\(\int \frac {(d+e x)^5}{(a+c x^2)^{5/2}} \, dx\) [577]

Optimal result

Integrand size = 19, antiderivative size = 191 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {(a e-c d x) (d+e x)^4}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {2 (d+e x)^2 \left (2 a^2 e^3-c d \left (c d^2+3 a e^2\right ) x\right )}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {e \left (4 \left (c^2 d^4+4 a c d^2 e^2-2 a^2 e^4\right )+c d e \left (2 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{3 a^2 c^3}+\frac {5 d e^4 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{5/2}} \]

[Out]

-1/3*(-c*d*x+a*e)*(e*x+d)^4/a/c/(c*x^2+a)^(3/2)+5*d*e^4*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/c^(5/2)-2/3*(e*x+d)
^2*(2*a^2*e^3-c*d*(3*a*e^2+c*d^2)*x)/a^2/c^2/(c*x^2+a)^(1/2)-1/3*e*(-8*a^2*e^4+16*a*c*d^2*e^2+4*c^2*d^4+c*d*e*
(7*a*e^2+2*c*d^2)*x)*(c*x^2+a)^(1/2)/a^2/c^3

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {753, 833, 794, 223, 212} \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^2 \left (2 a^2 e^3-c d x \left (3 a e^2+c d^2\right )\right )}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {e \sqrt {a+c x^2} \left (4 \left (-2 a^2 e^4+4 a c d^2 e^2+c^2 d^4\right )+c d e x \left (7 a e^2+2 c d^2\right )\right )}{3 a^2 c^3}+\frac {5 d e^4 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{5/2}}-\frac {(d+e x)^4 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \]

[In]

Int[(d + e*x)^5/(a + c*x^2)^(5/2),x]

[Out]

-1/3*((a*e - c*d*x)*(d + e*x)^4)/(a*c*(a + c*x^2)^(3/2)) - (2*(d + e*x)^2*(2*a^2*e^3 - c*d*(c*d^2 + 3*a*e^2)*x
))/(3*a^2*c^2*Sqrt[a + c*x^2]) - (e*(4*(c^2*d^4 + 4*a*c*d^2*e^2 - 2*a^2*e^4) + c*d*e*(2*c*d^2 + 7*a*e^2)*x)*Sq
rt[a + c*x^2])/(3*a^2*c^3) + (5*d*e^4*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/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 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 753

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

Rule 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 833

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
- 1)*(a + c*x^2)^(p + 1)*((a*(e*f + d*g) - (c*d*f - a*e*g)*x)/(2*a*c*(p + 1))), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rubi steps \begin{align*} \text {integral}& = -\frac {(a e-c d x) (d+e x)^4}{3 a c \left (a+c x^2\right )^{3/2}}+\frac {\int \frac {(d+e x)^3 \left (2 \left (c d^2+2 a e^2\right )-2 c d e x\right )}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a c} \\ & = -\frac {(a e-c d x) (d+e x)^4}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {2 (d+e x)^2 \left (2 a^2 e^3-c d \left (c d^2+3 a e^2\right ) x\right )}{3 a^2 c^2 \sqrt {a+c x^2}}+\frac {\int \frac {(d+e x) \left (-2 a e^2 \left (c d^2-4 a e^2\right )-2 c d e \left (2 c d^2+7 a e^2\right ) x\right )}{\sqrt {a+c x^2}} \, dx}{3 a^2 c^2} \\ & = -\frac {(a e-c d x) (d+e x)^4}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {2 (d+e x)^2 \left (2 a^2 e^3-c d \left (c d^2+3 a e^2\right ) x\right )}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {e \left (4 \left (c^2 d^4+4 a c d^2 e^2-2 a^2 e^4\right )+c d e \left (2 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{3 a^2 c^3}+\frac {\left (5 d e^4\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{c^2} \\ & = -\frac {(a e-c d x) (d+e x)^4}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {2 (d+e x)^2 \left (2 a^2 e^3-c d \left (c d^2+3 a e^2\right ) x\right )}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {e \left (4 \left (c^2 d^4+4 a c d^2 e^2-2 a^2 e^4\right )+c d e \left (2 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{3 a^2 c^3}+\frac {\left (5 d e^4\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{c^2} \\ & = -\frac {(a e-c d x) (d+e x)^4}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {2 (d+e x)^2 \left (2 a^2 e^3-c d \left (c d^2+3 a e^2\right ) x\right )}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {e \left (4 \left (c^2 d^4+4 a c d^2 e^2-2 a^2 e^4\right )+c d e \left (2 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{3 a^2 c^3}+\frac {5 d e^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.94 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {8 a^4 e^5+2 c^4 d^5 x^3+a c^3 d^3 x \left (3 d^2+10 e^2 x^2\right )+a^3 c e^3 \left (-20 d^2-15 d e x+12 e^2 x^2\right )+a^2 c^2 e \left (-5 d^4-30 d^2 e^2 x^2-20 d e^3 x^3+3 e^4 x^4\right )-15 a^2 \sqrt {c} d e^4 \left (a+c x^2\right )^{3/2} \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{3 a^2 c^3 \left (a+c x^2\right )^{3/2}} \]

[In]

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

[Out]

(8*a^4*e^5 + 2*c^4*d^5*x^3 + a*c^3*d^3*x*(3*d^2 + 10*e^2*x^2) + a^3*c*e^3*(-20*d^2 - 15*d*e*x + 12*e^2*x^2) +
a^2*c^2*e*(-5*d^4 - 30*d^2*e^2*x^2 - 20*d*e^3*x^3 + 3*e^4*x^4) - 15*a^2*Sqrt[c]*d*e^4*(a + c*x^2)^(3/2)*Log[-(
Sqrt[c]*x) + Sqrt[a + c*x^2]])/(3*a^2*c^3*(a + c*x^2)^(3/2))

Maple [A] (verified)

Time = 2.36 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.47

method	result	size
default	\(d^{5} \left (\frac {x}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {c \,x^{2}+a}}\right )+e^{5} \left (\frac {x^{4}}{c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 a \left (-\frac {x^{2}}{c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 c^{2} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\right )}{c}\right )+5 d \,e^{4} \left (-\frac {x^{3}}{3 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{c \sqrt {c \,x^{2}+a}}+\frac {\ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}}{c}\right )-\frac {5 d^{4} e}{3 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+10 d^{2} e^{3} \left (-\frac {x^{2}}{c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 c^{2} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\right )+10 d^{3} e^{2} \left (-\frac {x}{2 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {c \,x^{2}+a}}\right )}{2 c}\right )\)	\(281\)
risch	\(\frac {e^{5} \sqrt {c \,x^{2}+a}}{c^{3}}+\frac {\frac {5 d \,e^{4} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{\sqrt {c}}+\frac {\left (-15 d \,e^{4} a^{2} c +10 d^{3} e^{2} c^{2} a -4 \sqrt {-a c}\, a^{2} e^{5}+20 \sqrt {-a c}\, a c \,d^{2} e^{3}+d^{5} c^{3}\right ) \sqrt {c \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}+2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}}{4 a^{2} c^{2} \left (x -\frac {\sqrt {-a c}}{c}\right )}-\frac {\left (-4 \sqrt {-a c}\, a^{2} e^{5}+20 \sqrt {-a c}\, a c \,d^{2} e^{3}+15 d \,e^{4} a^{2} c -10 d^{3} e^{2} c^{2} a -d^{5} c^{3}\right ) \sqrt {c \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}-2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}}{4 a^{2} c^{2} \left (x +\frac {\sqrt {-a c}}{c}\right )}+\frac {\left (-\sqrt {-a c}\, a^{2} e^{5}+10 \sqrt {-a c}\, a c \,d^{2} e^{3}-5 \sqrt {-a c}\, c^{2} d^{4} e -5 d \,e^{4} a^{2} c +10 d^{3} e^{2} c^{2} a -d^{5} c^{3}\right ) \left (-\frac {\sqrt {c \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}+2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}}{3 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}}-\frac {\sqrt {c \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}+2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}}{3 a \left (x -\frac {\sqrt {-a c}}{c}\right )}\right )}{4 a \,c^{2}}+\frac {\left (\sqrt {-a c}\, a^{2} e^{5}-10 \sqrt {-a c}\, a c \,d^{2} e^{3}+5 \sqrt {-a c}\, c^{2} d^{4} e -5 d \,e^{4} a^{2} c +10 d^{3} e^{2} c^{2} a -d^{5} c^{3}\right ) \left (\frac {\sqrt {c \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}-2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}}{3 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}}-\frac {\sqrt {c \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}-2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}}{3 a \left (x +\frac {\sqrt {-a c}}{c}\right )}\right )}{4 a \,c^{2}}}{c^{2}}\)	\(707\)

[In]

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

[Out]

d^5*(1/3*x/a/(c*x^2+a)^(3/2)+2/3*x/a^2/(c*x^2+a)^(1/2))+e^5*(x^4/c/(c*x^2+a)^(3/2)-4*a/c*(-x^2/c/(c*x^2+a)^(3/
2)-2/3*a/c^2/(c*x^2+a)^(3/2)))+5*d*e^4*(-1/3*x^3/c/(c*x^2+a)^(3/2)+1/c*(-x/c/(c*x^2+a)^(1/2)+1/c^(3/2)*ln(c^(1
/2)*x+(c*x^2+a)^(1/2))))-5/3*d^4*e/c/(c*x^2+a)^(3/2)+10*d^2*e^3*(-x^2/c/(c*x^2+a)^(3/2)-2/3*a/c^2/(c*x^2+a)^(3
/2))+10*d^3*e^2*(-1/2*x/c/(c*x^2+a)^(3/2)+1/2*a/c*(1/3*x/a/(c*x^2+a)^(3/2)+2/3*x/a^2/(c*x^2+a)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 488, normalized size of antiderivative = 2.55 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=\left [\frac {15 \, {\left (a^{2} c^{2} d e^{4} x^{4} + 2 \, a^{3} c d e^{4} x^{2} + a^{4} d e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (3 \, a^{2} c^{2} e^{5} x^{4} - 5 \, a^{2} c^{2} d^{4} e - 20 \, a^{3} c d^{2} e^{3} + 8 \, a^{4} e^{5} + 2 \, {\left (c^{4} d^{5} + 5 \, a c^{3} d^{3} e^{2} - 10 \, a^{2} c^{2} d e^{4}\right )} x^{3} - 6 \, {\left (5 \, a^{2} c^{2} d^{2} e^{3} - 2 \, a^{3} c e^{5}\right )} x^{2} + 3 \, {\left (a c^{3} d^{5} - 5 \, a^{3} c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{6 \, {\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )}}, -\frac {15 \, {\left (a^{2} c^{2} d e^{4} x^{4} + 2 \, a^{3} c d e^{4} x^{2} + a^{4} d e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (3 \, a^{2} c^{2} e^{5} x^{4} - 5 \, a^{2} c^{2} d^{4} e - 20 \, a^{3} c d^{2} e^{3} + 8 \, a^{4} e^{5} + 2 \, {\left (c^{4} d^{5} + 5 \, a c^{3} d^{3} e^{2} - 10 \, a^{2} c^{2} d e^{4}\right )} x^{3} - 6 \, {\left (5 \, a^{2} c^{2} d^{2} e^{3} - 2 \, a^{3} c e^{5}\right )} x^{2} + 3 \, {\left (a c^{3} d^{5} - 5 \, a^{3} c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{3 \, {\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )}}\right ] \]

[In]

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

[Out]

[1/6*(15*(a^2*c^2*d*e^4*x^4 + 2*a^3*c*d*e^4*x^2 + a^4*d*e^4)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*
x - a) + 2*(3*a^2*c^2*e^5*x^4 - 5*a^2*c^2*d^4*e - 20*a^3*c*d^2*e^3 + 8*a^4*e^5 + 2*(c^4*d^5 + 5*a*c^3*d^3*e^2
- 10*a^2*c^2*d*e^4)*x^3 - 6*(5*a^2*c^2*d^2*e^3 - 2*a^3*c*e^5)*x^2 + 3*(a*c^3*d^5 - 5*a^3*c*d*e^4)*x)*sqrt(c*x^
2 + a))/(a^2*c^5*x^4 + 2*a^3*c^4*x^2 + a^4*c^3), -1/3*(15*(a^2*c^2*d*e^4*x^4 + 2*a^3*c*d*e^4*x^2 + a^4*d*e^4)*
sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (3*a^2*c^2*e^5*x^4 - 5*a^2*c^2*d^4*e - 20*a^3*c*d^2*e^3 + 8*a^4*
e^5 + 2*(c^4*d^5 + 5*a*c^3*d^3*e^2 - 10*a^2*c^2*d*e^4)*x^3 - 6*(5*a^2*c^2*d^2*e^3 - 2*a^3*c*e^5)*x^2 + 3*(a*c^
3*d^5 - 5*a^3*c*d*e^4)*x)*sqrt(c*x^2 + a))/(a^2*c^5*x^4 + 2*a^3*c^4*x^2 + a^4*c^3)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.47 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {5}{3} \, d e^{4} x {\left (\frac {3 \, x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {2 \, a}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}}\right )} + \frac {e^{5} x^{4}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} - \frac {10 \, d^{2} e^{3} x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {4 \, a e^{5} x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}} + \frac {2 \, d^{5} x}{3 \, \sqrt {c x^{2} + a} a^{2}} + \frac {d^{5} x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {10 \, d^{3} e^{2} x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {10 \, d^{3} e^{2} x}{3 \, \sqrt {c x^{2} + a} a c} - \frac {5 \, d e^{4} x}{3 \, \sqrt {c x^{2} + a} c^{2}} + \frac {5 \, d e^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{c^{\frac {5}{2}}} - \frac {5 \, d^{4} e}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c} - \frac {20 \, a d^{2} e^{3}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}} + \frac {8 \, a^{2} e^{5}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{3}} \]

[In]

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

[Out]

-5/3*d*e^4*x*(3*x^2/((c*x^2 + a)^(3/2)*c) + 2*a/((c*x^2 + a)^(3/2)*c^2)) + e^5*x^4/((c*x^2 + a)^(3/2)*c) - 10*
d^2*e^3*x^2/((c*x^2 + a)^(3/2)*c) + 4*a*e^5*x^2/((c*x^2 + a)^(3/2)*c^2) + 2/3*d^5*x/(sqrt(c*x^2 + a)*a^2) + 1/
3*d^5*x/((c*x^2 + a)^(3/2)*a) - 10/3*d^3*e^2*x/((c*x^2 + a)^(3/2)*c) + 10/3*d^3*e^2*x/(sqrt(c*x^2 + a)*a*c) -
5/3*d*e^4*x/(sqrt(c*x^2 + a)*c^2) + 5*d*e^4*arcsinh(c*x/sqrt(a*c))/c^(5/2) - 5/3*d^4*e/((c*x^2 + a)^(3/2)*c) -
20/3*a*d^2*e^3/((c*x^2 + a)^(3/2)*c^2) + 8/3*a^2*e^5/((c*x^2 + a)^(3/2)*c^3)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.08 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {5 \, d e^{4} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{c^{\frac {5}{2}}} + \frac {{\left ({\left ({\left (\frac {3 \, e^{5} x}{c} + \frac {2 \, {\left (c^{6} d^{5} + 5 \, a c^{5} d^{3} e^{2} - 10 \, a^{2} c^{4} d e^{4}\right )}}{a^{2} c^{5}}\right )} x - \frac {6 \, {\left (5 \, a^{2} c^{4} d^{2} e^{3} - 2 \, a^{3} c^{3} e^{5}\right )}}{a^{2} c^{5}}\right )} x + \frac {3 \, {\left (a c^{5} d^{5} - 5 \, a^{3} c^{3} d e^{4}\right )}}{a^{2} c^{5}}\right )} x - \frac {5 \, a^{2} c^{4} d^{4} e + 20 \, a^{3} c^{3} d^{2} e^{3} - 8 \, a^{4} c^{2} e^{5}}{a^{2} c^{5}}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} \]

[In]

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

[Out]

-5*d*e^4*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2) + 1/3*((((3*e^5*x/c + 2*(c^6*d^5 + 5*a*c^5*d^3*e^2 - 1
0*a^2*c^4*d*e^4)/(a^2*c^5))*x - 6*(5*a^2*c^4*d^2*e^3 - 2*a^3*c^3*e^5)/(a^2*c^5))*x + 3*(a*c^5*d^5 - 5*a^3*c^3*
d*e^4)/(a^2*c^5))*x - (5*a^2*c^4*d^4*e + 20*a^3*c^3*d^2*e^3 - 8*a^4*c^2*e^5)/(a^2*c^5))/(c*x^2 + a)^(3/2)

Mupad [F(-1)]

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

[In]

int((d + e*x)^5/(a + c*x^2)^(5/2),x)

[Out]

int((d + e*x)^5/(a + c*x^2)^(5/2), x)