3.4.17 \(\int \frac {x^3 \sqrt {a+c x^2}}{d+e x} \, dx\) [317]

Optimal. Leaf size=211 \[ -\frac {\left (8 c d^3-e \left (4 c d^2-a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 c e^4}-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}+\frac {\left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2} e^5}+\frac {d^3 \sqrt {c d^2+a e^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^5} \]

[Out]

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

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Rubi [A]
time = 0.24, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1668, 829, 858, 223, 212, 739} \begin {gather*} \frac {\left (-a^2 e^4+4 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 c^{3/2} e^5}+\frac {d^3 \sqrt {a e^2+c d^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^5}-\frac {\sqrt {a+c x^2} \left (8 c d^3-e x \left (4 c d^2-a e^2\right )\right )}{8 c e^4}-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {\left (a+c x^2\right )^{3/2} (d+e x)}{4 c e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*((8*c*d^3 - e*(4*c*d^2 - a*e^2)*x)*Sqrt[a + c*x^2])/(c*e^4) - (7*d*(a + c*x^2)^(3/2))/(12*c*e^2) + ((d +
e*x)*(a + c*x^2)^(3/2))/(4*c*e^2) + ((8*c^2*d^4 + 4*a*c*d^2*e^2 - a^2*e^4)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]
])/(8*c^(3/2)*e^5) + (d^3*Sqrt[c*d^2 + a*e^2]*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/e^
5

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 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]

Rule 1668

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rubi steps

\begin {align*} \int \frac {x^3 \sqrt {a+c x^2}}{d+e x} \, dx &=\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}+\frac {\int \frac {\sqrt {a+c x^2} \left (-a d e^2-e \left (3 c d^2+a e^2\right ) x-7 c d e^2 x^2\right )}{d+e x} \, dx}{4 c e^3}\\ &=-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}+\frac {\int \frac {\left (-3 a c d e^4+3 c e^3 \left (4 c d^2-a e^2\right ) x\right ) \sqrt {a+c x^2}}{d+e x} \, dx}{12 c^2 e^5}\\ &=-\frac {\left (8 c d^3-e \left (4 c d^2-a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 c e^4}-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}+\frac {\int \frac {-3 a c^2 d e^4 \left (4 c d^2+a e^2\right )+3 c^2 e^3 \left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{24 c^3 e^7}\\ &=-\frac {\left (8 c d^3-e \left (4 c d^2-a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 c e^4}-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}-\frac {\left (d^3 \left (c d^2+a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^5}+\frac {\left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{8 c e^5}\\ &=-\frac {\left (8 c d^3-e \left (4 c d^2-a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 c e^4}-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}+\frac {\left (d^3 \left (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 )}{e^5}+\frac {\left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{8 c e^5}\\ &=-\frac {\left (8 c d^3-e \left (4 c d^2-a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 c e^4}-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}+\frac {\left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2} e^5}+\frac {d^3 \sqrt {c d^2+a e^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^5}\\ \end {align*}

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Mathematica [A]
time = 0.48, size = 199, normalized size = 0.94 \begin {gather*} \frac {\sqrt {c} e \sqrt {a+c x^2} \left (a e^2 (-8 d+3 e x)+c \left (-24 d^3+12 d^2 e x-8 d e^2 x^2+6 e^3 x^3\right )\right )-48 c^{3/2} d^3 \sqrt {-c d^2-a e^2} \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )-3 \left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{24 c^{3/2} e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c]*e*Sqrt[a + c*x^2]*(a*e^2*(-8*d + 3*e*x) + c*(-24*d^3 + 12*d^2*e*x - 8*d*e^2*x^2 + 6*e^3*x^3)) - 48*c^
(3/2)*d^3*Sqrt[-(c*d^2) - a*e^2]*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]] - 3*(8
*c^2*d^4 + 4*a*c*d^2*e^2 - a^2*e^4)*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/(24*c^(3/2)*e^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(386\) vs. \(2(185)=370\).
time = 0.09, size = 387, normalized size = 1.83

method	result	size
default	\(\frac {\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4 c}-\frac {a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4 c}}{e}-\frac {d \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{3 c \,e^{2}}+\frac {d^{2} \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{e^{3}}-\frac {d^{3} \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}-\frac {\sqrt {c}\, d \ln \left (\frac {-\frac {c d}{e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\right )}{e}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\right )}{e^{4}}\)	\(387\)
risch	\(-\frac {\left (-6 c \,e^{3} x^{3}+8 d \,e^{2} c \,x^{2}-3 a \,e^{3} x -12 c \,d^{2} e x +8 a d \,e^{2}+24 c \,d^{3}\right ) \sqrt {c \,x^{2}+a}}{24 c \,e^{4}}-\frac {\ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) a^{2}}{8 c^{\frac {3}{2}} e}+\frac {\ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) a \,d^{2}}{2 \sqrt {c}\, e^{3}}+\frac {\sqrt {c}\, \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) d^{4}}{e^{5}}+\frac {d^{3} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right ) a}{e^{4} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}+\frac {c \,d^{5} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{6} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\)	\(406\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(c*x^2+a)^(1/2)/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

1/e*(1/4*x*(c*x^2+a)^(3/2)/c-1/4*a/c*(1/2*x*(c*x^2+a)^(1/2)+1/2*a/c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))))-1/3*
d*(c*x^2+a)^(3/2)/c/e^2+d^2/e^3*(1/2*x*(c*x^2+a)^(1/2)+1/2*a/c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2)))-d^3/e^4*((
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 = 0.32, size = 200, normalized size = 0.95 \begin {gather*} \sqrt {c} d^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-5\right )} - \sqrt {c d^{2} e^{\left (-2\right )} + a} d^{3} \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 (-4\right )} + \frac {1}{2} \, \sqrt {c x^{2} + a} d^{2} x e^{\left (-3\right )} + \frac {a d^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-3\right )}}{2 \, \sqrt {c}} - \sqrt {c x^{2} + a} d^{3} e^{\left (-4\right )} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} x e^{\left (-1\right )}}{4 \, c} - \frac {\sqrt {c x^{2} + a} a x e^{\left (-1\right )}}{8 \, c} - \frac {a^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-1\right )}}{8 \, c^{\frac {3}{2}}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} d e^{\left (-2\right )}}{3 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c*x^2+a)^(1/2)/(e*x+d),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 10.54, size = 920, normalized size = 4.36 \begin {gather*} \left [\frac {{\left (24 \, \sqrt {c d^{2} + a e^{2}} c^{2} d^{3} \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 ) - 3 \, {\left (8 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (12 \, c^{2} d^{2} x e^{2} - 24 \, c^{2} d^{3} e + 3 \, {\left (2 \, c^{2} x^{3} + a c x\right )} e^{4} - 8 \, {\left (c^{2} d x^{2} + a c d\right )} e^{3}\right )} \sqrt {c x^{2} + a}\right )} e^{\left (-5\right )}}{48 \, c^{2}}, -\frac {{\left (48 \, \sqrt {-c d^{2} - a e^{2}} c^{2} d^{3} \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 ) + 3 \, {\left (8 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (12 \, c^{2} d^{2} x e^{2} - 24 \, c^{2} d^{3} e + 3 \, {\left (2 \, c^{2} x^{3} + a c x\right )} e^{4} - 8 \, {\left (c^{2} d x^{2} + a c d\right )} e^{3}\right )} \sqrt {c x^{2} + a}\right )} e^{\left (-5\right )}}{48 \, c^{2}}, \frac {{\left (12 \, \sqrt {c d^{2} + a e^{2}} c^{2} d^{3} \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 ) - 3 \, {\left (8 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (12 \, c^{2} d^{2} x e^{2} - 24 \, c^{2} d^{3} e + 3 \, {\left (2 \, c^{2} x^{3} + a c x\right )} e^{4} - 8 \, {\left (c^{2} d x^{2} + a c d\right )} e^{3}\right )} \sqrt {c x^{2} + a}\right )} e^{\left (-5\right )}}{24 \, c^{2}}, -\frac {{\left (24 \, \sqrt {-c d^{2} - a e^{2}} c^{2} d^{3} \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 ) + 3 \, {\left (8 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (12 \, c^{2} d^{2} x e^{2} - 24 \, c^{2} d^{3} e + 3 \, {\left (2 \, c^{2} x^{3} + a c x\right )} e^{4} - 8 \, {\left (c^{2} d x^{2} + a c d\right )} e^{3}\right )} \sqrt {c x^{2} + a}\right )} e^{\left (-5\right )}}{24 \, c^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c*x^2+a)^(1/2)/(e*x+d),x, algorithm="fricas")

[Out]

[1/48*(24*sqrt(c*d^2 + a*e^2)*c^2*d^3*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)) - 3*(8*c^2*d^4 + 4*a*c*d^2*e^2 -
a^2*e^4)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(12*c^2*d^2*x*e^2 - 24*c^2*d^3*e + 3*(2*
c^2*x^3 + a*c*x)*e^4 - 8*(c^2*d*x^2 + a*c*d)*e^3)*sqrt(c*x^2 + a))*e^(-5)/c^2, -1/48*(48*sqrt(-c*d^2 - a*e^2)*
c^2*d^3*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)) + 3*(8*c^2*d^4 + 4*a*c*d^2*e^2 - a^2*e^4)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(12*
c^2*d^2*x*e^2 - 24*c^2*d^3*e + 3*(2*c^2*x^3 + a*c*x)*e^4 - 8*(c^2*d*x^2 + a*c*d)*e^3)*sqrt(c*x^2 + a))*e^(-5)/
c^2, 1/24*(12*sqrt(c*d^2 + a*e^2)*c^2*d^3*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)) - 3*(8*c^2*d^4 + 4*a*c*d^2*e
^2 - a^2*e^4)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (12*c^2*d^2*x*e^2 - 24*c^2*d^3*e + 3*(2*c^2*x^3 +
a*c*x)*e^4 - 8*(c^2*d*x^2 + a*c*d)*e^3)*sqrt(c*x^2 + a))*e^(-5)/c^2, -1/24*(24*sqrt(-c*d^2 - a*e^2)*c^2*d^3*ar
ctan(-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)) + 3*(8
*c^2*d^4 + 4*a*c*d^2*e^2 - a^2*e^4)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (12*c^2*d^2*x*e^2 - 24*c^2*d
^3*e + 3*(2*c^2*x^3 + a*c*x)*e^4 - 8*(c^2*d*x^2 + a*c*d)*e^3)*sqrt(c*x^2 + a))*e^(-5)/c^2]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \sqrt {a + c x^{2}}}{d + e x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(c*x**2+a)**(1/2)/(e*x+d),x)

[Out]

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

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Giac [A]
time = 0.98, size = 201, normalized size = 0.95 \begin {gather*} -\frac {2 \, {\left (c d^{5} + a d^{3} 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 (-5\right )}}{\sqrt {-c d^{2} - a e^{2}}} + \frac {1}{24} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left (3 \, x e^{\left (-1\right )} - 4 \, d e^{\left (-2\right )}\right )} x + \frac {3 \, {\left (4 \, c^{2} d^{2} e^{12} + a c e^{14}\right )} e^{\left (-15\right )}}{c^{2}}\right )} x - \frac {8 \, {\left (3 \, c^{2} d^{3} e^{11} + a c d e^{13}\right )} e^{\left (-15\right )}}{c^{2}}\right )} - \frac {{\left (8 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} e^{\left (-5\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{8 \, c^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c*x^2+a)^(1/2)/(e*x+d),x, algorithm="giac")

[Out]

-2*(c*d^5 + a*d^3*e^2)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))*e^(-5)/sqrt
(-c*d^2 - a*e^2) + 1/24*sqrt(c*x^2 + a)*((2*(3*x*e^(-1) - 4*d*e^(-2))*x + 3*(4*c^2*d^2*e^12 + a*c*e^14)*e^(-15
)/c^2)*x - 8*(3*c^2*d^3*e^11 + a*c*d*e^13)*e^(-15)/c^2) - 1/8*(8*c^2*d^4 + 4*a*c*d^2*e^2 - a^2*e^4)*e^(-5)*log
(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(3/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3\,\sqrt {c\,x^2+a}}{d+e\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(a + c*x^2)^(1/2))/(d + e*x),x)

[Out]

int((x^3*(a + c*x^2)^(1/2))/(d + e*x), x)

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