3.5.72 \(\int \frac {(a+c x^2)^{5/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=222 \[ \frac {5 c^{3/2} \left (a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 e^6}+\frac {5 c^2 d \left (3 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 e^6 \sqrt {a e^2+c d^2}}-\frac {5 c \sqrt {a+c x^2} \left (a e^2+4 c d^2+2 c d e x\right )}{2 e^5 (d+e x)}+\frac {5 c \left (a+c x^2\right )^{3/2} (2 d+e x)}{6 e^3 (d+e x)^2}-\frac {\left (a+c x^2\right )^{5/2}}{3 e (d+e x)^3} \]

________________________________________________________________________________________

Rubi [A]  time = 0.22, antiderivative size = 222, 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 = {733, 813, 844, 217, 206, 725} \begin {gather*} \frac {5 c^{3/2} \left (a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 e^6}+\frac {5 c^2 d \left (3 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 e^6 \sqrt {a e^2+c d^2}}-\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 (d+e x)}+\frac {5 c \left (a+c x^2\right )^{3/2} (2 d+e x)}{6 e^3 (d+e x)^2}-\frac {\left (a+c x^2\right )^{5/2}}{3 e (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^(5/2)/(d + e*x)^4,x]

[Out]

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

Rule 206

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

Rule 217

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 725

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 733

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 813

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 844

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)^4} \, dx &=-\frac {\left (a+c x^2\right )^{5/2}}{3 e (d+e x)^3}+\frac {(5 c) \int \frac {x \left (a+c x^2\right )^{3/2}}{(d+e x)^3} \, dx}{3 e}\\ &=\frac {5 c (2 d+e x) \left (a+c x^2\right )^{3/2}}{6 e^3 (d+e x)^2}-\frac {\left (a+c x^2\right )^{5/2}}{3 e (d+e x)^3}-\frac {(5 c) \int \frac {(-4 a e+8 c d x) \sqrt {a+c x^2}}{(d+e x)^2} \, dx}{8 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 (d+e x)}+\frac {5 c (2 d+e x) \left (a+c x^2\right )^{3/2}}{6 e^3 (d+e x)^2}-\frac {\left (a+c x^2\right )^{5/2}}{3 e (d+e x)^3}+\frac {(5 c) \int \frac {-16 a c d e+8 c \left (4 c d^2+a e^2\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{16 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 (d+e x)}+\frac {5 c (2 d+e x) \left (a+c x^2\right )^{3/2}}{6 e^3 (d+e x)^2}-\frac {\left (a+c x^2\right )^{5/2}}{3 e (d+e x)^3}+\frac {\left (5 c^2 \left (4 c d^2+a e^2\right )\right ) \int \frac {1}{\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}{(d+e x) \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 (d+e x)}+\frac {5 c (2 d+e x) \left (a+c x^2\right )^{3/2}}{6 e^3 (d+e x)^2}-\frac {\left (a+c x^2\right )^{5/2}}{3 e (d+e x)^3}+\frac {\left (5 c^2 \left (4 c d^2+a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {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 ) \operatorname {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 {5 c \left (4 c d^2+a e^2+2 c d e x\right ) \sqrt {a+c x^2}}{2 e^5 (d+e x)}+\frac {5 c (2 d+e x) \left (a+c x^2\right )^{3/2}}{6 e^3 (d+e x)^2}-\frac {\left (a+c x^2\right )^{5/2}}{3 e (d+e x)^3}+\frac {5 c^{3/2} \left (4 c d^2+a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 e^6}+\frac {5 c^2 d \left (4 c d^2+3 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 \sqrt {c d^2+a e^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.29, size = 260, normalized size = 1.17 \begin {gather*} \frac {-\frac {e \sqrt {a+c x^2} \left (2 a^2 e^4+a c e^2 \left (5 d^2+15 d e x+14 e^2 x^2\right )+c^2 \left (60 d^4+150 d^3 e x+110 d^2 e^2 x^2+15 d e^3 x^3-3 e^4 x^4\right )\right )}{(d+e x)^3}+15 c^{3/2} \left (a e^2+4 c d^2\right ) \log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right )+\frac {15 c^2 d \left (3 a e^2+4 c d^2\right ) \log \left (\sqrt {a+c x^2} \sqrt {a e^2+c d^2}+a e-c d x\right )}{\sqrt {a e^2+c d^2}}-\frac {15 c^2 d \left (3 a e^2+4 c d^2\right ) \log (d+e x)}{\sqrt {a e^2+c d^2}}}{6 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^(5/2)/(d + e*x)^4,x]

[Out]

(-((e*Sqrt[a + c*x^2]*(2*a^2*e^4 + a*c*e^2*(5*d^2 + 15*d*e*x + 14*e^2*x^2) + c^2*(60*d^4 + 150*d^3*e*x + 110*d
^2*e^2*x^2 + 15*d*e^3*x^3 - 3*e^4*x^4)))/(d + e*x)^3) - (15*c^2*d*(4*c*d^2 + 3*a*e^2)*Log[d + e*x])/Sqrt[c*d^2
+ a*e^2] + 15*c^(3/2)*(4*c*d^2 + a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + (15*c^2*d*(4*c*d^2 + 3*a*e^2)*Lo
g[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/Sqrt[c*d^2 + a*e^2])/(6*e^6)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 2.47, size = 271, normalized size = 1.22 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-2 a^2 e^4-5 a c d^2 e^2-15 a c d e^3 x-14 a c e^4 x^2-60 c^2 d^4-150 c^2 d^3 e x-110 c^2 d^2 e^2 x^2-15 c^2 d e^3 x^3+3 c^2 e^4 x^4\right )}{6 e^5 (d+e x)^3}-\frac {5 \left (a c^{3/2} e^2+4 c^{5/2} d^2\right ) \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{2 e^6}-\frac {5 \sqrt {-a e^2-c d^2} \left (3 a c^2 d e^2+4 c^3 d^3\right ) \tan ^{-1}\left (\frac {-e \sqrt {a+c x^2}+\sqrt {c} d+\sqrt {c} e x}{\sqrt {-a e^2-c d^2}}\right )}{e^6 \left (a e^2+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + c*x^2)^(5/2)/(d + e*x)^4,x]

[Out]

(Sqrt[a + c*x^2]*(-60*c^2*d^4 - 5*a*c*d^2*e^2 - 2*a^2*e^4 - 150*c^2*d^3*e*x - 15*a*c*d*e^3*x - 110*c^2*d^2*e^2
*x^2 - 14*a*c*e^4*x^2 - 15*c^2*d*e^3*x^3 + 3*c^2*e^4*x^4))/(6*e^5*(d + e*x)^3) - (5*Sqrt[-(c*d^2) - a*e^2]*(4*
c^3*d^3 + 3*a*c^2*d*e^2)*ArcTan[(Sqrt[c]*d + Sqrt[c]*e*x - e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/(e^6*(c
*d^2 + a*e^2)) - (5*(4*c^(5/2)*d^2 + a*c^(3/2)*e^2)*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/(2*e^6)

________________________________________________________________________________________

fricas [B]  time = 3.59, size = 2501, normalized size = 11.27

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

[1/12*(15*(4*c^3*d^7 + 5*a*c^2*d^5*e^2 + a^2*c*d^3*e^4 + (4*c^3*d^4*e^3 + 5*a*c^2*d^2*e^5 + a^2*c*e^7)*x^3 + 3
*(4*c^3*d^5*e^2 + 5*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x^2 + 3*(4*c^3*d^6*e + 5*a*c^2*d^4*e^3 + a^2*c*d^2*e^5)*x)*sq
rt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 15*(4*c^3*d^6 + 3*a*c^2*d^4*e^2 + (4*c^3*d^3*e^3 + 3*a
*c^2*d*e^5)*x^3 + 3*(4*c^3*d^4*e^2 + 3*a*c^2*d^2*e^4)*x^2 + 3*(4*c^3*d^5*e + 3*a*c^2*d^3*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*(60*c^3*d^6*e + 65*a*c^2*d^4*e^3 + 7*a^2*c*d^2*e^5 + 2*a^3*e
^7 - 3*(c^3*d^2*e^5 + a*c^2*e^7)*x^4 + 15*(c^3*d^3*e^4 + a*c^2*d*e^6)*x^3 + 2*(55*c^3*d^4*e^3 + 62*a*c^2*d^2*e
^5 + 7*a^2*c*e^7)*x^2 + 15*(10*c^3*d^5*e^2 + 11*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c*d^5*e^6 +
a*d^3*e^8 + (c*d^2*e^9 + a*e^11)*x^3 + 3*(c*d^3*e^8 + a*d*e^10)*x^2 + 3*(c*d^4*e^7 + a*d^2*e^9)*x), 1/12*(30*(
4*c^3*d^6 + 3*a*c^2*d^4*e^2 + (4*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 3*(4*c^3*d^4*e^2 + 3*a*c^2*d^2*e^4)*x^2 +
3*(4*c^3*d^5*e + 3*a*c^2*d^3*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^3*d^7 + 5*a*c^2*d^5*e^2 + a^2*c*d^3*e^4 + (4*c^
3*d^4*e^3 + 5*a*c^2*d^2*e^5 + a^2*c*e^7)*x^3 + 3*(4*c^3*d^5*e^2 + 5*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x^2 + 3*(4*c^
3*d^6*e + 5*a*c^2*d^4*e^3 + a^2*c*d^2*e^5)*x)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(60*
c^3*d^6*e + 65*a*c^2*d^4*e^3 + 7*a^2*c*d^2*e^5 + 2*a^3*e^7 - 3*(c^3*d^2*e^5 + a*c^2*e^7)*x^4 + 15*(c^3*d^3*e^4
+ a*c^2*d*e^6)*x^3 + 2*(55*c^3*d^4*e^3 + 62*a*c^2*d^2*e^5 + 7*a^2*c*e^7)*x^2 + 15*(10*c^3*d^5*e^2 + 11*a*c^2*
d^3*e^4 + a^2*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c*d^5*e^6 + a*d^3*e^8 + (c*d^2*e^9 + a*e^11)*x^3 + 3*(c*d^3*e^8 +
a*d*e^10)*x^2 + 3*(c*d^4*e^7 + a*d^2*e^9)*x), -1/12*(30*(4*c^3*d^7 + 5*a*c^2*d^5*e^2 + a^2*c*d^3*e^4 + (4*c^3*
d^4*e^3 + 5*a*c^2*d^2*e^5 + a^2*c*e^7)*x^3 + 3*(4*c^3*d^5*e^2 + 5*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x^2 + 3*(4*c^3*
d^6*e + 5*a*c^2*d^4*e^3 + a^2*c*d^2*e^5)*x)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - 15*(4*c^3*d^6 + 3*a*
c^2*d^4*e^2 + (4*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 3*(4*c^3*d^4*e^2 + 3*a*c^2*d^2*e^4)*x^2 + 3*(4*c^3*d^5*e +
3*a*c^2*d^3*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*(60*c^3*d^6*e + 65*a*c^2*d
^4*e^3 + 7*a^2*c*d^2*e^5 + 2*a^3*e^7 - 3*(c^3*d^2*e^5 + a*c^2*e^7)*x^4 + 15*(c^3*d^3*e^4 + a*c^2*d*e^6)*x^3 +
2*(55*c^3*d^4*e^3 + 62*a*c^2*d^2*e^5 + 7*a^2*c*e^7)*x^2 + 15*(10*c^3*d^5*e^2 + 11*a*c^2*d^3*e^4 + a^2*c*d*e^6)
*x)*sqrt(c*x^2 + a))/(c*d^5*e^6 + a*d^3*e^8 + (c*d^2*e^9 + a*e^11)*x^3 + 3*(c*d^3*e^8 + a*d*e^10)*x^2 + 3*(c*d
^4*e^7 + a*d^2*e^9)*x), 1/6*(15*(4*c^3*d^6 + 3*a*c^2*d^4*e^2 + (4*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 3*(4*c^3*
d^4*e^2 + 3*a*c^2*d^2*e^4)*x^2 + 3*(4*c^3*d^5*e + 3*a*c^2*d^3*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^3*d^7 + 5*a*c^
2*d^5*e^2 + a^2*c*d^3*e^4 + (4*c^3*d^4*e^3 + 5*a*c^2*d^2*e^5 + a^2*c*e^7)*x^3 + 3*(4*c^3*d^5*e^2 + 5*a*c^2*d^3
*e^4 + a^2*c*d*e^6)*x^2 + 3*(4*c^3*d^6*e + 5*a*c^2*d^4*e^3 + a^2*c*d^2*e^5)*x)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt
(c*x^2 + a)) - (60*c^3*d^6*e + 65*a*c^2*d^4*e^3 + 7*a^2*c*d^2*e^5 + 2*a^3*e^7 - 3*(c^3*d^2*e^5 + a*c^2*e^7)*x^
4 + 15*(c^3*d^3*e^4 + a*c^2*d*e^6)*x^3 + 2*(55*c^3*d^4*e^3 + 62*a*c^2*d^2*e^5 + 7*a^2*c*e^7)*x^2 + 15*(10*c^3*
d^5*e^2 + 11*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c*d^5*e^6 + a*d^3*e^8 + (c*d^2*e^9 + a*e^11)*x^
3 + 3*(c*d^3*e^8 + a*d*e^10)*x^2 + 3*(c*d^4*e^7 + a*d^2*e^9)*x)]

________________________________________________________________________________________

giac [B]  time = 0.50, size = 575, normalized size = 2.59 \begin {gather*} -\frac {5}{2} \, {\left (4 \, c^{\frac {5}{2}} d^{2} + a c^{\frac {3}{2}} e^{2}\right )} e^{\left (-6\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right ) - \frac {5 \, {\left (4 \, c^{3} d^{3} + 3 \, a c^{2} d 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 ) e^{\left (-6\right )}}{\sqrt {-c d^{2} - a e^{2}}} + \frac {1}{2} \, {\left (c^{2} x e^{\left (-4\right )} - 8 \, c^{2} d e^{\left (-5\right )}\right )} \sqrt {c x^{2} + a} - \frac {{\left (210 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} c^{\frac {7}{2}} d^{4} e + 188 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{4} d^{5} + 60 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} c^{3} d^{3} e^{2} - 354 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {7}{2}} d^{4} e - 226 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c^{3} d^{3} e^{2} + 27 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a c^{\frac {5}{2}} d^{2} e^{3} + 27 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a c^{2} d e^{4} + 222 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c^{3} d^{3} e^{2} - 84 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{2} c^{2} d e^{4} - 18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{2} c^{\frac {3}{2}} e^{5} - 47 \, a^{3} c^{\frac {5}{2}} d^{2} e^{3} + 57 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{3} c^{2} d e^{4} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{3} c^{\frac {3}{2}} e^{5} - 14 \, a^{4} c^{\frac {3}{2}} e^{5}\right )} e^{\left (-6\right )}}{3 \, {\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 )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

-5/2*(4*c^(5/2)*d^2 + a*c^(3/2)*e^2)*e^(-6)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a))) - 5*(4*c^3*d^3 + 3*a*c^2*d*
e^2)*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) +
1/2*(c^2*x*e^(-4) - 8*c^2*d*e^(-5))*sqrt(c*x^2 + a) - 1/3*(210*(sqrt(c)*x - sqrt(c*x^2 + a))^4*c^(7/2)*d^4*e
+ 188*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^4*d^5 + 60*(sqrt(c)*x - sqrt(c*x^2 + a))^5*c^3*d^3*e^2 - 354*(sqrt(c)*
x - sqrt(c*x^2 + a))^2*a*c^(7/2)*d^4*e - 226*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c^3*d^3*e^2 + 27*(sqrt(c)*x - s
qrt(c*x^2 + a))^4*a*c^(5/2)*d^2*e^3 + 27*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^2*d*e^4 + 222*(sqrt(c)*x - sqrt(c
*x^2 + a))*a^2*c^3*d^3*e^2 - 84*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*c^2*d*e^4 - 18*(sqrt(c)*x - sqrt(c*x^2 + a
))^4*a^2*c^(3/2)*e^5 - 47*a^3*c^(5/2)*d^2*e^3 + 57*(sqrt(c)*x - sqrt(c*x^2 + a))*a^3*c^2*d*e^4 + 24*(sqrt(c)*x
- sqrt(c*x^2 + a))^2*a^3*c^(3/2)*e^5 - 14*a^4*c^(3/2)*e^5)*e^(-6)/((sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqr
t(c)*x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^3

________________________________________________________________________________________

maple [B]  time = 0.06, size = 3789, normalized size = 17.07 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^(5/2)/(e*x+d)^4,x)

[Out]

15/8/e^2*c^3*d^2/(a*e^2+c*d^2)^2*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)*x+15/4/e^4*c^4*d^4/(a*
e^2+c*d^2)^2*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)*x-15/e^3*c^3*d^3/(a*e^2+c*d^2)^2*(-2*(x+d/
e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)*a+15/2/e^7*c^5*d^7/(a*e^2+c*d^2)^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln(
(-2*(x+d/e)*c*d/e+2*(a*e^2+c*d^2)/e^2+2*((a*e^2+c*d^2)/e^2)^(1/2)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/
e^2)^(1/2))/(x+d/e))+1/6/e*c*d/(a*e^2+c*d^2)^2/(x+d/e)^2*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(7/2
)+5/8/e^2*c^4*d^4/(a*e^2+c*d^2)^3*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)*x-5/6/e*c^3*d^3/(a*e^
2+c*d^2)^3*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)*a+5/4/e^4*c^5*d^6/(a*e^2+c*d^2)^3*(-2*(x+d/e
)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)*x-5/2/e*c^2*d/(a*e^2+c*d^2)^2*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^
2+c*d^2)/e^2)^(3/2)*a+5/2/(a*e^2+c*d^2)^2*c^2*a^2*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)*x-4/3
/(a*e^2+c*d^2)^2*c/(x+d/e)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(7/2)+4/3/(a*e^2+c*d^2)^2*c^2*(-2*
(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(5/2)*x+5/2/(a*e^2+c*d^2)^2*c^(3/2)*a^3*ln((-c*d/e+(x+d/e)*c)/c^(
1/2)+(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))-1/3/e^2/(a*e^2+c*d^2)/(x+d/e)^3*(-2*(x+d/e)*c*d/e
+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(7/2)-15/2/e*c^2*d/(a*e^2+c*d^2)^2*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)
/e^2)^(1/2)*a^2+75/4/e^4*c^(7/2)*d^4/(a*e^2+c*d^2)^2*ln((-c*d/e+(x+d/e)*c)/c^(1/2)+(-2*(x+d/e)*c*d/e+(x+d/e)^2
*c+(a*e^2+c*d^2)/e^2)^(1/2))*a+25/4/e^4*c^(9/2)*d^6/(a*e^2+c*d^2)^3*ln((-c*d/e+(x+d/e)*c)/c^(1/2)+(-2*(x+d/e)*
c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))*a-5/2/e*c^3*d^3/(a*e^2+c*d^2)^3*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e
^2+c*d^2)/e^2)^(1/2)*a^2+5/8*c^3*d^2/(a*e^2+c*d^2)^3*a*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)*
x+15/16*c^3*d^2/(a*e^2+c*d^2)^3*a^2*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)*x-5/e^3*c^4*d^5/(a*
e^2+c*d^2)^3*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)*a+5/2/e^7*c^6*d^9/(a*e^2+c*d^2)^3/((a*e^2+
c*d^2)/e^2)^(1/2)*ln((-2*(x+d/e)*c*d/e+2*(a*e^2+c*d^2)/e^2+2*((a*e^2+c*d^2)/e^2)^(1/2)*(-2*(x+d/e)*c*d/e+(x+d/
e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))+75/16/e^2*c^(7/2)*d^4/(a*e^2+c*d^2)^3*ln((-c*d/e+(x+d/e)*c)/c^(1/2)+
(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))*a^2+225/16/e^2*c^(5/2)*d^2/(a*e^2+c*d^2)^2*ln((-c*d/e+
(x+d/e)*c)/c^(1/2)+(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))*a^2-5/2/e^5*c^5*d^7/(a*e^2+c*d^2)^3
*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)-5/2/e^3*c^3*d^3/(a*e^2+c*d^2)^2*(-2*(x+d/e)*c*d/e+(x+d
/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)+5/2/e^6*c^(11/2)*d^8/(a*e^2+c*d^2)^3*ln((-c*d/e+(x+d/e)*c)/c^(1/2)+(-2*(x+d/e
)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))+15/2/e^6*c^(9/2)*d^6/(a*e^2+c*d^2)^2*ln((-c*d/e+(x+d/e)*c)/c^(1/
2)+(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))-15/2/e^5*c^4*d^5/(a*e^2+c*d^2)^2*(-2*(x+d/e)*c*d/e+
(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)+15/16*c^(5/2)*d^2/(a*e^2+c*d^2)^3*a^3*ln((-c*d/e+(x+d/e)*c)/c^(1/2)+(-2*(
x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))+5/3/(a*e^2+c*d^2)^2*c^2*a*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*
e^2+c*d^2)/e^2)^(3/2)*x-5/6/e^3*c^4*d^5/(a*e^2+c*d^2)^3*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)
-3/2/e*c^2*d/(a*e^2+c*d^2)^2*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(5/2)+45/2/e^3*c^3*d^3/(a*e^2+c*
d^2)^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((-2*(x+d/e)*c*d/e+2*(a*e^2+c*d^2)/e^2+2*((a*e^2+c*d^2)/e^2)^(1/2)*(-2*(x+d
/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))*a^2+45/2/e^5*c^4*d^5/(a*e^2+c*d^2)^2/((a*e^2+c*d^2)/e
^2)^(1/2)*ln((-2*(x+d/e)*c*d/e+2*(a*e^2+c*d^2)/e^2+2*((a*e^2+c*d^2)/e^2)^(1/2)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(
a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))*a+35/16/e^2*c^4*d^4/(a*e^2+c*d^2)^3*a*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c
*d^2)/e^2)^(1/2)*x+15/2/e^3*c^4*d^5/(a*e^2+c*d^2)^3/((a*e^2+c*d^2)/e^2)^(1/2)*ln((-2*(x+d/e)*c*d/e+2*(a*e^2+c*
d^2)/e^2+2*((a*e^2+c*d^2)/e^2)^(1/2)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))*a^2+15/2
/e^5*c^5*d^7/(a*e^2+c*d^2)^3/((a*e^2+c*d^2)/e^2)^(1/2)*ln((-2*(x+d/e)*c*d/e+2*(a*e^2+c*d^2)/e^2+2*((a*e^2+c*d^
2)/e^2)^(1/2)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))*a+105/16/e^2*c^3*d^2/(a*e^2+c*d
^2)^2*a*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)*x+15/2/e*c^2*d/(a*e^2+c*d^2)^2/((a*e^2+c*d^2)/e
^2)^(1/2)*ln((-2*(x+d/e)*c*d/e+2*(a*e^2+c*d^2)/e^2+2*((a*e^2+c*d^2)/e^2)^(1/2)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(
a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))*a^3+5/2/e*c^3*d^3/(a*e^2+c*d^2)^3/((a*e^2+c*d^2)/e^2)^(1/2)*ln((-2*(x+d/e)*c
*d/e+2*(a*e^2+c*d^2)/e^2+2*((a*e^2+c*d^2)/e^2)^(1/2)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))/(
x+d/e))*a^3-1/2*c^2*d^2/(a*e^2+c*d^2)^3/(x+d/e)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(7/2)+1/2*c^3
*d^2/(a*e^2+c*d^2)^3*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(5/2)*x-1/2/e*c^3*d^3/(a*e^2+c*d^2)^3*(-
2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(5/2)

________________________________________________________________________________________

maxima [B]  time = 2.79, size = 1023, normalized size = 4.61

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

-5/4*c^5*d^6*arcsinh(c*x/sqrt(a*c))/(c^(5/2)*d^4*e^6 + 2*a*c^(3/2)*d^2*e^8 + a^2*sqrt(c)*e^10) - 5/4*a*c^4*d^4
*arcsinh(c*x/sqrt(a*c))/(c^(5/2)*d^4*e^4 + 2*a*c^(3/2)*d^2*e^6 + a^2*sqrt(c)*e^8) + 5/4*sqrt(c*x^2 + a)*c^4*d^
4*x/(c^2*d^4*e^4 + 2*a*c*d^2*e^6 + a^2*e^8) - 5/6*(c*x^2 + a)^(3/2)*c^3*d^3/(c^2*d^4*e^3 + 2*a*c*d^2*e^5 + a^2
*e^7) + 5/6*(c*x^2 + a)^(3/2)*c^3*d^2*x/(c^2*d^4*e^2 + 2*a*c*d^2*e^4 + a^2*e^6) + 5/4*sqrt(c*x^2 + a)*a*c^3*d^
2*x/(c^2*d^4*e^2 + 2*a*c*d^2*e^4 + a^2*e^6) - 5/4*a*c^3*d^2*arcsinh(c*x/sqrt(a*c))/(c^(3/2)*d^2*e^4 + a*sqrt(c
)*e^6) - 1/2*(c*x^2 + a)^(5/2)*c^2*d^2/(c^2*d^4*e^2*x + 2*a*c*d^2*e^4*x + a^2*e^6*x + c^2*d^5*e + 2*a*c*d^3*e^
3 + a^2*d*e^5) - 5/2*sqrt(c*x^2 + a)*c^3*d^3/(c*d^2*e^5 + a*e^7) + 15/4*sqrt(c*x^2 + a)*c^3*d^2*x/(c*d^2*e^4 +
a*e^6) + 1/6*(c*x^2 + a)^(7/2)*c*d/(c^2*d^4*e*x^2 + 2*a*c*d^2*e^3*x^2 + a^2*e^5*x^2 + 2*c^2*d^5*x + 4*a*c*d^3
*e^2*x + 2*a^2*d*e^4*x + c^2*d^6/e + 2*a*c*d^4*e + a^2*d^2*e^3) - 1/6*(c*x^2 + a)^(5/2)*c^2*d/(c^2*d^4*e + 2*a
*c*d^2*e^3 + a^2*e^5) - 5/2*(c*x^2 + a)^(3/2)*c^2*d/(c*d^2*e^3 + a*e^5) + 5/3*(c*x^2 + a)^(3/2)*c^2*x/(c*d^2*e
^2 + a*e^4) + 5/2*sqrt(c*x^2 + a)*a*c^2*x/(c*d^2*e^2 + a*e^4) - 1/3*(c*x^2 + a)^(7/2)/(c*d^2*e^2*x^3 + a*e^4*x
^3 + 3*c*d^3*e*x^2 + 3*a*d*e^3*x^2 + 3*c*d^4*x + 3*a*d^2*e^2*x + c*d^5/e + a*d^3*e) - 4/3*(c*x^2 + a)^(5/2)*c/
(c*d^2*e^2*x + a*e^4*x + c*d^3*e + a*d*e^3) + 45/4*c^(5/2)*d^2*arcsinh(c*x/sqrt(a*c))/e^6 + 5/2*a*c^(3/2)*arcs
inh(c*x/sqrt(a*c))/e^4 - 5/2*c^3*d^3*arcsinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(e*x + d)))/(s
qrt(a + c*d^2/e^2)*e^7) - 15/2*sqrt(a + c*d^2/e^2)*c^2*d*arcsinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*
c)*abs(e*x + d)))/e^5 - 15/2*sqrt(c*x^2 + a)*c^2*d/e^5

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,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)^4,x)

[Out]

int((a + c*x^2)^(5/2)/(d + e*x)^4, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(5/2)/(e*x+d)**4,x)

[Out]

Integral((a + c*x**2)**(5/2)/(d + e*x)**4, x)

________________________________________________________________________________________