3.6.49 \(\int \frac {(a+c x^2)^{5/2}}{d+e x} \, dx\) [549]

Optimal. Leaf size=226 \[ \frac {\left (8 \left (c d^2+a e^2\right )^2-c d e \left (4 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5}+\frac {\left (4 \left (c d^2+a e^2\right )-3 c d e x\right ) \left (a+c x^2\right )^{3/2}}{12 e^3}+\frac {\left (a+c x^2\right )^{5/2}}{5 e}-\frac {\sqrt {c} d \left (8 c^2 d^4+20 a c d^2 e^2+15 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 e^6}-\frac {\left (c d^2+a e^2\right )^{5/2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^6} \]

[Out]

1/12*(-3*c*d*e*x+4*a*e^2+4*c*d^2)*(c*x^2+a)^(3/2)/e^3+1/5*(c*x^2+a)^(5/2)/e-(a*e^2+c*d^2)^(5/2)*arctanh((-c*d*
x+a*e)/(a*e^2+c*d^2)^(1/2)/(c*x^2+a)^(1/2))/e^6-1/8*d*(15*a^2*e^4+20*a*c*d^2*e^2+8*c^2*d^4)*arctanh(x*c^(1/2)/
(c*x^2+a)^(1/2))*c^(1/2)/e^6+1/8*(8*(a*e^2+c*d^2)^2-c*d*e*(7*a*e^2+4*c*d^2)*x)*(c*x^2+a)^(1/2)/e^5

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Rubi [A]
time = 0.17, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {749, 829, 858, 223, 212, 739} \begin {gather*} -\frac {\sqrt {c} d \left (15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 e^6}-\frac {\left (a e^2+c d^2\right )^{5/2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^6}+\frac {\sqrt {a+c x^2} \left (8 \left (a e^2+c d^2\right )^2-c d e x \left (7 a e^2+4 c d^2\right )\right )}{8 e^5}+\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (a e^2+c d^2\right )-3 c d e x\right )}{12 e^3}+\frac {\left (a+c x^2\right )^{5/2}}{5 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^(5/2)/(d + e*x),x]

[Out]

((8*(c*d^2 + a*e^2)^2 - c*d*e*(4*c*d^2 + 7*a*e^2)*x)*Sqrt[a + c*x^2])/(8*e^5) + ((4*(c*d^2 + a*e^2) - 3*c*d*e*
x)*(a + c*x^2)^(3/2))/(12*e^3) + (a + c*x^2)^(5/2)/(5*e) - (Sqrt[c]*d*(8*c^2*d^4 + 20*a*c*d^2*e^2 + 15*a^2*e^4
)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(8*e^6) - ((c*d^2 + a*e^2)^(5/2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a
*e^2]*Sqrt[a + c*x^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 749

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 + 2*p + 1))), x] + Dist[2*(p/(e*(m + 2*p + 1))), Int[(d + e*x)^m*Simp[a*e - c*d*x, x]*(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] && NeQ[m + 2*p + 1, 0] && ( !Ration
alQ[m] || LtQ[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

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*} \int \frac {\left (a+c x^2\right )^{5/2}}{d+e x} \, dx &=\frac {\left (a+c x^2\right )^{5/2}}{5 e}+\frac {\int \frac {(a e-c d x) \left (a+c x^2\right )^{3/2}}{d+e x} \, dx}{e}\\ &=\frac {\left (4 \left (c d^2+a e^2\right )-3 c d e x\right ) \left (a+c x^2\right )^{3/2}}{12 e^3}+\frac {\left (a+c x^2\right )^{5/2}}{5 e}+\frac {\int \frac {\left (a c e \left (c d^2+4 a e^2\right )-c^2 d \left (4 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{d+e x} \, dx}{4 c e^3}\\ &=\frac {\left (8 \left (c d^2+a e^2\right )^2-c d e \left (4 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5}+\frac {\left (4 \left (c d^2+a e^2\right )-3 c d e x\right ) \left (a+c x^2\right )^{3/2}}{12 e^3}+\frac {\left (a+c x^2\right )^{5/2}}{5 e}+\frac {\int \frac {a c^2 e \left (4 c^2 d^4+9 a c d^2 e^2+8 a^2 e^4\right )-c^3 d \left (8 c^2 d^4+20 a c d^2 e^2+15 a^2 e^4\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{8 c^2 e^5}\\ &=\frac {\left (8 \left (c d^2+a e^2\right )^2-c d e \left (4 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5}+\frac {\left (4 \left (c d^2+a e^2\right )-3 c d e x\right ) \left (a+c x^2\right )^{3/2}}{12 e^3}+\frac {\left (a+c x^2\right )^{5/2}}{5 e}+\frac {\left (c d^2+a e^2\right )^3 \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^6}-\frac {\left (c d \left (8 c^2 d^4+20 a c d^2 e^2+15 a^2 e^4\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{8 e^6}\\ &=\frac {\left (8 \left (c d^2+a e^2\right )^2-c d e \left (4 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5}+\frac {\left (4 \left (c d^2+a e^2\right )-3 c d e x\right ) \left (a+c x^2\right )^{3/2}}{12 e^3}+\frac {\left (a+c x^2\right )^{5/2}}{5 e}-\frac {\left (c d^2+a e^2\right )^3 \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 )}{e^6}-\frac {\left (c d \left (8 c^2 d^4+20 a c d^2 e^2+15 a^2 e^4\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{8 e^6}\\ &=\frac {\left (8 \left (c d^2+a e^2\right )^2-c d e \left (4 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5}+\frac {\left (4 \left (c d^2+a e^2\right )-3 c d e x\right ) \left (a+c x^2\right )^{3/2}}{12 e^3}+\frac {\left (a+c x^2\right )^{5/2}}{5 e}-\frac {\sqrt {c} d \left (8 c^2 d^4+20 a c d^2 e^2+15 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 e^6}-\frac {\left (c d^2+a e^2\right )^{5/2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^6}\\ \end {align*}

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Mathematica [A]
time = 0.65, size = 221, normalized size = 0.98 \begin {gather*} \frac {e \sqrt {a+c x^2} \left (184 a^2 e^4+a c e^2 \left (280 d^2-135 d e x+88 e^2 x^2\right )+2 c^2 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )+240 \left (-c d^2-a e^2\right )^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )+15 \sqrt {c} d \left (8 c^2 d^4+20 a c d^2 e^2+15 a^2 e^4\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{120 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*Sqrt[a + c*x^2]*(184*a^2*e^4 + a*c*e^2*(280*d^2 - 135*d*e*x + 88*e^2*x^2) + 2*c^2*(60*d^4 - 30*d^3*e*x + 20
*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)) + 240*(-(c*d^2) - a*e^2)^(5/2)*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a
+ c*x^2])/Sqrt[-(c*d^2) - a*e^2]] + 15*Sqrt[c]*d*(8*c^2*d^4 + 20*a*c*d^2*e^2 + 15*a^2*e^4)*Log[-(Sqrt[c]*x) +
Sqrt[a + c*x^2]])/(120*e^6)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(831\) vs. \(2(199)=398\).
time = 0.48, size = 832, normalized size = 3.68

method	result	size
risch	\(\frac {\left (24 c^{2} e^{4} x^{4}-30 c^{2} d \,e^{3} x^{3}+88 a c \,e^{4} x^{2}+40 c^{2} d^{2} e^{2} x^{2}-135 a c d \,e^{3} x -60 c^{2} d^{3} e x +184 a^{2} e^{4}+280 a c \,d^{2} e^{2}+120 c^{2} d^{4}\right ) \sqrt {c \,x^{2}+a}}{120 e^{5}}-\frac {15 \sqrt {c}\, d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right ) a^{2}}{8 e^{2}}-\frac {5 c^{\frac {3}{2}} d^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right ) a}{2 e^{4}}-\frac {c^{\frac {5}{2}} d^{5} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{e^{6}}-\frac {\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 ) a^{3}}{e \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {3 \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 ) d^{2} a^{2} c}{e^{3} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {3 \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 ) d^{4} c^{2} a}{e^{5} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {\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 ) d^{6} c^{3}}{e^{7} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\)	\(719\)
default	\(\frac {\frac {\left (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}}\right )^{\frac {5}{2}}}{5}-\frac {c d \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right ) \left (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}}\right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right ) \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}}}}{4 c}+\frac {\left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \ln \left (\frac {-\frac {c d}{e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\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}}}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{e}+\frac {\left (e^{2} a +c \,d^{2}\right ) \left (\frac {\left (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}}\right )^{\frac {3}{2}}}{3}-\frac {c d \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right ) \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}}}}{4 c}+\frac {\left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \ln \left (\frac {-\frac {c d}{e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\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}}}\right )}{8 c^{\frac {3}{2}}}\right )}{e}+\frac {\left (e^{2} a +c \,d^{2}\right ) \left (\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}}}-\frac {\sqrt {c}\, d \ln \left (\frac {-\frac {c d}{e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\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}}}\right )}{e}-\frac {\left (e^{2} a +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}}}}\right )}{e^{2}}\right )}{e^{2}}}{e}\)	\(832\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.35, size = 261, normalized size = 1.15 \begin {gather*} -c^{\frac {5}{2}} d^{5} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-6\right )} - \frac {1}{2} \, \sqrt {c x^{2} + a} c^{2} d^{3} x e^{\left (-4\right )} - \frac {5}{2} \, a c^{\frac {3}{2}} d^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-4\right )} + \sqrt {c x^{2} + a} c^{2} d^{4} e^{\left (-5\right )} - \frac {1}{4} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c d x e^{\left (-2\right )} - \frac {7}{8} \, \sqrt {c x^{2} + a} a c d x e^{\left (-2\right )} - \frac {15}{8} \, a^{2} \sqrt {c} d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-2\right )} + \frac {1}{3} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c d^{2} e^{\left (-3\right )} + 2 \, \sqrt {c x^{2} + a} a c d^{2} e^{\left (-3\right )} + {\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {5}{2}} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-1\right )} + \frac {1}{5} \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} e^{\left (-1\right )} + \frac {1}{3} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a e^{\left (-1\right )} + \sqrt {c x^{2} + a} a^{2} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^(5/2)*d^5*arcsinh(c*x/sqrt(a*c))*e^(-6) - 1/2*sqrt(c*x^2 + a)*c^2*d^3*x*e^(-4) - 5/2*a*c^(3/2)*d^3*arcsinh(
c*x/sqrt(a*c))*e^(-4) + sqrt(c*x^2 + a)*c^2*d^4*e^(-5) - 1/4*(c*x^2 + a)^(3/2)*c*d*x*e^(-2) - 7/8*sqrt(c*x^2 +
a)*a*c*d*x*e^(-2) - 15/8*a^2*sqrt(c)*d*arcsinh(c*x/sqrt(a*c))*e^(-2) + 1/3*(c*x^2 + a)^(3/2)*c*d^2*e^(-3) + 2
*sqrt(c*x^2 + a)*a*c*d^2*e^(-3) + (c*d^2*e^(-2) + a)^(5/2)*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(
a*c)*abs(x*e + d)))*e^(-1) + 1/5*(c*x^2 + a)^(5/2)*e^(-1) + 1/3*(c*x^2 + a)^(3/2)*a*e^(-1) + sqrt(c*x^2 + a)*a
^2*e^(-1)

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Fricas [A]
time = 57.05, size = 1113, normalized size = 4.92 \begin {gather*} \left [\frac {1}{240} \, {\left (15 \, {\left (8 \, c^{2} d^{5} + 20 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 120 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (-\frac {2 \, c^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a} + {\left (a c x^{2} + 2 \, a^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) - 2 \, {\left (60 \, c^{2} d^{3} x e^{2} - 120 \, c^{2} d^{4} e - 8 \, {\left (3 \, c^{2} x^{4} + 11 \, a c x^{2} + 23 \, a^{2}\right )} e^{5} + 15 \, {\left (2 \, c^{2} d x^{3} + 9 \, a c d x\right )} e^{4} - 40 \, {\left (c^{2} d^{2} x^{2} + 7 \, a c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}\right )} e^{\left (-6\right )}, \frac {1}{120} \, {\left (15 \, {\left (8 \, c^{2} d^{5} + 20 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + 60 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (-\frac {2 \, c^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a} + {\left (a c x^{2} + 2 \, a^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) - {\left (60 \, c^{2} d^{3} x e^{2} - 120 \, c^{2} d^{4} e - 8 \, {\left (3 \, c^{2} x^{4} + 11 \, a c x^{2} + 23 \, a^{2}\right )} e^{5} + 15 \, {\left (2 \, c^{2} d x^{3} + 9 \, a c d x\right )} e^{4} - 40 \, {\left (c^{2} d^{2} x^{2} + 7 \, a c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}\right )} e^{\left (-6\right )}, \frac {1}{240} \, {\left (240 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (-\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{2}}\right ) + 15 \, {\left (8 \, c^{2} d^{5} + 20 \, a c d^{3} e^{2} + 15 \, a^{2} 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 (60 \, c^{2} d^{3} x e^{2} - 120 \, c^{2} d^{4} e - 8 \, {\left (3 \, c^{2} x^{4} + 11 \, a c x^{2} + 23 \, a^{2}\right )} e^{5} + 15 \, {\left (2 \, c^{2} d x^{3} + 9 \, a c d x\right )} e^{4} - 40 \, {\left (c^{2} d^{2} x^{2} + 7 \, a c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}\right )} e^{\left (-6\right )}, \frac {1}{120} \, {\left (120 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (-\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{2}}\right ) + 15 \, {\left (8 \, c^{2} d^{5} + 20 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (60 \, c^{2} d^{3} x e^{2} - 120 \, c^{2} d^{4} e - 8 \, {\left (3 \, c^{2} x^{4} + 11 \, a c x^{2} + 23 \, a^{2}\right )} e^{5} + 15 \, {\left (2 \, c^{2} d x^{3} + 9 \, a c d x\right )} e^{4} - 40 \, {\left (c^{2} d^{2} x^{2} + 7 \, a c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}\right )} e^{\left (-6\right )}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/240*(15*(8*c^2*d^5 + 20*a*c*d^3*e^2 + 15*a^2*d*e^4)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a)
+ 120*(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(c*d^2 + a*e^2)*log(-(2*c^2*d^2*x^2 - 2*a*c*d*x*e + a*c*d^2 + 2
*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a) + (a*c*x^2 + 2*a^2)*e^2)/(x^2*e^2 + 2*d*x*e + d^2)) - 2*(60
*c^2*d^3*x*e^2 - 120*c^2*d^4*e - 8*(3*c^2*x^4 + 11*a*c*x^2 + 23*a^2)*e^5 + 15*(2*c^2*d*x^3 + 9*a*c*d*x)*e^4 -
40*(c^2*d^2*x^2 + 7*a*c*d^2)*e^3)*sqrt(c*x^2 + a))*e^(-6), 1/120*(15*(8*c^2*d^5 + 20*a*c*d^3*e^2 + 15*a^2*d*e^
4)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + 60*(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(c*d^2 + a*e^2)*lo
g(-(2*c^2*d^2*x^2 - 2*a*c*d*x*e + a*c*d^2 + 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a) + (a*c*x^2 + 2
*a^2)*e^2)/(x^2*e^2 + 2*d*x*e + d^2)) - (60*c^2*d^3*x*e^2 - 120*c^2*d^4*e - 8*(3*c^2*x^4 + 11*a*c*x^2 + 23*a^2
)*e^5 + 15*(2*c^2*d*x^3 + 9*a*c*d*x)*e^4 - 40*(c^2*d^2*x^2 + 7*a*c*d^2)*e^3)*sqrt(c*x^2 + a))*e^(-6), 1/240*(2
40*(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*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)/(c^2*d^2*x^2 + a*c*d^2 + (a*c*x^2 + a^2)*e^2)) + 15*(8*c^2*d^5 + 20*a*c*d^3*e^2 + 15*a^2*d*e^4)*sqrt(
c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(60*c^2*d^3*x*e^2 - 120*c^2*d^4*e - 8*(3*c^2*x^4 + 11*a
*c*x^2 + 23*a^2)*e^5 + 15*(2*c^2*d*x^3 + 9*a*c*d*x)*e^4 - 40*(c^2*d^2*x^2 + 7*a*c*d^2)*e^3)*sqrt(c*x^2 + a))*e
^(-6), 1/120*(120*(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*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)/(c^2*d^2*x^2 + a*c*d^2 + (a*c*x^2 + a^2)*e^2)) + 15*(8*c^2*d^5 + 20*a*c*d^3*e^2 + 15*a
^2*d*e^4)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (60*c^2*d^3*x*e^2 - 120*c^2*d^4*e - 8*(3*c^2*x^4 + 11*
a*c*x^2 + 23*a^2)*e^5 + 15*(2*c^2*d*x^3 + 9*a*c*d*x)*e^4 - 40*(c^2*d^2*x^2 + 7*a*c*d^2)*e^3)*sqrt(c*x^2 + a))*
e^(-6)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.36, size = 282, normalized size = 1.25 \begin {gather*} \frac {1}{8} \, {\left (8 \, c^{\frac {5}{2}} d^{5} + 20 \, a c^{\frac {3}{2}} d^{3} e^{2} + 15 \, a^{2} \sqrt {c} d e^{4}\right )} e^{\left (-6\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right ) + \frac {2 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\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 ) e^{\left (-6\right )}}{\sqrt {-c d^{2} - a e^{2}}} + \frac {1}{120} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left (3 \, {\left (4 \, c^{2} x e^{\left (-1\right )} - 5 \, c^{2} d e^{\left (-2\right )}\right )} x + \frac {4 \, {\left (5 \, c^{5} d^{2} e^{18} + 11 \, a c^{4} e^{20}\right )} e^{\left (-21\right )}}{c^{3}}\right )} x - \frac {15 \, {\left (4 \, c^{5} d^{3} e^{17} + 9 \, a c^{4} d e^{19}\right )} e^{\left (-21\right )}}{c^{3}}\right )} x + \frac {8 \, {\left (15 \, c^{5} d^{4} e^{16} + 35 \, a c^{4} d^{2} e^{18} + 23 \, a^{2} c^{3} e^{20}\right )} e^{\left (-21\right )}}{c^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(8*c^(5/2)*d^5 + 20*a*c^(3/2)*d^3*e^2 + 15*a^2*sqrt(c)*d*e^4)*e^(-6)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a))
) + 2*(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(
c)*d)/sqrt(-c*d^2 - a*e^2))*e^(-6)/sqrt(-c*d^2 - a*e^2) + 1/120*sqrt(c*x^2 + a)*((2*(3*(4*c^2*x*e^(-1) - 5*c^2
*d*e^(-2))*x + 4*(5*c^5*d^2*e^18 + 11*a*c^4*e^20)*e^(-21)/c^3)*x - 15*(4*c^5*d^3*e^17 + 9*a*c^4*d*e^19)*e^(-21
)/c^3)*x + 8*(15*c^5*d^4*e^16 + 35*a*c^4*d^2*e^18 + 23*a^2*c^3*e^20)*e^(-21)/c^3)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+a\right )}^{5/2}}{d+e\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^(5/2)/(d + e*x),x)

[Out]

int((a + c*x^2)^(5/2)/(d + e*x), x)

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