3.5.60 \(\int \frac {(a+c x^2)^{3/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=161 \[ -\frac {3 c^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{e^4}-\frac {3 c \left (a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 e^4 \sqrt {a e^2+c d^2}}+\frac {3 c \sqrt {a+c x^2} (2 d+e x)}{2 e^3 (d+e x)}-\frac {\left (a+c x^2\right )^{3/2}}{2 e (d+e x)^2} \]

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Rubi [A]  time = 0.12, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {3 c^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{e^4}-\frac {3 c \left (a e^2+2 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^4 \sqrt {a e^2+c d^2}}+\frac {3 c \sqrt {a+c x^2} (2 d+e x)}{2 e^3 (d+e x)}-\frac {\left (a+c x^2\right )^{3/2}}{2 e (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^(3/2)/(d + e*x)^3,x]

[Out]

(3*c*(2*d + e*x)*Sqrt[a + c*x^2])/(2*e^3*(d + e*x)) - (a + c*x^2)^(3/2)/(2*e*(d + e*x)^2) - (3*c^(3/2)*d*ArcTa
nh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/e^4 - (3*c*(2*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^4*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 )^{3/2}}{(d+e x)^3} \, dx &=-\frac {\left (a+c x^2\right )^{3/2}}{2 e (d+e x)^2}+\frac {(3 c) \int \frac {x \sqrt {a+c x^2}}{(d+e x)^2} \, dx}{2 e}\\ &=\frac {3 c (2 d+e x) \sqrt {a+c x^2}}{2 e^3 (d+e x)}-\frac {\left (a+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac {(3 c) \int \frac {-2 a e+4 c d x}{(d+e x) \sqrt {a+c x^2}} \, dx}{4 e^3}\\ &=\frac {3 c (2 d+e x) \sqrt {a+c x^2}}{2 e^3 (d+e x)}-\frac {\left (a+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac {\left (3 c^2 d\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{e^4}+\frac {\left (3 c \left (2 c d^2+a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{2 e^4}\\ &=\frac {3 c (2 d+e x) \sqrt {a+c x^2}}{2 e^3 (d+e x)}-\frac {\left (a+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac {\left (3 c^2 d\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{e^4}-\frac {\left (3 c \left (2 c d^2+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^4}\\ &=\frac {3 c (2 d+e x) \sqrt {a+c x^2}}{2 e^3 (d+e x)}-\frac {\left (a+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac {3 c^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{e^4}-\frac {3 c \left (2 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^4 \sqrt {c d^2+a e^2}}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 189, normalized size = 1.17 \begin {gather*} \frac {-6 c^{3/2} d \log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right )+\frac {e \sqrt {a+c x^2} \left (c \left (6 d^2+9 d e x+2 e^2 x^2\right )-a e^2\right )}{(d+e x)^2}-\frac {3 c \left (a e^2+2 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 {3 c \left (a e^2+2 c d^2\right ) \log (d+e x)}{\sqrt {a e^2+c d^2}}}{2 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((e*Sqrt[a + c*x^2]*(-(a*e^2) + c*(6*d^2 + 9*d*e*x + 2*e^2*x^2)))/(d + e*x)^2 + (3*c*(2*c*d^2 + a*e^2)*Log[d +
e*x])/Sqrt[c*d^2 + a*e^2] - 6*c^(3/2)*d*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] - (3*c*(2*c*d^2 + a*e^2)*Log[a*e -
c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/Sqrt[c*d^2 + a*e^2])/(2*e^4)

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IntegrateAlgebraic [A]  time = 1.12, size = 185, normalized size = 1.15 \begin {gather*} \frac {3 c^{3/2} d \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{e^4}+\frac {3 \sqrt {-a e^2-c d^2} \left (a c e^2+2 c^2 d^2\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^4 \left (a e^2+c d^2\right )}+\frac {\sqrt {a+c x^2} \left (-a e^2+6 c d^2+9 c d e x+2 c e^2 x^2\right )}{2 e^3 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + c*x^2]*(6*c*d^2 - a*e^2 + 9*c*d*e*x + 2*c*e^2*x^2))/(2*e^3*(d + e*x)^2) + (3*Sqrt[-(c*d^2) - a*e^2]*
(2*c^2*d^2 + a*c*e^2)*ArcTan[(Sqrt[c]*d + Sqrt[c]*e*x - e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/(e^4*(c*d^
2 + a*e^2)) + (3*c^(3/2)*d*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/e^4

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fricas [B]  time = 0.95, size = 1545, normalized size = 9.60

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(6*(c^2*d^5 + a*c*d^3*e^2 + (c^2*d^3*e^2 + a*c*d*e^4)*x^2 + 2*(c^2*d^4*e + a*c*d^2*e^3)*x)*sqrt(c)*log(-2
*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 3*(2*c^2*d^4 + a*c*d^2*e^2 + (2*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(2*
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*(6*c^2*d^4*e + 5*a*c*
d^2*e^3 - a^2*e^5 + 2*(c^2*d^2*e^3 + a*c*e^5)*x^2 + 9*(c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c*d^4*e^4
+ a*d^2*e^6 + (c*d^2*e^6 + a*e^8)*x^2 + 2*(c*d^3*e^5 + a*d*e^7)*x), 1/4*(12*(c^2*d^5 + a*c*d^3*e^2 + (c^2*d^3
*e^2 + a*c*d*e^4)*x^2 + 2*(c^2*d^4*e + a*c*d^2*e^3)*x)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + 3*(2*c^2*
d^4 + a*c*d^2*e^2 + (2*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(2*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*(6*c^2*d^4*e + 5*a*c*d^2*e^3 - a^2*e^5 + 2*(c^2*d^2*e^3 + a*c*e^5)*x^2 + 9
*(c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c*d^4*e^4 + a*d^2*e^6 + (c*d^2*e^6 + a*e^8)*x^2 + 2*(c*d^3*e^5
+ a*d*e^7)*x), -1/2*(3*(2*c^2*d^4 + a*c*d^2*e^2 + (2*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(2*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)) - 3*(c^2*d^5 + a*c*d^3*e^2 + (c^2*d^3*e^2 + a*c*d*e^4)*x^2 + 2*(c^2*d^4*e + a*c*d^2*e^3)*x
)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - (6*c^2*d^4*e + 5*a*c*d^2*e^3 - a^2*e^5 + 2*(c^2*d^
2*e^3 + a*c*e^5)*x^2 + 9*(c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c*d^4*e^4 + a*d^2*e^6 + (c*d^2*e^6 + a
*e^8)*x^2 + 2*(c*d^3*e^5 + a*d*e^7)*x), -1/2*(3*(2*c^2*d^4 + a*c*d^2*e^2 + (2*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(
2*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)) - 6*(c^2*d^5 + a*c*d^3*e^2 + (c^2*d^3*e^2 + a*c*d*e^4)*x^2 + 2*(c^
2*d^4*e + a*c*d^2*e^3)*x)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (6*c^2*d^4*e + 5*a*c*d^2*e^3 - a^2*e^5
+ 2*(c^2*d^2*e^3 + a*c*e^5)*x^2 + 9*(c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c*d^4*e^4 + a*d^2*e^6 + (c
*d^2*e^6 + a*e^8)*x^2 + 2*(c*d^3*e^5 + a*d*e^7)*x)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to divide, perhaps due to rounding error%%%{-1,[6,0,7,0]%%%}+%%%{%%{[6,0]:[1,0,%%%{-1,[1]%%%}]%%},[5,1,6,0]
%%%}+%%%{%%%{-12,[1]%%%},[4,2,5,0]%%%}+%%%{3,[4,0,7,1]%%%}+%%%{%%{[%%%{8,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3
,3,4,0]%%%}+%%%{%%{[-12,0]:[1,0,%%%{-1,[1]%%%}]%%},[3,1,6,1]%%%}+%%%{%%%{12,[1]%%%},[2,2,5,1]%%%}+%%%{-3,[2,0,
7,2]%%%}+%%%{%%{[6,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,1,6,2]%%%}+%%%{1,[0,0,7,3]%%%} / %%%{%%{poly1[%%%{1,[1]%%%},0
]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,3,0]%%%}+%%%{%%%{-6,[2]%%%},[5,1,2,0]%%%}+%%%{%%{[%%%{12,[2]%%%},0]:[1,0,%%%{-1
,[1]%%%}]%%},[4,2,1,0]%%%}+%%%{%%{poly1[%%%{-3,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,3,1]%%%}+%%%{%%%{-8,[3]
%%%},[3,3,0,0]%%%}+%%%{%%%{12,[2]%%%},[3,1,2,1]%%%}+%%%{%%{[%%%{-12,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,2,1,
1]%%%}+%%%{%%{poly1[%%%{3,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,3,2]%%%}+%%%{%%%{-6,[2]%%%},[1,1,2,2]%%%}+%%
%{%%{poly1[%%%{-1,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,3,3]%%%} Error: Bad Argument Value

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maple [B]  time = 0.06, size = 2117, normalized size = 13.15

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^(3/2)/(e*x+d)^3,x)

[Out]

-1/2/e/(a*e^2+c*d^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)^(5/2)+1/2*c*d/(a*e^2+c*d^2)^2/
(x+d/e)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(5/2)+1/2/e*c^2*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)-3/4/e^2*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)*x-9/4/e^2*c^(5/2)*d^3/(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+3/2/e*c^2*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)^
(1/2)*a+3/2/e^3*c^3*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)-3/2/e^4*c^(7/2)
*d^5/(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))-3/2
/e*c^2*d^2/(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^2-3/e^3*c^3*d^4/(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-3/2/e^5*c^4*d^6/(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))-1/2*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)*x-3/
4*c^2*d/(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-3/4*c^(3/2)*d/(a*e^2+c*d^2)
^2*a^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))+1/2/e/(a*e^2+c*d^
2)*c*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)-3/4/e^2/(a*e^2+c*d^2)*c^2*d*(-2*(x+d/e)*c*d/e+(x+d
/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)*x-9/4/e^2/(a*e^2+c*d^2)*c^(3/2)*d*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+3/2/e/(a*e^2+c*d^2)*c*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)
/e^2)^(1/2)*a+3/2/e^3/(a*e^2+c*d^2)*c^2*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)*d^2-3/2/e^4/(a*
e^2+c*d^2)*c^(5/2)*d^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))-3
/2/e/(a*e^2+c*d^2)*c/((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^2-3/e^3/(a*e^2+c*d^2)*c^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*d^2-3/2/e^5/(a*e^2+c*d^2)*c^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))*d^4

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maxima [B]  time = 1.89, size = 341, normalized size = 2.12 \begin {gather*} \frac {3 \, \sqrt {c x^{2} + a} c^{2} d^{2}}{2 \, {\left (c d^{2} e^{3} + a e^{5}\right )}} - \frac {3 \, \sqrt {c x^{2} + a} c^{2} d x}{2 \, {\left (c d^{2} e^{2} + a e^{4}\right )}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} c d}{2 \, {\left (c d^{2} e^{2} x + a e^{4} x + c d^{3} e + a d e^{3}\right )}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{2 \, {\left (c d^{2} e x^{2} + a e^{3} x^{2} + 2 \, c d^{3} x + 2 \, a d e^{2} x + \frac {c d^{4}}{e} + a d^{2} e\right )}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} c}{2 \, {\left (c d^{2} e + a e^{3}\right )}} - \frac {3 \, c^{\frac {3}{2}} d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{e^{4}} + \frac {3 \, c^{2} d^{2} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{2 \, \sqrt {a + \frac {c d^{2}}{e^{2}}} e^{5}} + \frac {3 \, \sqrt {a + \frac {c d^{2}}{e^{2}}} c \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{2 \, e^{3}} + \frac {3 \, \sqrt {c x^{2} + a} c}{2 \, e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*sqrt(c*x^2 + a)*c^2*d^2/(c*d^2*e^3 + a*e^5) - 3/2*sqrt(c*x^2 + a)*c^2*d*x/(c*d^2*e^2 + a*e^4) + 1/2*(c*x^2
+ a)^(3/2)*c*d/(c*d^2*e^2*x + a*e^4*x + c*d^3*e + a*d*e^3) - 1/2*(c*x^2 + a)^(5/2)/(c*d^2*e*x^2 + a*e^3*x^2 +
2*c*d^3*x + 2*a*d*e^2*x + c*d^4/e + a*d^2*e) + 1/2*(c*x^2 + a)^(3/2)*c/(c*d^2*e + a*e^3) - 3*c^(3/2)*d*arcsin
h(c*x/sqrt(a*c))/e^4 + 3/2*c^2*d^2*arcsinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(e*x + d)))/(sqr
t(a + c*d^2/e^2)*e^5) + 3/2*sqrt(a + c*d^2/e^2)*c*arcsinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(
e*x + d)))/e^3 + 3/2*sqrt(c*x^2 + a)*c/e^3

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2+a\right )}^{3/2}}{{\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)^(3/2)/(d + e*x)^3,x)

[Out]

int((a + c*x^2)^(3/2)/(d + e*x)^3, x)

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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 {3}{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)**(3/2)/(e*x+d)**3,x)

[Out]

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

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