3.7.94 \(\int \frac {x^4 (c+d x^2)^{5/2}}{a+b x^2} \, dx\) [694]

Optimal. Leaf size=291 \[ \frac {\left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x \sqrt {c+d x^2}}{128 b^4 d}+\frac {\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt {c+d x^2}}{192 b^3}+\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {a^{3/2} (b c-a d)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^5}-\frac {\left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 b^5 d^{3/2}} \]

[Out]

1/8*d*x^5*(d*x^2+c)^(3/2)/b+a^(3/2)*(-a*d+b*c)^(5/2)*arctan(x*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^2+c)^(1/2))/b^5-1/
128*(-128*a^4*d^4+320*a^3*b*c*d^3-240*a^2*b^2*c^2*d^2+40*a*b^3*c^3*d+5*b^4*c^4)*arctanh(x*d^(1/2)/(d*x^2+c)^(1
/2))/b^5/d^(3/2)+1/128*(-64*a^3*d^3+144*a^2*b*c*d^2-88*a*b^2*c^2*d+5*b^3*c^3)*x*(d*x^2+c)^(1/2)/b^4/d+1/192*(4
8*a^2*d^2-104*a*b*c*d+59*b^2*c^2)*x^3*(d*x^2+c)^(1/2)/b^3+1/48*d*(-8*a*d+11*b*c)*x^5*(d*x^2+c)^(1/2)/b^2

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Rubi [A]
time = 0.37, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {488, 595, 596, 537, 223, 212, 385, 211} \begin {gather*} \frac {a^{3/2} (b c-a d)^{5/2} \text {ArcTan}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^5}+\frac {x^3 \sqrt {c+d x^2} \left (48 a^2 d^2-104 a b c d+59 b^2 c^2\right )}{192 b^3}+\frac {x \sqrt {c+d x^2} \left (-64 a^3 d^3+144 a^2 b c d^2-88 a b^2 c^2 d+5 b^3 c^3\right )}{128 b^4 d}-\frac {\left (-128 a^4 d^4+320 a^3 b c d^3-240 a^2 b^2 c^2 d^2+40 a b^3 c^3 d+5 b^4 c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 b^5 d^{3/2}}+\frac {d x^5 \sqrt {c+d x^2} (11 b c-8 a d)}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((5*b^3*c^3 - 88*a*b^2*c^2*d + 144*a^2*b*c*d^2 - 64*a^3*d^3)*x*Sqrt[c + d*x^2])/(128*b^4*d) + ((59*b^2*c^2 - 1
04*a*b*c*d + 48*a^2*d^2)*x^3*Sqrt[c + d*x^2])/(192*b^3) + (d*(11*b*c - 8*a*d)*x^5*Sqrt[c + d*x^2])/(48*b^2) +
(d*x^5*(c + d*x^2)^(3/2))/(8*b) + (a^(3/2)*(b*c - a*d)^(5/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^
2])])/b^5 - ((5*b^4*c^4 + 40*a*b^3*c^3*d - 240*a^2*b^2*c^2*d^2 + 320*a^3*b*c*d^3 - 128*a^4*d^4)*ArcTanh[(Sqrt[
d]*x)/Sqrt[c + d*x^2]])/(128*b^5*d^(3/2))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 488

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*(e*x)^
(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*e*(m + n*(p + q) + 1))), x] + Dist[1/(b*(m + n*(p + q) + 1
)), Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*((c*b - a*d)*(m + 1) + c*b*n*(p + q)) + (d*(c*b - a*d
)*(m + 1) + d*n*(q - 1)*(b*c - a*d) + c*b*d*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && N
eQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 537

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 595

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[f*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*g*(m + n*(p + q + 1) + 1))), x] + Dis
t[1/(b*(m + n*(p + q + 1) + 1)), Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*((b*e - a*f)*(m + 1) + b
*e*n*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*n*q*(b*c - a*d) + b*e*d*n*(p + q + 1))*x^n, x], x], x] /; FreeQ
[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 596

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q +
1) + 1))), x] - Dist[g^n/(b*d*(m + n*(p + q + 1) + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*
f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*(f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x]
/; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 1]

Rubi steps

\begin {align*} \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx &=\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {\int \frac {x^4 \sqrt {c+d x^2} \left (c (8 b c-5 a d)+d (11 b c-8 a d) x^2\right )}{a+b x^2} \, dx}{8 b}\\ &=\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {\int \frac {x^4 \left (c \left (48 b^2 c^2-85 a b c d+40 a^2 d^2\right )+d \left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^2\right )}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{48 b^2}\\ &=\frac {\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt {c+d x^2}}{192 b^3}+\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}-\frac {\int \frac {x^2 \left (3 a c d \left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right )-3 d \left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x^2\right )}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{192 b^3 d}\\ &=\frac {\left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x \sqrt {c+d x^2}}{128 b^4 d}+\frac {\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt {c+d x^2}}{192 b^3}+\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {\int \frac {-3 a c d \left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right )-3 d \left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{384 b^4 d^2}\\ &=\frac {\left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x \sqrt {c+d x^2}}{128 b^4 d}+\frac {\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt {c+d x^2}}{192 b^3}+\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {\left (a^2 (b c-a d)^3\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{b^5}-\frac {\left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{128 b^5 d}\\ &=\frac {\left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x \sqrt {c+d x^2}}{128 b^4 d}+\frac {\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt {c+d x^2}}{192 b^3}+\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {\left (a^2 (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{b^5}-\frac {\left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{128 b^5 d}\\ &=\frac {\left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x \sqrt {c+d x^2}}{128 b^4 d}+\frac {\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt {c+d x^2}}{192 b^3}+\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {a^{3/2} (b c-a d)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^5}-\frac {\left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 b^5 d^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.64, size = 266, normalized size = 0.91 \begin {gather*} \frac {\frac {b x \sqrt {c+d x^2} \left (-192 a^3 d^3+48 a^2 b d^2 \left (9 c+2 d x^2\right )-8 a b^2 d \left (33 c^2+26 c d x^2+8 d^2 x^4\right )+b^3 \left (15 c^3+118 c^2 d x^2+136 c d^2 x^4+48 d^3 x^6\right )\right )}{d}-384 a^{3/2} (b c-a d)^{5/2} \tan ^{-1}\left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )+\frac {3 \left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{d^{3/2}}}{384 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*x*Sqrt[c + d*x^2]*(-192*a^3*d^3 + 48*a^2*b*d^2*(9*c + 2*d*x^2) - 8*a*b^2*d*(33*c^2 + 26*c*d*x^2 + 8*d^2*x^
4) + b^3*(15*c^3 + 118*c^2*d*x^2 + 136*c*d^2*x^4 + 48*d^3*x^6)))/d - 384*a^(3/2)*(b*c - a*d)^(5/2)*ArcTan[(a*S
qrt[d] + b*x*(Sqrt[d]*x - Sqrt[c + d*x^2]))/(Sqrt[a]*Sqrt[b*c - a*d])] + (3*(5*b^4*c^4 + 40*a*b^3*c^3*d - 240*
a^2*b^2*c^2*d^2 + 320*a^3*b*c*d^3 - 128*a^4*d^4)*Log[-(Sqrt[d]*x) + Sqrt[c + d*x^2]])/d^(3/2))/(384*b^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2251\) vs. \(2(257)=514\).
time = 0.14, size = 2252, normalized size = 7.74

method	result	size
risch	\(\text {Expression too large to display}\)	\(1560\)
default	\(\text {Expression too large to display}\)	\(2252\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(d*x^2+c)^(5/2)/(b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

1/b*(1/8*x*(d*x^2+c)^(7/2)/d-1/8*c/d*(1/6*x*(d*x^2+c)^(5/2)+5/6*c*(1/4*x*(d*x^2+c)^(3/2)+3/4*c*(1/2*x*(d*x^2+c
)^(1/2)+1/2*c/d^(1/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))))))-a/b^2*(1/6*x*(d*x^2+c)^(5/2)+5/6*c*(1/4*x*(d*x^2+c)^(3
/2)+3/4*c*(1/2*x*(d*x^2+c)^(1/2)+1/2*c/d^(1/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2)))))+1/2/b^2*a^2/(-a*b)^(1/2)*(1/5*
(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(5/2)+d*(-a*b)^(1/2)/b*(1/8*(2*
d*(x-1/b*(-a*b)^(1/2))+2*d*(-a*b)^(1/2)/b)/d*(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))
-(a*d-b*c)/b)^(3/2)+3/16*(-4*d*(a*d-b*c)/b+4*d^2*a/b)/d*(1/4*(2*d*(x-1/b*(-a*b)^(1/2))+2*d*(-a*b)^(1/2)/b)/d*(
d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/8*(-4*d*(a*d-b*c)/b+4*d^
2*a/b)/d^(3/2)*ln((d*(-a*b)^(1/2)/b+d*(x-1/b*(-a*b)^(1/2)))/d^(1/2)+(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)
/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))))-(a*d-b*c)/b*(1/3*(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x
-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+d*(-a*b)^(1/2)/b*(1/4*(2*d*(x-1/b*(-a*b)^(1/2))+2*d*(-a*b)^(1/2)/b)/d*(d
*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/8*(-4*d*(a*d-b*c)/b+4*d^2
*a/b)/d^(3/2)*ln((d*(-a*b)^(1/2)/b+d*(x-1/b*(-a*b)^(1/2)))/d^(1/2)+(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/
b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)))-(a*d-b*c)/b*((d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(
-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+d^(1/2)*(-a*b)^(1/2)/b*ln((d*(-a*b)^(1/2)/b+d*(x-1/b*(-a*b)^(1/2)))/d^(1/2)+(d
*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))+(a*d-b*c)/b/(-(a*d-b*c)/b)
^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*(d*(x-1/b*(-a*b)^(1/2
))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2))))))-1/2/b^2*a^2/(-a*b)^(
1/2)*(1/5*(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(5/2)-d*(-a*b)^(1/2)/
b*(1/8*(2*d*(x+1/b*(-a*b)^(1/2))-2*d*(-a*b)^(1/2)/b)/d*(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a
*b)^(1/2))-(a*d-b*c)/b)^(3/2)+3/16*(-4*d*(a*d-b*c)/b+4*d^2*a/b)/d*(1/4*(2*d*(x+1/b*(-a*b)^(1/2))-2*d*(-a*b)^(1
/2)/b)/d*(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/8*(-4*d*(a*d-b
*c)/b+4*d^2*a/b)/d^(3/2)*ln((-d*(-a*b)^(1/2)/b+d*(x+1/b*(-a*b)^(1/2)))/d^(1/2)+(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(
-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))))-(a*d-b*c)/b*(1/3*(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)
^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)-d*(-a*b)^(1/2)/b*(1/4*(2*d*(x+1/b*(-a*b)^(1/2))-2*d*(-a*b)^(1
/2)/b)/d*(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/8*(-4*d*(a*d-b
*c)/b+4*d^2*a/b)/d^(3/2)*ln((-d*(-a*b)^(1/2)/b+d*(x+1/b*(-a*b)^(1/2)))/d^(1/2)+(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(
-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)))-(a*d-b*c)/b*((d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2
)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-d^(1/2)*(-a*b)^(1/2)/b*ln((-d*(-a*b)^(1/2)/b+d*(x+1/b*(-a*b)^(1/2)
))/d^(1/2)+(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))+(a*d-b*c)/b/(
-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*(d*(x+1/
b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2))))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 12.58, size = 1443, normalized size = 4.96 \begin {gather*} \left [-\frac {3 \, {\left (5 \, b^{4} c^{4} + 40 \, a b^{3} c^{3} d - 240 \, a^{2} b^{2} c^{2} d^{2} + 320 \, a^{3} b c d^{3} - 128 \, a^{4} d^{4}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 192 \, {\left (a b^{2} c^{2} d^{2} - 2 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 2 \, {\left (48 \, b^{4} d^{4} x^{7} + 8 \, {\left (17 \, b^{4} c d^{3} - 8 \, a b^{3} d^{4}\right )} x^{5} + 2 \, {\left (59 \, b^{4} c^{2} d^{2} - 104 \, a b^{3} c d^{3} + 48 \, a^{2} b^{2} d^{4}\right )} x^{3} + 3 \, {\left (5 \, b^{4} c^{3} d - 88 \, a b^{3} c^{2} d^{2} + 144 \, a^{2} b^{2} c d^{3} - 64 \, a^{3} b d^{4}\right )} x\right )} \sqrt {d x^{2} + c}}{768 \, b^{5} d^{2}}, \frac {3 \, {\left (5 \, b^{4} c^{4} + 40 \, a b^{3} c^{3} d - 240 \, a^{2} b^{2} c^{2} d^{2} + 320 \, a^{3} b c d^{3} - 128 \, a^{4} d^{4}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + 96 \, {\left (a b^{2} c^{2} d^{2} - 2 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + {\left (48 \, b^{4} d^{4} x^{7} + 8 \, {\left (17 \, b^{4} c d^{3} - 8 \, a b^{3} d^{4}\right )} x^{5} + 2 \, {\left (59 \, b^{4} c^{2} d^{2} - 104 \, a b^{3} c d^{3} + 48 \, a^{2} b^{2} d^{4}\right )} x^{3} + 3 \, {\left (5 \, b^{4} c^{3} d - 88 \, a b^{3} c^{2} d^{2} + 144 \, a^{2} b^{2} c d^{3} - 64 \, a^{3} b d^{4}\right )} x\right )} \sqrt {d x^{2} + c}}{384 \, b^{5} d^{2}}, \frac {384 \, {\left (a b^{2} c^{2} d^{2} - 2 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 3 \, {\left (5 \, b^{4} c^{4} + 40 \, a b^{3} c^{3} d - 240 \, a^{2} b^{2} c^{2} d^{2} + 320 \, a^{3} b c d^{3} - 128 \, a^{4} d^{4}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (48 \, b^{4} d^{4} x^{7} + 8 \, {\left (17 \, b^{4} c d^{3} - 8 \, a b^{3} d^{4}\right )} x^{5} + 2 \, {\left (59 \, b^{4} c^{2} d^{2} - 104 \, a b^{3} c d^{3} + 48 \, a^{2} b^{2} d^{4}\right )} x^{3} + 3 \, {\left (5 \, b^{4} c^{3} d - 88 \, a b^{3} c^{2} d^{2} + 144 \, a^{2} b^{2} c d^{3} - 64 \, a^{3} b d^{4}\right )} x\right )} \sqrt {d x^{2} + c}}{768 \, b^{5} d^{2}}, \frac {192 \, {\left (a b^{2} c^{2} d^{2} - 2 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) + 3 \, {\left (5 \, b^{4} c^{4} + 40 \, a b^{3} c^{3} d - 240 \, a^{2} b^{2} c^{2} d^{2} + 320 \, a^{3} b c d^{3} - 128 \, a^{4} d^{4}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (48 \, b^{4} d^{4} x^{7} + 8 \, {\left (17 \, b^{4} c d^{3} - 8 \, a b^{3} d^{4}\right )} x^{5} + 2 \, {\left (59 \, b^{4} c^{2} d^{2} - 104 \, a b^{3} c d^{3} + 48 \, a^{2} b^{2} d^{4}\right )} x^{3} + 3 \, {\left (5 \, b^{4} c^{3} d - 88 \, a b^{3} c^{2} d^{2} + 144 \, a^{2} b^{2} c d^{3} - 64 \, a^{3} b d^{4}\right )} x\right )} \sqrt {d x^{2} + c}}{384 \, b^{5} d^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/768*(3*(5*b^4*c^4 + 40*a*b^3*c^3*d - 240*a^2*b^2*c^2*d^2 + 320*a^3*b*c*d^3 - 128*a^4*d^4)*sqrt(d)*log(-2*d
*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) - 192*(a*b^2*c^2*d^2 - 2*a^2*b*c*d^3 + a^3*d^4)*sqrt(-a*b*c + a^2*d)*l
og(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 + 4*((b*c - 2*a*d)*x^3 - a
*c*x)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 2*(48*b^4*d^4*x^7 + 8*(17*b^4*c*d^3
- 8*a*b^3*d^4)*x^5 + 2*(59*b^4*c^2*d^2 - 104*a*b^3*c*d^3 + 48*a^2*b^2*d^4)*x^3 + 3*(5*b^4*c^3*d - 88*a*b^3*c^
2*d^2 + 144*a^2*b^2*c*d^3 - 64*a^3*b*d^4)*x)*sqrt(d*x^2 + c))/(b^5*d^2), 1/384*(3*(5*b^4*c^4 + 40*a*b^3*c^3*d
- 240*a^2*b^2*c^2*d^2 + 320*a^3*b*c*d^3 - 128*a^4*d^4)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) + 96*(a*b^2
*c^2*d^2 - 2*a^2*b*c*d^3 + a^3*d^4)*sqrt(-a*b*c + a^2*d)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2
- 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 + 4*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c))/(b^2*x^4
+ 2*a*b*x^2 + a^2)) + (48*b^4*d^4*x^7 + 8*(17*b^4*c*d^3 - 8*a*b^3*d^4)*x^5 + 2*(59*b^4*c^2*d^2 - 104*a*b^3*c*
d^3 + 48*a^2*b^2*d^4)*x^3 + 3*(5*b^4*c^3*d - 88*a*b^3*c^2*d^2 + 144*a^2*b^2*c*d^3 - 64*a^3*b*d^4)*x)*sqrt(d*x^
2 + c))/(b^5*d^2), 1/768*(384*(a*b^2*c^2*d^2 - 2*a^2*b*c*d^3 + a^3*d^4)*sqrt(a*b*c - a^2*d)*arctan(1/2*sqrt(a*
b*c - a^2*d)*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)/((a*b*c*d - a^2*d^2)*x^3 + (a*b*c^2 - a^2*c*d)*x)) - 3*
(5*b^4*c^4 + 40*a*b^3*c^3*d - 240*a^2*b^2*c^2*d^2 + 320*a^3*b*c*d^3 - 128*a^4*d^4)*sqrt(d)*log(-2*d*x^2 - 2*sq
rt(d*x^2 + c)*sqrt(d)*x - c) + 2*(48*b^4*d^4*x^7 + 8*(17*b^4*c*d^3 - 8*a*b^3*d^4)*x^5 + 2*(59*b^4*c^2*d^2 - 10
4*a*b^3*c*d^3 + 48*a^2*b^2*d^4)*x^3 + 3*(5*b^4*c^3*d - 88*a*b^3*c^2*d^2 + 144*a^2*b^2*c*d^3 - 64*a^3*b*d^4)*x)
*sqrt(d*x^2 + c))/(b^5*d^2), 1/384*(192*(a*b^2*c^2*d^2 - 2*a^2*b*c*d^3 + a^3*d^4)*sqrt(a*b*c - a^2*d)*arctan(1
/2*sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)/((a*b*c*d - a^2*d^2)*x^3 + (a*b*c^2 - a^2*c*d
)*x)) + 3*(5*b^4*c^4 + 40*a*b^3*c^3*d - 240*a^2*b^2*c^2*d^2 + 320*a^3*b*c*d^3 - 128*a^4*d^4)*sqrt(-d)*arctan(s
qrt(-d)*x/sqrt(d*x^2 + c)) + (48*b^4*d^4*x^7 + 8*(17*b^4*c*d^3 - 8*a*b^3*d^4)*x^5 + 2*(59*b^4*c^2*d^2 - 104*a*
b^3*c*d^3 + 48*a^2*b^2*d^4)*x^3 + 3*(5*b^4*c^3*d - 88*a*b^3*c^2*d^2 + 144*a^2*b^2*c*d^3 - 64*a^3*b*d^4)*x)*sqr
t(d*x^2 + c))/(b^5*d^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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