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

Optimal. Leaf size=213 \[ \frac {5 c \left (4 c d^2+a e^2-2 c d e x\right ) \sqrt {a+c x^2}}{2 e^5}+\frac {5 c (4 d+e x) \left (a+c x^2\right )^{3/2}}{6 e^3 (d+e x)}-\frac {\left (a+c x^2\right )^{5/2}}{2 e (d+e x)^2}-\frac {5 c^{3/2} d \left (4 c d^2+3 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 e^6}-\frac {5 c \sqrt {c d^2+a e^2} \left (4 c d^2+a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 e^6} \]

[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

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Rubi [A]
time = 0.16, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {747, 827, 829, 858, 223, 212, 739} \begin {gather*} -\frac {5 c^{3/2} d \left (3 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 \sqrt {a e^2+c d^2} \left (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}+\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} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 1.23, size = 229, normalized size = 1.08 \begin {gather*} \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 ) \tan ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2243\) vs. \(2(185)=370\).
time = 0.49, size = 2244, normalized size = 10.54

method	result	size
default	\(\text {Expression too large to display}\)	\(2244\)
risch	\(\text {Expression too large to display}\)	\(2497\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e^3*(-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)^(7/2)-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)^(7/2)-5*c*d*e/(a*e^2+c*d
^2)*(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)))))+6*c/(a*e^2+c*d^2)*e^2*(1/12*(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)^(5/2)+5/24*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^
2/e^2)/c*(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))))))+5/2*c/(a*e^2+c*d^2)*e^2*(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 [B] Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (182) = 364\).
time = 0.37, size = 542, normalized size = 2.54 \begin {gather*} \frac {15 \, c^{4} d^{5} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{4 \, {\left (c^{\frac {3}{2}} d^{2} e^{6} + a \sqrt {c} e^{8}\right )}} - \frac {55}{4} \, c^{\frac {5}{2}} d^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-6\right )} + \frac {15 \, a c^{3} d^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{4 \, {\left (c^{\frac {3}{2}} d^{2} e^{4} + a \sqrt {c} e^{6}\right )}} - \frac {15 \, \sqrt {c x^{2} + a} c^{3} d^{3} x}{4 \, {\left (c d^{2} e^{4} + a e^{6}\right )}} + \frac {15}{2} \, \sqrt {c d^{2} e^{\left (-2\right )} + a} c^{2} d^{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 (-5\right )} - \frac {5}{4} \, \sqrt {c x^{2} + a} c^{2} d x e^{\left (-4\right )} - \frac {15}{2} \, a c^{\frac {3}{2}} d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-4\right )} + 10 \, \sqrt {c x^{2} + a} c^{2} d^{2} e^{\left (-5\right )} + \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2} d^{2}}{2 \, {\left (c d^{2} e^{3} + a e^{5}\right )}} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2} d x}{2 \, {\left (c d^{2} e^{2} + a e^{4}\right )}} - \frac {15 \, \sqrt {c x^{2} + a} a c^{2} d x}{4 \, {\left (c d^{2} e^{2} + a e^{4}\right )}} + \frac {5}{2} \, {\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {3}{2}} c \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 (-3\right )} + \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} c d}{2 \, {\left (c d^{2} x e^{2} + c d^{3} e + a x e^{4} + a d e^{3}\right )}} + \frac {5}{6} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c e^{\left (-3\right )} + \frac {5}{2} \, \sqrt {c x^{2} + a} a c e^{\left (-3\right )} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}}}{2 \, {\left (c d^{2} x^{2} e + c d^{4} e^{\left (-1\right )} + 2 \, c d^{3} x + a x^{2} e^{3} + 2 \, a d x e^{2} + a d^{2} e\right )}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} c}{2 \, {\left (c d^{2} e + a e^{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)^3,x, algorithm="maxima")

[Out]

15/4*c^4*d^5*arcsinh(c*x/sqrt(a*c))/(c^(3/2)*d^2*e^6 + a*sqrt(c)*e^8) - 55/4*c^(5/2)*d^3*arcsinh(c*x/sqrt(a*c)
)*e^(-6) + 15/4*a*c^3*d^3*arcsinh(c*x/sqrt(a*c))/(c^(3/2)*d^2*e^4 + a*sqrt(c)*e^6) - 15/4*sqrt(c*x^2 + a)*c^3*
d^3*x/(c*d^2*e^4 + a*e^6) + 15/2*sqrt(c*d^2*e^(-2) + a)*c^2*d^2*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(
sqrt(a*c)*abs(x*e + d)))*e^(-5) - 5/4*sqrt(c*x^2 + a)*c^2*d*x*e^(-4) - 15/2*a*c^(3/2)*d*arcsinh(c*x/sqrt(a*c))
*e^(-4) + 10*sqrt(c*x^2 + a)*c^2*d^2*e^(-5) + 5/2*(c*x^2 + a)^(3/2)*c^2*d^2/(c*d^2*e^3 + a*e^5) - 5/2*(c*x^2 +
a)^(3/2)*c^2*d*x/(c*d^2*e^2 + a*e^4) - 15/4*sqrt(c*x^2 + a)*a*c^2*d*x/(c*d^2*e^2 + a*e^4) + 5/2*(c*d^2*e^(-2)
+ a)^(3/2)*c*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(a*c)*abs(x*e + d)))*e^(-3) + 3/2*(c*x^2 + a)^
(5/2)*c*d/(c*d^2*x*e^2 + c*d^3*e + a*x*e^4 + a*d*e^3) + 5/6*(c*x^2 + a)^(3/2)*c*e^(-3) + 5/2*sqrt(c*x^2 + a)*a
*c*e^(-3) - 1/2*(c*x^2 + a)^(7/2)/(c*d^2*x^2*e + c*d^4*e^(-1) + 2*c*d^3*x + a*x^2*e^3 + 2*a*d*x*e^2 + a*d^2*e)
+ 1/2*(c*x^2 + a)^(5/2)*c/(c*d^2*e + a*e^3)

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Fricas [A]
time = 5.52, size = 1464, normalized size = 6.87 \begin {gather*} \left [\frac {15 \, {\left (8 \, c^{2} d^{4} x e + 4 \, c^{2} d^{5} + 3 \, a c d x^{2} e^{4} + 6 \, a c d^{2} x e^{3} + {\left (4 \, c^{2} d^{3} x^{2} + 3 \, a c d^{3}\right )} e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 15 \, {\left (8 \, c^{2} d^{3} x e + 4 \, c^{2} d^{4} + a c x^{2} e^{4} + 2 \, a c d x e^{3} + {\left (4 \, c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}\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 (90 \, c^{2} d^{3} x e^{2} + 60 \, c^{2} d^{4} e + {\left (2 \, c^{2} x^{4} + 14 \, a c x^{2} - 3 \, a^{2}\right )} e^{5} - 5 \, {\left (c^{2} d x^{3} - 11 \, a c d x\right )} e^{4} + 5 \, {\left (4 \, c^{2} d^{2} x^{2} + 7 \, a c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{12 \, {\left (x^{2} e^{8} + 2 \, d x e^{7} + d^{2} e^{6}\right )}}, \frac {30 \, {\left (8 \, c^{2} d^{4} x e + 4 \, c^{2} d^{5} + 3 \, a c d x^{2} e^{4} + 6 \, a c d^{2} x e^{3} + {\left (4 \, c^{2} d^{3} x^{2} + 3 \, a c d^{3}\right )} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + 15 \, {\left (8 \, c^{2} d^{3} x e + 4 \, c^{2} d^{4} + a c x^{2} e^{4} + 2 \, a c d x e^{3} + {\left (4 \, c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}\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 (90 \, c^{2} d^{3} x e^{2} + 60 \, c^{2} d^{4} e + {\left (2 \, c^{2} x^{4} + 14 \, a c x^{2} - 3 \, a^{2}\right )} e^{5} - 5 \, {\left (c^{2} d x^{3} - 11 \, a c d x\right )} e^{4} + 5 \, {\left (4 \, c^{2} d^{2} x^{2} + 7 \, a c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{12 \, {\left (x^{2} e^{8} + 2 \, d x e^{7} + d^{2} e^{6}\right )}}, \frac {30 \, {\left (8 \, c^{2} d^{3} x e + 4 \, c^{2} d^{4} + a c x^{2} e^{4} + 2 \, a c d x e^{3} + {\left (4 \, c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}\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^{4} x e + 4 \, c^{2} d^{5} + 3 \, a c d x^{2} e^{4} + 6 \, a c d^{2} x e^{3} + {\left (4 \, c^{2} d^{3} x^{2} + 3 \, a c d^{3}\right )} e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (90 \, c^{2} d^{3} x e^{2} + 60 \, c^{2} d^{4} e + {\left (2 \, c^{2} x^{4} + 14 \, a c x^{2} - 3 \, a^{2}\right )} e^{5} - 5 \, {\left (c^{2} d x^{3} - 11 \, a c d x\right )} e^{4} + 5 \, {\left (4 \, c^{2} d^{2} x^{2} + 7 \, a c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{12 \, {\left (x^{2} e^{8} + 2 \, d x e^{7} + d^{2} e^{6}\right )}}, \frac {15 \, {\left (8 \, c^{2} d^{3} x e + 4 \, c^{2} d^{4} + a c x^{2} e^{4} + 2 \, a c d x e^{3} + {\left (4 \, c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}\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^{4} x e + 4 \, c^{2} d^{5} + 3 \, a c d x^{2} e^{4} + 6 \, a c d^{2} x e^{3} + {\left (4 \, c^{2} d^{3} x^{2} + 3 \, a c d^{3}\right )} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (90 \, c^{2} d^{3} x e^{2} + 60 \, c^{2} d^{4} e + {\left (2 \, c^{2} x^{4} + 14 \, a c x^{2} - 3 \, a^{2}\right )} e^{5} - 5 \, {\left (c^{2} d x^{3} - 11 \, a c d x\right )} e^{4} + 5 \, {\left (4 \, c^{2} d^{2} x^{2} + 7 \, a c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{6 \, {\left (x^{2} e^{8} + 2 \, d x e^{7} + d^{2} e^{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)^3,x, algorithm="fricas")

[Out]

[1/12*(15*(8*c^2*d^4*x*e + 4*c^2*d^5 + 3*a*c*d*x^2*e^4 + 6*a*c*d^2*x*e^3 + (4*c^2*d^3*x^2 + 3*a*c*d^3)*e^2)*sq
rt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 15*(8*c^2*d^3*x*e + 4*c^2*d^4 + a*c*x^2*e^4 + 2*a*c*d*
x*e^3 + (4*c^2*d^2*x^2 + a*c*d^2)*e^2)*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*sqr
t(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*(90*c^2
*d^3*x*e^2 + 60*c^2*d^4*e + (2*c^2*x^4 + 14*a*c*x^2 - 3*a^2)*e^5 - 5*(c^2*d*x^3 - 11*a*c*d*x)*e^4 + 5*(4*c^2*d
^2*x^2 + 7*a*c*d^2)*e^3)*sqrt(c*x^2 + a))/(x^2*e^8 + 2*d*x*e^7 + d^2*e^6), 1/12*(30*(8*c^2*d^4*x*e + 4*c^2*d^5
+ 3*a*c*d*x^2*e^4 + 6*a*c*d^2*x*e^3 + (4*c^2*d^3*x^2 + 3*a*c*d^3)*e^2)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2
+ a)) + 15*(8*c^2*d^3*x*e + 4*c^2*d^4 + a*c*x^2*e^4 + 2*a*c*d*x*e^3 + (4*c^2*d^2*x^2 + a*c*d^2)*e^2)*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*(90*c^2*d^3*x*e^2 + 60*c^2*d^4*e + (2*c^2*x^4 + 14*a*c*
x^2 - 3*a^2)*e^5 - 5*(c^2*d*x^3 - 11*a*c*d*x)*e^4 + 5*(4*c^2*d^2*x^2 + 7*a*c*d^2)*e^3)*sqrt(c*x^2 + a))/(x^2*e
^8 + 2*d*x*e^7 + d^2*e^6), 1/12*(30*(8*c^2*d^3*x*e + 4*c^2*d^4 + a*c*x^2*e^4 + 2*a*c*d*x*e^3 + (4*c^2*d^2*x^2
+ a*c*d^2)*e^2)*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^4*x*e + 4*c^2*d^5 + 3*a*c*d*x^2*e^4 + 6*a*c*d^2*x*e^3 + (4*c^2*
d^3*x^2 + 3*a*c*d^3)*e^2)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(90*c^2*d^3*x*e^2 + 60*c
^2*d^4*e + (2*c^2*x^4 + 14*a*c*x^2 - 3*a^2)*e^5 - 5*(c^2*d*x^3 - 11*a*c*d*x)*e^4 + 5*(4*c^2*d^2*x^2 + 7*a*c*d^
2)*e^3)*sqrt(c*x^2 + a))/(x^2*e^8 + 2*d*x*e^7 + d^2*e^6), 1/6*(15*(8*c^2*d^3*x*e + 4*c^2*d^4 + a*c*x^2*e^4 + 2
*a*c*d*x*e^3 + (4*c^2*d^2*x^2 + a*c*d^2)*e^2)*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^4*x*e + 4*c^2*d^5 + 3*a*c*d*x^2*e
^4 + 6*a*c*d^2*x*e^3 + (4*c^2*d^3*x^2 + 3*a*c*d^3)*e^2)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (90*c^2*
d^3*x*e^2 + 60*c^2*d^4*e + (2*c^2*x^4 + 14*a*c*x^2 - 3*a^2)*e^5 - 5*(c^2*d*x^3 - 11*a*c*d*x)*e^4 + 5*(4*c^2*d^
2*x^2 + 7*a*c*d^2)*e^3)*sqrt(c*x^2 + a))/(x^2*e^8 + 2*d*x*e^7 + d^2*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}}}{\left (d + e x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (182) = 364\).
time = 1.44, size = 521, normalized size = 2.45 \begin {gather*} \frac {5}{2} \, {\left (4 \, c^{\frac {5}{2}} d^{3} + 3 \, a c^{\frac {3}{2}} d 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^{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 ) e^{\left (-6\right )}}{\sqrt {-c d^{2} - a e^{2}}} + \frac {1}{6} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, c^{2} x e^{\left (-3\right )} - 9 \, c^{2} d e^{\left (-4\right )}\right )} x + \frac {2 \, {\left (18 \, c^{3} d^{2} e^{13} + 7 \, a c^{2} e^{15}\right )} e^{\left (-18\right )}}{c}\right )} + \frac {{\left (10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{3} d^{4} e + 18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} c^{\frac {7}{2}} d^{5} - 26 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a c^{3} d^{4} e + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {5}{2}} d^{3} e^{2} + 11 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c^{2} d^{2} e^{3} + 9 \, a^{2} c^{\frac {5}{2}} d^{3} e^{2} - 25 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c^{2} d^{2} e^{3} - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{2} c^{\frac {3}{2}} d e^{4} + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{2} c e^{5} + 9 \, a^{3} c^{\frac {3}{2}} d e^{4} + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{3} c e^{5}\right )} e^{\left (-6\right )}}{{\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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}}{{\left (d+e\,x\right )}^3} \,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)^3,x)

[Out]

int((a + c*x^2)^(5/2)/(d + e*x)^3, x)

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