3*d*x^3 + 3*b^3*e*x - 6*b^2*c*d*x - 24*b*c^2*d*x^2 + 12*b^2*c*e*x^2 + 8*b*c^2*e*x^3))/(3*b^4
*(b*x + c*x^2)^(3/2))

________________________________________________________________________________________