\(\int \frac {(a+c x^2)^{5/2}}{(d+e x)^3} \, dx\) [551]

Optimal result

Integrand size = 19, antiderivative size = 213 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {5 c \left (4 c d^2+a e^2-2 c d e x\right ) \sqrt {a+c x^2}}{2 e^5}+\frac {5 c (4 d+e x) \left (a+c x^2\right )^{3/2}}{6 e^3 (d+e x)}-\frac {\left (a+c x^2\right )^{5/2}}{2 e (d+e x)^2}-\frac {5 c^{3/2} d \left (4 c d^2+3 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 e^6}-\frac {5 c \sqrt {c d^2+a e^2} \left (4 c d^2+a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 e^6} \]

[Out]

5/6*c*(e*x+4*d)*(c*x^2+a)^(3/2)/e^3/(e*x+d)-1/2*(c*x^2+a)^(5/2)/e/(e*x+d)^2-5/2*c^(3/2)*d*(3*a*e^2+4*c*d^2)*ar
ctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/e^6-5/2*c*(a*e^2+4*c*d^2)*arctanh((-c*d*x+a*e)/(a*e^2+c*d^2)^(1/2)/(c*x^2+a)^
(1/2))*(a*e^2+c*d^2)^(1/2)/e^6+5/2*c*(-2*c*d*e*x+a*e^2+4*c*d^2)*(c*x^2+a)^(1/2)/e^5

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {747, 827, 829, 858, 223, 212, 739} \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=-\frac {5 c^{3/2} d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a e^2+4 c d^2\right )}{2 e^6}-\frac {5 c \sqrt {a e^2+c d^2} \left (a e^2+4 c d^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 e^6}+\frac {5 c \sqrt {a+c x^2} \left (a e^2+4 c d^2-2 c d e x\right )}{2 e^5}+\frac {5 c \left (a+c x^2\right )^{3/2} (4 d+e x)}{6 e^3 (d+e x)}-\frac {\left (a+c x^2\right )^{5/2}}{2 e (d+e x)^2} \]

[In]

Int[(a + c*x^2)^(5/2)/(d + e*x)^3,x]

[Out]

(5*c*(4*c*d^2 + a*e^2 - 2*c*d*e*x)*Sqrt[a + c*x^2])/(2*e^5) + (5*c*(4*d + e*x)*(a + c*x^2)^(3/2))/(6*e^3*(d +
e*x)) - (a + c*x^2)^(5/2)/(2*e*(d + e*x)^2) - (5*c^(3/2)*d*(4*c*d^2 + 3*a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*
x^2]])/(2*e^6) - (5*c*Sqrt[c*d^2 + a*e^2]*(4*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^6)

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 829

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
+ 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m +
2*p + 2))), x] + Dist[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 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 )^{5/2}}{2 e (d+e x)^2}+\frac {(5 c) \int \frac {x \left (a+c x^2\right )^{3/2}}{(d+e x)^2} \, dx}{2 e} \\ & = \frac {5 c (4 d+e x) \left (a+c x^2\right )^{3/2}}{6 e^3 (d+e x)}-\frac {\left (a+c x^2\right )^{5/2}}{2 e (d+e x)^2}-\frac {(5 c) \int \frac {(-2 a e+8 c d x) \sqrt {a+c x^2}}{d+e x} \, dx}{4 e^3} \\ & = \frac {5 c \left (4 c d^2+a e^2-2 c d e x\right ) \sqrt {a+c x^2}}{2 e^5}+\frac {5 c (4 d+e x) \left (a+c x^2\right )^{3/2}}{6 e^3 (d+e x)}-\frac {\left (a+c x^2\right )^{5/2}}{2 e (d+e x)^2}-\frac {5 \int \frac {-4 a c e \left (2 c d^2+a e^2\right )+4 c^2 d \left (4 c d^2+3 a e^2\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{8 e^5} \\ & = \frac {5 c \left (4 c d^2+a e^2-2 c d e x\right ) \sqrt {a+c x^2}}{2 e^5}+\frac {5 c (4 d+e x) \left (a+c x^2\right )^{3/2}}{6 e^3 (d+e x)}-\frac {\left (a+c x^2\right )^{5/2}}{2 e (d+e x)^2}+\frac {\left (5 c \left (c d^2+a e^2\right ) \left (4 c d^2+a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{2 e^6}-\frac {\left (5 c^2 d \left (4 c d^2+3 a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 e^6} \\ & = \frac {5 c \left (4 c d^2+a e^2-2 c d e x\right ) \sqrt {a+c x^2}}{2 e^5}+\frac {5 c (4 d+e x) \left (a+c x^2\right )^{3/2}}{6 e^3 (d+e x)}-\frac {\left (a+c x^2\right )^{5/2}}{2 e (d+e x)^2}-\frac {\left (5 c \left (c d^2+a e^2\right ) \left (4 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^6}-\frac {\left (5 c^2 d \left (4 c d^2+3 a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 e^6} \\ & = \frac {5 c \left (4 c d^2+a e^2-2 c d e x\right ) \sqrt {a+c x^2}}{2 e^5}+\frac {5 c (4 d+e x) \left (a+c x^2\right )^{3/2}}{6 e^3 (d+e x)}-\frac {\left (a+c x^2\right )^{5/2}}{2 e (d+e x)^2}-\frac {5 c^{3/2} d \left (4 c d^2+3 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 e^6}-\frac {5 c \sqrt {c d^2+a e^2} \left (4 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^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.66 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {-\frac {e \sqrt {a+c x^2} \left (3 a^2 e^4-a c e^2 \left (35 d^2+55 d e x+14 e^2 x^2\right )-c^2 \left (60 d^4+90 d^3 e x+20 d^2 e^2 x^2-5 d e^3 x^3+2 e^4 x^4\right )\right )}{(d+e x)^2}+30 c \sqrt {-c d^2-a e^2} \left (4 c d^2+a e^2\right ) \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )+15 c^{3/2} d \left (4 c d^2+3 a e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{6 e^6} \]

[In]

Integrate[(a + c*x^2)^(5/2)/(d + e*x)^3,x]

[Out]

(-((e*Sqrt[a + c*x^2]*(3*a^2*e^4 - a*c*e^2*(35*d^2 + 55*d*e*x + 14*e^2*x^2) - c^2*(60*d^4 + 90*d^3*e*x + 20*d^
2*e^2*x^2 - 5*d*e^3*x^3 + 2*e^4*x^4)))/(d + e*x)^2) + 30*c*Sqrt[-(c*d^2) - a*e^2]*(4*c*d^2 + a*e^2)*ArcTan[(Sq
rt[c]*(d + e*x) - e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]] + 15*c^(3/2)*d*(4*c*d^2 + 3*a*e^2)*Log[-(Sqrt[c]*
x) + Sqrt[a + c*x^2]])/(6*e^6)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(967\) vs. \(2(185)=370\).

Time = 2.11 (sec) , antiderivative size = 968, normalized size of antiderivative = 4.54

method	result	size
risch	\(\frac {c \left (2 c \,x^{2} e^{2}-9 x c d e +14 e^{2} a +36 c \,d^{2}\right ) \sqrt {c \,x^{2}+a}}{6 e^{5}}-\frac {\frac {5 c^{\frac {3}{2}} d \left (3 e^{2} a +4 c \,d^{2}\right ) \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{e}+\frac {6 c \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\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 {12 c d \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}}}}{\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 (-2 e^{6} a^{3}-6 d^{2} e^{4} a^{2} c -6 d^{4} e^{2} c^{2} a -2 c^{3} d^{6}\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}}}{2 e^{5}}\)	\(968\)
default	\(\text {Expression too large to display}\)	\(2244\)

[In]

int((c*x^2+a)^(5/2)/(e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

1/6*c*(2*c*e^2*x^2-9*c*d*e*x+14*a*e^2+36*c*d^2)*(c*x^2+a)^(1/2)/e^5-1/2/e^5*(5*c^(3/2)*d*(3*a*e^2+4*c*d^2)/e*l
n(c^(1/2)*x+(c*x^2+a)^(1/2))+6*c*(a^2*e^4+6*a*c*d^2*e^2+5*c^2*d^4)/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))+12*c*d*(a^2*e^4+2*a*c*d^2*e^2+c^2*d^4)/e^3*(-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*(-2*a^3*e
^6-6*a^2*c*d^2*e^4-6*a*c^2*d^4*e^2-2*c^3*d^6)*(-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 [A] (verification not implemented)

none

Time = 1.93 (sec) , antiderivative size = 1553, normalized size of antiderivative = 7.29 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^3,x, algorithm="fricas")

[Out]

[1/12*(15*(4*c^2*d^5 + 3*a*c*d^3*e^2 + (4*c^2*d^3*e^2 + 3*a*c*d*e^4)*x^2 + 2*(4*c^2*d^4*e + 3*a*c*d^2*e^3)*x)*
sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 15*(4*c^2*d^4 + a*c*d^2*e^2 + (4*c^2*d^2*e^2 + a*c*e
^4)*x^2 + 2*(4*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*(2*c^2
*e^5*x^4 - 5*c^2*d*e^4*x^3 + 60*c^2*d^4*e + 35*a*c*d^2*e^3 - 3*a^2*e^5 + 2*(10*c^2*d^2*e^3 + 7*a*c*e^5)*x^2 +
5*(18*c^2*d^3*e^2 + 11*a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(e^8*x^2 + 2*d*e^7*x + d^2*e^6), 1/12*(30*(4*c^2*d^5 + 3
*a*c*d^3*e^2 + (4*c^2*d^3*e^2 + 3*a*c*d*e^4)*x^2 + 2*(4*c^2*d^4*e + 3*a*c*d^2*e^3)*x)*sqrt(-c)*arctan(sqrt(-c)
*x/sqrt(c*x^2 + a)) + 15*(4*c^2*d^4 + a*c*d^2*e^2 + (4*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(4*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*(2*c^2*e^5*x^4 - 5*c^2*d*e^4*x^3 + 60*c^2*
d^4*e + 35*a*c*d^2*e^3 - 3*a^2*e^5 + 2*(10*c^2*d^2*e^3 + 7*a*c*e^5)*x^2 + 5*(18*c^2*d^3*e^2 + 11*a*c*d*e^4)*x)
*sqrt(c*x^2 + a))/(e^8*x^2 + 2*d*e^7*x + d^2*e^6), -1/12*(30*(4*c^2*d^4 + a*c*d^2*e^2 + (4*c^2*d^2*e^2 + a*c*e
^4)*x^2 + 2*(4*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)) - 15*(4*c^2*d^5 + 3*a*c*d^3*e^2 + (4*c^2*d^3*e^2 + 3*
a*c*d*e^4)*x^2 + 2*(4*c^2*d^4*e + 3*a*c*d^2*e^3)*x)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) -
2*(2*c^2*e^5*x^4 - 5*c^2*d*e^4*x^3 + 60*c^2*d^4*e + 35*a*c*d^2*e^3 - 3*a^2*e^5 + 2*(10*c^2*d^2*e^3 + 7*a*c*e^5
)*x^2 + 5*(18*c^2*d^3*e^2 + 11*a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(e^8*x^2 + 2*d*e^7*x + d^2*e^6), -1/6*(15*(4*c^2
*d^4 + a*c*d^2*e^2 + (4*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(4*c^2*d^3*e + a*c*d*e^3)*x)*sqrt(-c*d^2 - a*e^2)*arcta
n(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)) - 15*(4*c^
2*d^5 + 3*a*c*d^3*e^2 + (4*c^2*d^3*e^2 + 3*a*c*d*e^4)*x^2 + 2*(4*c^2*d^4*e + 3*a*c*d^2*e^3)*x)*sqrt(-c)*arctan
(sqrt(-c)*x/sqrt(c*x^2 + a)) - (2*c^2*e^5*x^4 - 5*c^2*d*e^4*x^3 + 60*c^2*d^4*e + 35*a*c*d^2*e^3 - 3*a^2*e^5 +
2*(10*c^2*d^2*e^3 + 7*a*c*e^5)*x^2 + 5*(18*c^2*d^3*e^2 + 11*a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(e^8*x^2 + 2*d*e^7*
x + d^2*e^6)]

Sympy [F]

\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{3}}\, dx \]

[In]

integrate((c*x**2+a)**(5/2)/(e*x+d)**3,x)

[Out]

Integral((a + c*x**2)**(5/2)/(d + e*x)**3, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c*x^2+a)^(5/2)/(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. 537 vs. \(2 (187) = 374\).

Time = 0.61 (sec) , antiderivative size = 537, normalized size of antiderivative = 2.52 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {1}{6} \, \sqrt {c x^{2} + a} {\left (x {\left (\frac {2 \, c^{2} x}{e^{3}} - \frac {9 \, c^{2} d}{e^{4}}\right )} + \frac {2 \, {\left (18 \, c^{3} d^{2} e^{13} + 7 \, a c^{2} e^{15}\right )}}{c e^{18}}\right )} + \frac {5 \, {\left (4 \, c^{\frac {5}{2}} d^{3} + 3 \, a c^{\frac {3}{2}} d e^{2}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{2 \, e^{6}} + \frac {5 \, {\left (4 \, c^{3} d^{4} + 5 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{\sqrt {-c d^{2} - a e^{2}} e^{6}} + \frac {10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{3} d^{4} e + 11 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c^{2} d^{2} e^{3} + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{2} c e^{5} + 18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} c^{\frac {7}{2}} d^{5} + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {5}{2}} d^{3} e^{2} - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{2} c^{\frac {3}{2}} d e^{4} - 26 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a c^{3} d^{4} e - 25 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c^{2} d^{2} e^{3} + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{3} c e^{5} + 9 \, a^{2} c^{\frac {5}{2}} d^{3} e^{2} + 9 \, a^{3} c^{\frac {3}{2}} d e^{4}}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} \sqrt {c} d - a e\right )}^{2} e^{6}} \]

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^3,x, algorithm="giac")

[Out]

1/6*sqrt(c*x^2 + a)*(x*(2*c^2*x/e^3 - 9*c^2*d/e^4) + 2*(18*c^3*d^2*e^13 + 7*a*c^2*e^15)/(c*e^18)) + 5/2*(4*c^(
5/2)*d^3 + 3*a*c^(3/2)*d*e^2)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/e^6 + 5*(4*c^3*d^4 + 5*a*c^2*d^2*e^2 + a^
2*c*e^4)*arctan(-((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^6
) + (10*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^3*d^4*e + 11*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c^2*d^2*e^3 + (sqrt(c
)*x - sqrt(c*x^2 + a))^3*a^2*c*e^5 + 18*(sqrt(c)*x - sqrt(c*x^2 + a))^2*c^(7/2)*d^5 + 9*(sqrt(c)*x - sqrt(c*x^
2 + a))^2*a*c^(5/2)*d^3*e^2 - 9*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^2*c^(3/2)*d*e^4 - 26*(sqrt(c)*x - sqrt(c*x^2
+ a))*a*c^3*d^4*e - 25*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c^2*d^2*e^3 + (sqrt(c)*x - sqrt(c*x^2 + a))*a^3*c*e^
5 + 9*a^2*c^(5/2)*d^3*e^2 + 9*a^3*c^(3/2)*d*e^4)/(((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^6)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^3} \,d x \]

[In]

int((a + c*x^2)^(5/2)/(d + e*x)^3,x)

[Out]

int((a + c*x^2)^(5/2)/(d + e*x)^3, x)