3.4.16 \(\int \frac {(d+e x)^2}{\sqrt {b x+c x^2}} \, dx\) [316]

Optimal. Leaf size=110 \[ \frac {3 e (2 c d-b e) \sqrt {b x+c x^2}}{4 c^2}+\frac {e (d+e x) \sqrt {b x+c x^2}}{2 c}+\frac {\left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{5/2}} \]

[Out]

1/4*(3*b^2*e^2-8*b*c*d*e+8*c^2*d^2)*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2))/c^(5/2)+3/4*e*(-b*e+2*c*d)*(c*x^2+b*x
)^(1/2)/c^2+1/2*e*(e*x+d)*(c*x^2+b*x)^(1/2)/c

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Rubi [A]
time = 0.05, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {756, 654, 634, 212} \begin {gather*} \frac {\left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{5/2}}+\frac {3 e \sqrt {b x+c x^2} (2 c d-b e)}{4 c^2}+\frac {e \sqrt {b x+c x^2} (d+e x)}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2/Sqrt[b*x + c*x^2],x]

[Out]

(3*e*(2*c*d - b*e)*Sqrt[b*x + c*x^2])/(4*c^2) + (e*(d + e*x)*Sqrt[b*x + c*x^2])/(2*c) + ((8*c^2*d^2 - 8*b*c*d*
e + 3*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*c^(5/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 634

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 654

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

Rule 756

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, 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]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rubi steps

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

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Mathematica [A]
time = 0.13, size = 109, normalized size = 0.99 \begin {gather*} \frac {\sqrt {c} e x (b+c x) (8 c d-3 b e+2 c e x)+\left (-8 c^2 d^2+8 b c d e-3 b^2 e^2\right ) \sqrt {x} \sqrt {b+c x} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{4 c^{5/2} \sqrt {x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c]*e*x*(b + c*x)*(8*c*d - 3*b*e + 2*c*e*x) + (-8*c^2*d^2 + 8*b*c*d*e - 3*b^2*e^2)*Sqrt[x]*Sqrt[b + c*x]*
Log[-(Sqrt[c]*Sqrt[x]) + Sqrt[b + c*x]])/(4*c^(5/2)*Sqrt[x*(b + c*x)])

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Maple [A]
time = 0.44, size = 157, normalized size = 1.43

method	result	size
risch	\(-\frac {\left (-2 c e x +3 b e -8 c d \right ) e x \left (c x +b \right )}{4 c^{2} \sqrt {x \left (c x +b \right )}}+\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) b^{2} e^{2}}{8 c^{\frac {5}{2}}}-\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) b d e}{c^{\frac {3}{2}}}+\frac {d^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{\sqrt {c}}\)	\(135\)
default	\(e^{2} \left (\frac {x \sqrt {c \,x^{2}+b x}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\right )+2 d e \left (\frac {\sqrt {c \,x^{2}+b x}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}\right )+\frac {d^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{\sqrt {c}}\)	\(157\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 153, normalized size = 1.39 \begin {gather*} \frac {d^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{\sqrt {c}} - \frac {b d e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{c^{\frac {3}{2}}} + \frac {\sqrt {c x^{2} + b x} x e^{2}}{2 \, c} + \frac {2 \, \sqrt {c x^{2} + b x} d e}{c} + \frac {3 \, b^{2} e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {5}{2}}} - \frac {3 \, \sqrt {c x^{2} + b x} b e^{2}}{4 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2/(c*x^2+b*x)^(1/2),x, algorithm="maxima")

[Out]

d^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/sqrt(c) - b*d*e*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/
c^(3/2) + 1/2*sqrt(c*x^2 + b*x)*x*e^2/c + 2*sqrt(c*x^2 + b*x)*d*e/c + 3/8*b^2*e^2*log(2*c*x + b + 2*sqrt(c*x^2
+ b*x)*sqrt(c))/c^(5/2) - 3/4*sqrt(c*x^2 + b*x)*b*e^2/c^2

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Fricas [A]
time = 1.29, size = 188, normalized size = 1.71 \begin {gather*} \left [\frac {{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (8 \, c^{2} d e + {\left (2 \, c^{2} x - 3 \, b c\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{8 \, c^{3}}, -\frac {{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (8 \, c^{2} d e + {\left (2 \, c^{2} x - 3 \, b c\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{4 \, c^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2/(c*x^2+b*x)^(1/2),x, algorithm="fricas")

[Out]

[1/8*((8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(8*c^2*d*e
+ (2*c^2*x - 3*b*c)*e^2)*sqrt(c*x^2 + b*x))/c^3, -1/4*((8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*sqrt(-c)*arctan(sqr
t(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (8*c^2*d*e + (2*c^2*x - 3*b*c)*e^2)*sqrt(c*x^2 + b*x))/c^3]

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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}}{\sqrt {x \left (b + c x\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2/(c*x**2+b*x)**(1/2),x)

[Out]

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

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Giac [A]
time = 1.53, size = 97, normalized size = 0.88 \begin {gather*} \frac {1}{4} \, \sqrt {c x^{2} + b x} {\left (\frac {2 \, x e^{2}}{c} + \frac {8 \, c d e - 3 \, b e^{2}}{c^{2}}\right )} - \frac {{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{8 \, c^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2/(c*x^2+b*x)^(1/2),x, algorithm="giac")

[Out]

1/4*sqrt(c*x^2 + b*x)*(2*x*e^2/c + (8*c*d*e - 3*b*e^2)/c^2) - 1/8*(8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*log(abs(
-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(5/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^2}{\sqrt {c\,x^2+b\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^2/(b*x + c*x^2)^(1/2),x)

[Out]

int((d + e*x)^2/(b*x + c*x^2)^(1/2), x)

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