3.6.50 \(\int \frac {(a+c x^2)^{5/2}}{(d+e x)^2} \, dx\) [550]

Optimal. Leaf size=219 \[ -\frac {5 c \left (8 d \left (c d^2+a e^2\right )-e \left (4 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5}-\frac {5 c (4 d-3 e x) \left (a+c x^2\right )^{3/2}}{12 e^3}-\frac {\left (a+c x^2\right )^{5/2}}{e (d+e x)}+\frac {5 \sqrt {c} \left (8 c^2 d^4+12 a c d^2 e^2+3 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 e^6}+\frac {5 c d \left (c d^2+a e^2\right )^{3/2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^6} \]

[Out]

-5/12*c*(-3*e*x+4*d)*(c*x^2+a)^(3/2)/e^3-(c*x^2+a)^(5/2)/e/(e*x+d)+5*c*d*(a*e^2+c*d^2)^(3/2)*arctanh((-c*d*x+a
*e)/(a*e^2+c*d^2)^(1/2)/(c*x^2+a)^(1/2))/e^6+5/8*(3*a^2*e^4+12*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^6-5/8*c*(8*d*(a*e^2+c*d^2)-e*(3*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 = 219, 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 = {747, 829, 858, 223, 212, 739} \begin {gather*} \frac {5 \sqrt {c} \left (3 a^2 e^4+12 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 {5 c d \left (a e^2+c d^2\right )^{3/2} \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 {5 c \sqrt {a+c x^2} \left (8 d \left (a e^2+c d^2\right )-e x \left (3 a e^2+4 c d^2\right )\right )}{8 e^5}-\frac {5 c \left (a+c x^2\right )^{3/2} (4 d-3 e x)}{12 e^3}-\frac {\left (a+c x^2\right )^{5/2}}{e (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^(5/2)/(d + e*x)^2,x]

[Out]

(-5*c*(8*d*(c*d^2 + a*e^2) - e*(4*c*d^2 + 3*a*e^2)*x)*Sqrt[a + c*x^2])/(8*e^5) - (5*c*(4*d - 3*e*x)*(a + c*x^2
)^(3/2))/(12*e^3) - (a + c*x^2)^(5/2)/(e*(d + e*x)) + (5*Sqrt[c]*(8*c^2*d^4 + 12*a*c*d^2*e^2 + 3*a^2*e^4)*ArcT
anh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(8*e^6) + (5*c*d*(c*d^2 + a*e^2)^(3/2)*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 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 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)^2} \, dx &=-\frac {\left (a+c x^2\right )^{5/2}}{e (d+e x)}+\frac {(5 c) \int \frac {x \left (a+c x^2\right )^{3/2}}{d+e x} \, dx}{e}\\ &=-\frac {5 c (4 d-3 e x) \left (a+c x^2\right )^{3/2}}{12 e^3}-\frac {\left (a+c x^2\right )^{5/2}}{e (d+e x)}+\frac {5 \int \frac {\left (-a c d e+c \left (4 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{d+e x} \, dx}{4 e^3}\\ &=-\frac {5 c \left (8 d \left (c d^2+a e^2\right )-e \left (4 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5}-\frac {5 c (4 d-3 e x) \left (a+c x^2\right )^{3/2}}{12 e^3}-\frac {\left (a+c x^2\right )^{5/2}}{e (d+e x)}+\frac {5 \int \frac {-a c^2 d e \left (4 c d^2+5 a e^2\right )+c^2 \left (8 c^2 d^4+12 a c d^2 e^2+3 a^2 e^4\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{8 c e^5}\\ &=-\frac {5 c \left (8 d \left (c d^2+a e^2\right )-e \left (4 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5}-\frac {5 c (4 d-3 e x) \left (a+c x^2\right )^{3/2}}{12 e^3}-\frac {\left (a+c x^2\right )^{5/2}}{e (d+e x)}-\frac {\left (5 c d \left (c d^2+a e^2\right )^2\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^6}+\frac {\left (5 c \left (8 c^2 d^4+12 a c d^2 e^2+3 a^2 e^4\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{8 e^6}\\ &=-\frac {5 c \left (8 d \left (c d^2+a e^2\right )-e \left (4 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5}-\frac {5 c (4 d-3 e x) \left (a+c x^2\right )^{3/2}}{12 e^3}-\frac {\left (a+c x^2\right )^{5/2}}{e (d+e x)}+\frac {\left (5 c d \left (c d^2+a e^2\right )^2\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 )}{e^6}+\frac {\left (5 c \left (8 c^2 d^4+12 a c d^2 e^2+3 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 {5 c \left (8 d \left (c d^2+a e^2\right )-e \left (4 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5}-\frac {5 c (4 d-3 e x) \left (a+c x^2\right )^{3/2}}{12 e^3}-\frac {\left (a+c x^2\right )^{5/2}}{e (d+e x)}+\frac {5 \sqrt {c} \left (8 c^2 d^4+12 a c d^2 e^2+3 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 e^6}+\frac {5 c d \left (c d^2+a e^2\right )^{3/2} \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 = 0.87, size = 229, normalized size = 1.05 \begin {gather*} -\frac {\frac {e \sqrt {a+c x^2} \left (24 a^2 e^4+a c e^2 \left (160 d^2+85 d e x-27 e^2 x^2\right )+2 c^2 \left (60 d^4+30 d^3 e x-10 d^2 e^2 x^2+5 d e^3 x^3-3 e^4 x^4\right )\right )}{d+e x}-240 c d \left (-c d^2-a e^2\right )^{3/2} \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} \left (8 c^2 d^4+12 a c d^2 e^2+3 a^2 e^4\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{24 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/24*((e*Sqrt[a + c*x^2]*(24*a^2*e^4 + a*c*e^2*(160*d^2 + 85*d*e*x - 27*e^2*x^2) + 2*c^2*(60*d^4 + 30*d^3*e*x
- 10*d^2*e^2*x^2 + 5*d*e^3*x^3 - 3*e^4*x^4)))/(d + e*x) - 240*c*d*(-(c*d^2) - a*e^2)^(3/2)*ArcTan[(Sqrt[c]*(d
+ e*x) - e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]] + 15*Sqrt[c]*(8*c^2*d^4 + 12*a*c*d^2*e^2 + 3*a^2*e^4)*Log
[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/e^6

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1308\) vs. \(2(195)=390\).
time = 0.48, size = 1309, normalized size = 5.98

method	result	size
default	\(\text {Expression too large to display}\)	\(1309\)
risch	\(\text {Expression too large to display}\)	\(1426\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e^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))))))

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Maxima [A]
time = 0.32, size = 238, normalized size = 1.09 \begin {gather*} 5 \, c^{\frac {5}{2}} d^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-6\right )} + \frac {5}{2} \, \sqrt {c x^{2} + a} c^{2} d^{2} x e^{\left (-4\right )} + \frac {15}{2} \, a c^{\frac {3}{2}} d^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-4\right )} - 5 \, \sqrt {c x^{2} + a} c^{2} d^{3} e^{\left (-5\right )} - 5 \, {\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {3}{2}} c d \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 {5}{4} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c x e^{\left (-2\right )} + \frac {15}{8} \, \sqrt {c x^{2} + a} a c x e^{\left (-2\right )} + \frac {15}{8} \, a^{2} \sqrt {c} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-2\right )} - \frac {5}{3} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c d e^{\left (-3\right )} - 5 \, \sqrt {c x^{2} + a} a c d e^{\left (-3\right )} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{x e^{2} + d e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*c^(5/2)*d^4*arcsinh(c*x/sqrt(a*c))*e^(-6) + 5/2*sqrt(c*x^2 + a)*c^2*d^2*x*e^(-4) + 15/2*a*c^(3/2)*d^2*arcsin
h(c*x/sqrt(a*c))*e^(-4) - 5*sqrt(c*x^2 + a)*c^2*d^3*e^(-5) - 5*(c*d^2*e^(-2) + a)^(3/2)*c*d*arcsinh(c*d*x/(sqr
t(a*c)*abs(x*e + d)) - a*e/(sqrt(a*c)*abs(x*e + d)))*e^(-3) + 5/4*(c*x^2 + a)^(3/2)*c*x*e^(-2) + 15/8*sqrt(c*x
^2 + a)*a*c*x*e^(-2) + 15/8*a^2*sqrt(c)*arcsinh(c*x/sqrt(a*c))*e^(-2) - 5/3*(c*x^2 + a)^(3/2)*c*d*e^(-3) - 5*s
qrt(c*x^2 + a)*a*c*d*e^(-3) - (c*x^2 + a)^(5/2)/(x*e^2 + d*e)

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Fricas [A]
time = 11.68, size = 1303, normalized size = 5.95 \begin {gather*} \left [\frac {15 \, {\left (8 \, c^{2} d^{4} x e + 8 \, c^{2} d^{5} + 12 \, a c d^{2} x e^{3} + 12 \, a c d^{3} e^{2} + 3 \, a^{2} x e^{5} + 3 \, 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^{3} x e + c^{2} d^{4} + a c d x e^{3} + a c d^{2} 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 (60 \, c^{2} d^{3} x e^{2} + 120 \, c^{2} d^{4} e - 3 \, {\left (2 \, c^{2} x^{4} + 9 \, a c x^{2} - 8 \, a^{2}\right )} e^{5} + 5 \, {\left (2 \, c^{2} d x^{3} + 17 \, a c d x\right )} e^{4} - 20 \, {\left (c^{2} d^{2} x^{2} - 8 \, a c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{48 \, {\left (x e^{7} + d e^{6}\right )}}, -\frac {240 \, {\left (c^{2} d^{3} x e + c^{2} d^{4} + a c d x e^{3} + a c d^{2} 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 + 8 \, c^{2} d^{5} + 12 \, a c d^{2} x e^{3} + 12 \, a c d^{3} e^{2} + 3 \, a^{2} x e^{5} + 3 \, 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 - 3 \, {\left (2 \, c^{2} x^{4} + 9 \, a c x^{2} - 8 \, a^{2}\right )} e^{5} + 5 \, {\left (2 \, c^{2} d x^{3} + 17 \, a c d x\right )} e^{4} - 20 \, {\left (c^{2} d^{2} x^{2} - 8 \, a c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{48 \, {\left (x e^{7} + d e^{6}\right )}}, -\frac {15 \, {\left (8 \, c^{2} d^{4} x e + 8 \, c^{2} d^{5} + 12 \, a c d^{2} x e^{3} + 12 \, a c d^{3} e^{2} + 3 \, a^{2} x e^{5} + 3 \, 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^{3} x e + c^{2} d^{4} + a c d x e^{3} + a c d^{2} 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 ) + {\left (60 \, c^{2} d^{3} x e^{2} + 120 \, c^{2} d^{4} e - 3 \, {\left (2 \, c^{2} x^{4} + 9 \, a c x^{2} - 8 \, a^{2}\right )} e^{5} + 5 \, {\left (2 \, c^{2} d x^{3} + 17 \, a c d x\right )} e^{4} - 20 \, {\left (c^{2} d^{2} x^{2} - 8 \, a c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{24 \, {\left (x e^{7} + d e^{6}\right )}}, -\frac {120 \, {\left (c^{2} d^{3} x e + c^{2} d^{4} + a c d x e^{3} + a c d^{2} 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 + 8 \, c^{2} d^{5} + 12 \, a c d^{2} x e^{3} + 12 \, a c d^{3} e^{2} + 3 \, a^{2} x e^{5} + 3 \, 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 - 3 \, {\left (2 \, c^{2} x^{4} + 9 \, a c x^{2} - 8 \, a^{2}\right )} e^{5} + 5 \, {\left (2 \, c^{2} d x^{3} + 17 \, a c d x\right )} e^{4} - 20 \, {\left (c^{2} d^{2} x^{2} - 8 \, a c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{24 \, {\left (x e^{7} + d 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)^2,x, algorithm="fricas")

[Out]

[1/48*(15*(8*c^2*d^4*x*e + 8*c^2*d^5 + 12*a*c*d^2*x*e^3 + 12*a*c*d^3*e^2 + 3*a^2*x*e^5 + 3*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^3*x*e + c^2*d^4 + a*c*d*x*e^3 + 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*(60*c^2*d^3*x*e^2 + 120*c^2*d^4*e - 3*(2*c^2*x^4 +
9*a*c*x^2 - 8*a^2)*e^5 + 5*(2*c^2*d*x^3 + 17*a*c*d*x)*e^4 - 20*(c^2*d^2*x^2 - 8*a*c*d^2)*e^3)*sqrt(c*x^2 + a)
)/(x*e^7 + d*e^6), -1/48*(240*(c^2*d^3*x*e + c^2*d^4 + a*c*d*x*e^3 + 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 + 8*c^2*d^5 + 12*a*c*d^2*x*e^3 + 12*a*c*d^3*e^2 + 3*a^2*x*e^5 + 3*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 - 3*(2*c^2*x^4 + 9*a*c*x^2 - 8*a^2)*e^5
+ 5*(2*c^2*d*x^3 + 17*a*c*d*x)*e^4 - 20*(c^2*d^2*x^2 - 8*a*c*d^2)*e^3)*sqrt(c*x^2 + a))/(x*e^7 + d*e^6), -1/24
*(15*(8*c^2*d^4*x*e + 8*c^2*d^5 + 12*a*c*d^2*x*e^3 + 12*a*c*d^3*e^2 + 3*a^2*x*e^5 + 3*a^2*d*e^4)*sqrt(-c)*arct
an(sqrt(-c)*x/sqrt(c*x^2 + a)) - 60*(c^2*d^3*x*e + c^2*d^4 + a*c*d*x*e^3 + a*c*d^2*e^2)*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 - 3*(2*c^2*x^4 + 9*a*c*x^2 - 8*a^2)*
e^5 + 5*(2*c^2*d*x^3 + 17*a*c*d*x)*e^4 - 20*(c^2*d^2*x^2 - 8*a*c*d^2)*e^3)*sqrt(c*x^2 + a))/(x*e^7 + d*e^6), -
1/24*(120*(c^2*d^3*x*e + c^2*d^4 + a*c*d*x*e^3 + 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 + 8*c^2*d^5
+ 12*a*c*d^2*x*e^3 + 12*a*c*d^3*e^2 + 3*a^2*x*e^5 + 3*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 - 3*(2*c^2*x^4 + 9*a*c*x^2 - 8*a^2)*e^5 + 5*(2*c^2*d*x^3 + 17*a*c*d*x)*e^4
- 20*(c^2*d^2*x^2 - 8*a*c*d^2)*e^3)*sqrt(c*x^2 + a))/(x*e^7 + d*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 )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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 )}^2} \,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)^2,x)

[Out]

int((a + c*x^2)^(5/2)/(d + e*x)^2, x)

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