\(\int \frac {(a+c x^2)^{3/2}}{(d+e x)^3} \, dx\) [540]

Optimal result

Integrand size = 19, antiderivative size = 161 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {3 c (2 d+e x) \sqrt {a+c x^2}}{2 e^3 (d+e x)}-\frac {\left (a+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac {3 c^{3/2} d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{e^4}-\frac {3 c \left (2 c d^2+a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 e^4 \sqrt {c d^2+a e^2}} \]

[Out]

-1/2*(c*x^2+a)^(3/2)/e/(e*x+d)^2-3*c^(3/2)*d*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/e^4-3/2*c*(a*e^2+2*c*d^2)*arct
anh((-c*d*x+a*e)/(a*e^2+c*d^2)^(1/2)/(c*x^2+a)^(1/2))/e^4/(a*e^2+c*d^2)^(1/2)+3/2*c*(e*x+2*d)*(c*x^2+a)^(1/2)/
e^3/(e*x+d)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {747, 827, 858, 223, 212, 739} \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=-\frac {3 c^{3/2} d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{e^4}-\frac {3 c \left (a e^2+2 c d^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 e^4 \sqrt {a e^2+c d^2}}+\frac {3 c \sqrt {a+c x^2} (2 d+e x)}{2 e^3 (d+e x)}-\frac {\left (a+c x^2\right )^{3/2}}{2 e (d+e x)^2} \]

[In]

Int[(a + c*x^2)^(3/2)/(d + e*x)^3,x]

[Out]

(3*c*(2*d + e*x)*Sqrt[a + c*x^2])/(2*e^3*(d + e*x)) - (a + c*x^2)^(3/2)/(2*e*(d + e*x)^2) - (3*c^(3/2)*d*ArcTa
nh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/e^4 - (3*c*(2*c*d^2 + a*e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[
a + c*x^2])])/(2*e^4*Sqrt[c*d^2 + a*e^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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 747

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + c*x^2)^p/(e
*(m + 1))), x] - Dist[2*c*(p/(e*(m + 1))), Int[x*(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c,
d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m +
2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 827

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
+ 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

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

Mathematica [A] (verified)

Time = 1.27 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {\frac {e \sqrt {a+c x^2} \left (6 c d^2-a e^2+9 c d e x+2 c e^2 x^2\right )}{(d+e x)^2}-\frac {6 c \left (2 c d^2+a e^2\right ) \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\sqrt {-c d^2-a e^2}}+6 c^{3/2} d \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{2 e^4} \]

[In]

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

[Out]

((e*Sqrt[a + c*x^2]*(6*c*d^2 - a*e^2 + 9*c*d*e*x + 2*c*e^2*x^2))/(d + e*x)^2 - (6*c*(2*c*d^2 + a*e^2)*ArcTan[(
Sqrt[c]*(d + e*x) - e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/Sqrt[-(c*d^2) - a*e^2] + 6*c^(3/2)*d*Log[-(Sqr
t[c]*x) + Sqrt[a + c*x^2]])/(2*e^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(883\) vs. \(2(139)=278\).

Time = 2.04 (sec) , antiderivative size = 884, normalized size of antiderivative = 5.49

method	result	size
risch	\(\frac {\sqrt {c \,x^{2}+a}\, c}{e^{3}}-\frac {\frac {3 c^{\frac {3}{2}} d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{e}+\frac {2 c \left (e^{2} a +3 c \,d^{2}\right ) \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}+\frac {4 c d \left (e^{2} a +c \,d^{2}\right ) \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c d e \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{3}}+\frac {\left (-a^{2} e^{4}-2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{2 \left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {3 c d e \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c d e \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{2 \left (e^{2} a +c \,d^{2}\right )}+\frac {c \,e^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{4}}}{e^{3}}\)	\(884\)
default	\(\text {Expression too large to display}\)	\(1471\)

[In]

int((c*x^2+a)^(3/2)/(e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (140) = 280\).

Time = 0.60 (sec) , antiderivative size = 1545, normalized size of antiderivative = 9.60 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^3,x, algorithm="fricas")

[Out]

[1/4*(6*(c^2*d^5 + a*c*d^3*e^2 + (c^2*d^3*e^2 + a*c*d*e^4)*x^2 + 2*(c^2*d^4*e + a*c*d^2*e^3)*x)*sqrt(c)*log(-2
*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 3*(2*c^2*d^4 + a*c*d^2*e^2 + (2*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(2*
c^2*d^3*e + a*c*d*e^3)*x)*sqrt(c*d^2 + a*e^2)*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x
^2 - 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(6*c^2*d^4*e + 5*a*c*
d^2*e^3 - a^2*e^5 + 2*(c^2*d^2*e^3 + a*c*e^5)*x^2 + 9*(c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c*d^4*e^4
+ a*d^2*e^6 + (c*d^2*e^6 + a*e^8)*x^2 + 2*(c*d^3*e^5 + a*d*e^7)*x), 1/4*(12*(c^2*d^5 + a*c*d^3*e^2 + (c^2*d^3
*e^2 + a*c*d*e^4)*x^2 + 2*(c^2*d^4*e + a*c*d^2*e^3)*x)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + 3*(2*c^2*
d^4 + a*c*d^2*e^2 + (2*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(2*c^2*d^3*e + a*c*d*e^3)*x)*sqrt(c*d^2 + a*e^2)*log((2*
a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 - 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 +
a))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(6*c^2*d^4*e + 5*a*c*d^2*e^3 - a^2*e^5 + 2*(c^2*d^2*e^3 + a*c*e^5)*x^2 + 9
*(c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c*d^4*e^4 + a*d^2*e^6 + (c*d^2*e^6 + a*e^8)*x^2 + 2*(c*d^3*e^5
+ a*d*e^7)*x), -1/2*(3*(2*c^2*d^4 + a*c*d^2*e^2 + (2*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(2*c^2*d^3*e + a*c*d*e^3)
*x)*sqrt(-c*d^2 - a*e^2)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a)/(a*c*d^2 + a^2*e^2 + (c^2*d
^2 + a*c*e^2)*x^2)) - 3*(c^2*d^5 + a*c*d^3*e^2 + (c^2*d^3*e^2 + a*c*d*e^4)*x^2 + 2*(c^2*d^4*e + a*c*d^2*e^3)*x
)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - (6*c^2*d^4*e + 5*a*c*d^2*e^3 - a^2*e^5 + 2*(c^2*d^
2*e^3 + a*c*e^5)*x^2 + 9*(c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c*d^4*e^4 + a*d^2*e^6 + (c*d^2*e^6 + a
*e^8)*x^2 + 2*(c*d^3*e^5 + a*d*e^7)*x), -1/2*(3*(2*c^2*d^4 + a*c*d^2*e^2 + (2*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(
2*c^2*d^3*e + a*c*d*e^3)*x)*sqrt(-c*d^2 - a*e^2)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a)/(a*
c*d^2 + a^2*e^2 + (c^2*d^2 + a*c*e^2)*x^2)) - 6*(c^2*d^5 + a*c*d^3*e^2 + (c^2*d^3*e^2 + a*c*d*e^4)*x^2 + 2*(c^
2*d^4*e + a*c*d^2*e^3)*x)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (6*c^2*d^4*e + 5*a*c*d^2*e^3 - a^2*e^5
+ 2*(c^2*d^2*e^3 + a*c*e^5)*x^2 + 9*(c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c*d^4*e^4 + a*d^2*e^6 + (c
*d^2*e^6 + a*e^8)*x^2 + 2*(c*d^3*e^5 + a*d*e^7)*x)]

Sympy [F]

\[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{3}}\, dx \]

[In]

integrate((c*x**2+a)**(3/2)/(e*x+d)**3,x)

[Out]

Integral((a + c*x**2)**(3/2)/(d + e*x)**3, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (140) = 280\).

Time = 0.31 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.17 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {3 \, c^{\frac {3}{2}} d \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{e^{4}} + \frac {\sqrt {c x^{2} + a} c}{e^{3}} + \frac {3 \, {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{\sqrt {-c d^{2} - a e^{2}} e^{4}} + \frac {6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{2} d^{2} e + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c e^{3} + 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} c^{\frac {5}{2}} d^{3} - 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {3}{2}} d e^{2} - 14 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a c^{2} d^{2} e + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c e^{3} + 5 \, a^{2} c^{\frac {3}{2}} d e^{2}}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} \sqrt {c} d - a e\right )}^{2} e^{4}} \]

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^3,x, algorithm="giac")

[Out]

3*c^(3/2)*d*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/e^4 + sqrt(c*x^2 + a)*c/e^3 + 3*(2*c^2*d^2 + a*c*e^2)*arcta
n(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/(sqrt(-c*d^2 - a*e^2)*e^4) + (6*(sqrt(c
)*x - sqrt(c*x^2 + a))^3*c^2*d^2*e + (sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c*e^3 + 10*(sqrt(c)*x - sqrt(c*x^2 + a)
)^2*c^(5/2)*d^3 - 5*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(3/2)*d*e^2 - 14*(sqrt(c)*x - sqrt(c*x^2 + a))*a*c^2*d
^2*e + (sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c*e^3 + 5*a^2*c^(3/2)*d*e^2)/(((sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*
(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^2*e^4)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{3/2}}{{\left (d+e\,x\right )}^3} \,d x \]

[In]

int((a + c*x^2)^(3/2)/(d + e*x)^3,x)

[Out]

int((a + c*x^2)^(3/2)/(d + e*x)^3, x)