3.4.31 \(\int \frac {(d+e x)^2}{(b x+c x^2)^{5/2}} \, dx\) [331]

Optimal. Leaf size=78 \[ -\frac {2 (b+2 c x) (d+e x)^2}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {8 (2 c d-b e) (b d+(2 c d-b e) x)}{3 b^4 \sqrt {b x+c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {742, 650} \begin {gather*} \frac {8 (2 c d-b e) (x (2 c d-b e)+b d)}{3 b^4 \sqrt {b x+c x^2}}-\frac {2 (b+2 c x) (d+e x)^2}{3 b^2 \left (b x+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 650

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

Rule 742

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

Rubi steps

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

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Mathematica [A]
time = 0.19, size = 95, normalized size = 1.22 \begin {gather*} \frac {32 c^3 d^2 x^3+16 b c^2 d x^2 (3 d-2 e x)-2 b^3 \left (d^2+6 d e x-3 e^2 x^2\right )+4 b^2 c x \left (3 d^2-12 d e x+e^2 x^2\right )}{3 b^4 (x (b+c x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(221\) vs. \(2(70)=140\).
time = 0.47, size = 222, normalized size = 2.85

method	result	size
risch	\(-\frac {2 d \left (c x +b \right ) \left (6 b e x -8 c d x +b d \right )}{3 b^{4} x \sqrt {x \left (c x +b \right )}}+\frac {2 x \left (2 b e x c -8 c^{2} d x +3 b^{2} e -9 b c d \right ) \left (b e -c d \right )}{3 \sqrt {x \left (c x +b \right )}\, \left (c x +b \right ) b^{4}}\)	\(94\)
gosper	\(-\frac {2 x \left (c x +b \right ) \left (-2 b^{2} c \,e^{2} x^{3}+16 b \,c^{2} d e \,x^{3}-16 c^{3} d^{2} x^{3}-3 b^{3} e^{2} x^{2}+24 b^{2} c d e \,x^{2}-24 b \,c^{2} d^{2} x^{2}+6 b^{3} d e x -6 b^{2} c \,d^{2} x +d^{2} b^{3}\right )}{3 b^{4} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}\)	\(117\)
trager	\(-\frac {2 \left (-2 b^{2} c \,e^{2} x^{3}+16 b \,c^{2} d e \,x^{3}-16 c^{3} d^{2} x^{3}-3 b^{3} e^{2} x^{2}+24 b^{2} c d e \,x^{2}-24 b \,c^{2} d^{2} x^{2}+6 b^{3} d e x -6 b^{2} c \,d^{2} x +d^{2} b^{3}\right ) \sqrt {c \,x^{2}+b x}}{3 b^{4} x^{2} \left (c x +b \right )^{2}}\)	\(121\)
default	\(e^{2} \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )+2 d e \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )+d^{2} \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )\)	\(222\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (73) = 146\).
time = 0.28, size = 203, normalized size = 2.60 \begin {gather*} -\frac {4 \, c d^{2} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, c^{2} d^{2} x}{3 \, \sqrt {c x^{2} + b x} b^{4}} + \frac {4 \, d x e}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {32 \, c d x e}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {2 \, d^{2}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, c d^{2}}{3 \, \sqrt {c x^{2} + b x} b^{3}} + \frac {4 \, x e^{2}}{3 \, \sqrt {c x^{2} + b x} b^{2}} - \frac {2 \, x e^{2}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} - \frac {16 \, d e}{3 \, \sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, e^{2}}{3 \, \sqrt {c x^{2} + b x} b c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.62, size = 137, normalized size = 1.76 \begin {gather*} \frac {2 \, {\left (16 \, c^{3} d^{2} x^{3} + 24 \, b c^{2} d^{2} x^{2} + 6 \, b^{2} c d^{2} x - b^{3} d^{2} + {\left (2 \, b^{2} c x^{3} + 3 \, b^{3} x^{2}\right )} e^{2} - 2 \, {\left (8 \, b c^{2} d x^{3} + 12 \, b^{2} c d x^{2} + 3 \, b^{3} d 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)^2/(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [A]
time = 1.29, size = 111, normalized size = 1.42 \begin {gather*} \frac {2 \, {\left ({\left (x {\left (\frac {2 \, {\left (8 \, c^{3} d^{2} - 8 \, b c^{2} d e + b^{2} c e^{2}\right )} x}{b^{4}} + \frac {3 \, {\left (8 \, b c^{2} d^{2} - 8 \, b^{2} c d e + b^{3} e^{2}\right )}}{b^{4}}\right )} + \frac {6 \, {\left (b^{2} c d^{2} - b^{3} d e\right )}}{b^{4}}\right )} x - \frac {d^{2}}{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)^2/(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.36, size = 111, normalized size = 1.42 \begin {gather*} \frac {2\,\left (-b^3\,d^2-6\,b^3\,d\,e\,x+3\,b^3\,e^2\,x^2+6\,b^2\,c\,d^2\,x-24\,b^2\,c\,d\,e\,x^2+2\,b^2\,c\,e^2\,x^3+24\,b\,c^2\,d^2\,x^2-16\,b\,c^2\,d\,e\,x^3+16\,c^3\,d^2\,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)^2/(b*x + c*x^2)^(5/2),x)

[Out]

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

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