3.5.75 \(\int \frac {(a+c x^2)^{5/2}}{(d+e x)^7} \, dx\)
Optimal. Leaf size=203 \[ -\frac {5 a^3 c^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{16 \left (a e^2+c d^2\right )^{7/2}}-\frac {5 a^2 c^2 \sqrt {a+c x^2} (a e-c d x)}{16 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac {5 a c \left (a+c x^2\right )^{3/2} (a e-c d x)}{24 (d+e x)^4 \left (a e^2+c d^2\right )^2}-\frac {\left (a+c x^2\right )^{5/2} (a e-c d x)}{6 (d+e x)^6 \left (a e^2+c d^2\right )} \]
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Rubi [A] time = 0.11, antiderivative size = 203, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used =
{721, 725, 206} \begin {gather*} -\frac {5 a^2 c^2 \sqrt {a+c x^2} (a e-c d x)}{16 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac {5 a^3 c^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{16 \left (a e^2+c d^2\right )^{7/2}}-\frac {5 a c \left (a+c x^2\right )^{3/2} (a e-c d x)}{24 (d+e x)^4 \left (a e^2+c d^2\right )^2}-\frac {\left (a+c x^2\right )^{5/2} (a e-c d x)}{6 (d+e x)^6 \left (a e^2+c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + c*x^2)^(5/2)/(d + e*x)^7,x]
[Out]
(-5*a^2*c^2*(a*e - c*d*x)*Sqrt[a + c*x^2])/(16*(c*d^2 + a*e^2)^3*(d + e*x)^2) - (5*a*c*(a*e - c*d*x)*(a + c*x^
2)^(3/2))/(24*(c*d^2 + a*e^2)^2*(d + e*x)^4) - ((a*e - c*d*x)*(a + c*x^2)^(5/2))/(6*(c*d^2 + a*e^2)*(d + e*x)^
6) - (5*a^3*c^3*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(16*(c*d^2 + a*e^2)^(7/2))
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 721
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(-2*a*e + (2*c*
d)*x)*(a + c*x^2)^p)/(2*(m + 1)*(c*d^2 + a*e^2)), x] - Dist[(4*a*c*p)/(2*(m + 1)*(c*d^2 + a*e^2)), Int[(d + e*
x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2,
0] && GtQ[p, 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]
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^7} \, dx &=-\frac {(a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}+\frac {(5 a c) \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{6 \left (c d^2+a e^2\right )}\\ &=-\frac {5 a c (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {(a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}+\frac {\left (5 a^2 c^2\right ) \int \frac {\sqrt {a+c x^2}}{(d+e x)^3} \, dx}{8 \left (c d^2+a e^2\right )^2}\\ &=-\frac {5 a^2 c^2 (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {5 a c (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {(a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}+\frac {\left (5 a^3 c^3\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{16 \left (c d^2+a e^2\right )^3}\\ &=-\frac {5 a^2 c^2 (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {5 a c (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {(a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {\left (5 a^3 c^3\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 )}{16 \left (c d^2+a e^2\right )^3}\\ &=-\frac {5 a^2 c^2 (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {5 a c (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {(a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {5 a^3 c^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{16 \left (c d^2+a e^2\right )^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 305, normalized size = 1.50 \begin {gather*} \frac {1}{48} \left (-\frac {15 a^3 c^3 \log \left (\sqrt {a+c x^2} \sqrt {a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{7/2}}+\frac {15 a^3 c^3 \log (d+e x)}{\left (a e^2+c d^2\right )^{7/2}}+\frac {\sqrt {a+c x^2} \left (-8 a^5 e^5-2 a^4 c e^3 \left (13 d^2+6 d e x+13 e^2 x^2\right )-a^3 c^2 e \left (33 d^4+54 d^3 e x+122 d^2 e^2 x^2+54 d e^3 x^3+33 e^4 x^4\right )+a^2 c^3 d x \left (33 d^4+54 d^3 e x+122 d^2 e^2 x^2+54 d e^3 x^3+33 e^4 x^4\right )+2 a c^4 d^3 x^3 \left (13 d^2+6 d e x+13 e^2 x^2\right )+8 c^5 d^5 x^5\right )}{(d+e x)^6 \left (a e^2+c d^2\right )^3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + c*x^2)^(5/2)/(d + e*x)^7,x]
[Out]
((Sqrt[a + c*x^2]*(-8*a^5*e^5 + 8*c^5*d^5*x^5 - 2*a^4*c*e^3*(13*d^2 + 6*d*e*x + 13*e^2*x^2) + 2*a*c^4*d^3*x^3*
(13*d^2 + 6*d*e*x + 13*e^2*x^2) - a^3*c^2*e*(33*d^4 + 54*d^3*e*x + 122*d^2*e^2*x^2 + 54*d*e^3*x^3 + 33*e^4*x^4
) + a^2*c^3*d*x*(33*d^4 + 54*d^3*e*x + 122*d^2*e^2*x^2 + 54*d*e^3*x^3 + 33*e^4*x^4)))/((c*d^2 + a*e^2)^3*(d +
e*x)^6) + (15*a^3*c^3*Log[d + e*x])/(c*d^2 + a*e^2)^(7/2) - (15*a^3*c^3*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*
Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(7/2))/48
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IntegrateAlgebraic [B] time = 85.10, size = 2613, normalized size = 12.87 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(a + c*x^2)^(5/2)/(d + e*x)^7,x]
[Out]
(8*a^9*e^11 - 48*a^8*Sqrt[c]*e^11*x*Sqrt[a + c*x^2] + c*(26*a^8*d^2*e^9 + 12*a^8*d*e^10*x + 178*a^8*e^11*x^2)
+ c^(3/2)*Sqrt[a + c*x^2]*(-156*a^7*d^2*e^9*x - 72*a^7*d*e^10*x^2 - 460*a^7*e^11*x^3) + c^2*(33*a^7*d^4*e^7 +
54*a^7*d^3*e^8*x + 616*a^7*d^2*e^9*x^2 + 282*a^7*d*e^10*x^3 + 1055*a^7*e^11*x^4) + c^(5/2)*Sqrt[a + c*x^2]*(-1
98*a^6*d^4*e^7*x - 324*a^6*d^3*e^8*x^2 - 1720*a^6*d^2*e^9*x^3 - 780*a^6*d*e^10*x^4 - 1698*a^6*e^11*x^5) + c^3*
(-33*a^6*d^5*e^6*x + 573*a^6*d^4*e^7*x^2 + 904*a^6*d^3*e^8*x^3 + 3980*a^6*d^2*e^9*x^4 + 1785*a^6*d*e^10*x^5 +
2983*a^6*e^11*x^6) + c^(7/2)*Sqrt[a + c*x^2]*(-33*a^5*d^7*e^4 - 198*a^5*d^6*e^5*x - 297*a^5*d^5*e^6*x^2 - 1590
*a^5*d^4*e^7*x^3 - 1815*a^5*d^3*e^8*x^4 - 6174*a^5*d^2*e^9*x^5 - 2655*a^5*d*e^10*x^6 - 3174*a^5*e^11*x^7) + c^
4*(198*a^5*d^7*e^4*x + 1188*a^5*d^6*e^5*x^2 + 2317*a^5*d^5*e^6*x^3 + 5100*a^5*d^4*e^7*x^4 + 4190*a^5*d^3*e^8*x
^5 + 10294*a^5*d^2*e^9*x^6 + 4095*a^5*d*e^10*x^7 + 4514*a^5*e^11*x^8) + c^(9/2)*Sqrt[a + c*x^2]*(-26*a^4*d^9*e
^2 - 156*a^4*d^8*e^3*x - 984*a^4*d^7*e^4*x^2 - 4084*a^4*d^6*e^5*x^3 - 7890*a^4*d^5*e^6*x^4 - 12024*a^4*d^4*e^7
*x^5 - 7600*a^4*d^3*e^8*x^6 - 10152*a^4*d^2*e^9*x^7 - 3180*a^4*d*e^10*x^8 - 2944*a^4*e^11*x^9) + c^5*(156*a^4*
d^9*e^2*x + 936*a^4*d^8*e^3*x^2 + 3594*a^4*d^7*e^4*x^3 + 10644*a^4*d^6*e^5*x^4 + 18470*a^4*d^5*e^6*x^5 + 24864
*a^4*d^4*e^7*x^6 + 14740*a^4*d^3*e^8*x^7 + 14552*a^4*d^2*e^9*x^8 + 3780*a^4*d*e^10*x^9 + 3472*a^4*e^11*x^10) +
c^(17/2)*Sqrt[a + c*x^2]*(-256*d^11*x^6 - 1536*d^10*e*x^7 - 3840*d^9*e^2*x^8 - 5120*d^8*e^3*x^9 - 3840*d^7*e^
4*x^10 - 1536*d^6*e^5*x^11) + c^(15/2)*Sqrt[a + c*x^2]*(-384*a*d^11*x^4 - 2304*a*d^10*e*x^5 - 6592*a*d^9*e^2*x
^6 - 12672*a*d^8*e^3*x^7 - 18240*a*d^7*e^4*x^8 - 18944*a*d^6*e^5*x^9 - 11520*a*d^5*e^6*x^10 - 4608*a*d^4*e^7*x
^11) + c^(13/2)*Sqrt[a + c*x^2]*(-144*a^2*d^11*x^2 - 864*a^2*d^10*e*x^3 - 3408*a^2*d^9*e^2*x^4 - 10368*a^2*d^8
*e^3*x^5 - 21936*a^2*d^7*e^4*x^6 - 32160*a^2*d^6*e^5*x^7 - 31680*a^2*d^5*e^6*x^8 - 26112*a^2*d^4*e^7*x^9 - 115
20*a^2*d^3*e^8*x^10 - 4608*a^2*d^2*e^9*x^11) + c^(11/2)*Sqrt[a + c*x^2]*(-8*a^3*d^11 - 48*a^3*d^10*e*x - 588*a
^3*d^9*e^2*x^2 - 2968*a^3*d^8*e^3*x^3 - 8724*a^3*d^7*e^4*x^4 - 18912*a^3*d^6*e^5*x^5 - 27640*a^3*d^5*e^6*x^6 -
31632*a^3*d^4*e^7*x^7 - 17160*a^3*d^3*e^8*x^8 - 9952*a^3*d^2*e^9*x^9 - 1200*a^3*d*e^10*x^10 - 1056*a^3*e^11*x
^11) + c^9*(256*d^11*x^7 + 1536*d^10*e*x^8 + 3840*d^9*e^2*x^9 + 5120*d^8*e^3*x^10 + 3840*d^7*e^4*x^11 + 1536*d
^6*e^5*x^12) + c^8*(512*a*d^11*x^5 + 3072*a*d^10*e*x^6 + 8512*a*d^9*e^2*x^7 + 15232*a*d^8*e^3*x^8 + 20160*a*d^
7*e^4*x^9 + 19712*a*d^6*e^5*x^10 + 11520*a*d^5*e^6*x^11 + 4608*a*d^4*e^7*x^12) + c^7*(304*a^2*d^11*x^3 + 1824*
a^2*d^10*e*x^4 + 6224*a^2*d^9*e^2*x^5 + 16064*a^2*d^8*e^3*x^6 + 30576*a^2*d^7*e^4*x^7 + 41440*a^2*d^6*e^5*x^8
+ 37440*a^2*d^5*e^6*x^9 + 28416*a^2*d^4*e^7*x^10 + 11520*a^2*d^3*e^8*x^11 + 4608*a^2*d^2*e^9*x^12) + c^6*(48*a
^3*d^11*x + 288*a^3*d^10*e*x^2 + 1708*a^3*d^9*e^2*x^3 + 6888*a^3*d^8*e^3*x^4 + 17652*a^3*d^7*e^4*x^5 + 32720*a
^3*d^6*e^5*x^6 + 42040*a^3*d^5*e^6*x^7 + 44112*a^3*d^4*e^7*x^8 + 22920*a^3*d^3*e^8*x^9 + 12256*a^3*d^2*e^9*x^1
0 + 1200*a^3*d*e^10*x^11 + 1056*a^3*e^11*x^12))/(288*a^6*Sqrt[c]*e^12*x*(d + e*x)^6 + 1536*c^(13/2)*d^6*e^6*x^
7*(d + e*x)^6 - 48*a^6*e^12*(d + e*x)^6*Sqrt[a + c*x^2] - 1536*c^6*d^6*e^6*x^6*(d + e*x)^6*Sqrt[a + c*x^2] + 4
8*c*e^6*(d + e*x)^6*Sqrt[a + c*x^2]*(-3*a^5*d^2*e^4 - 18*a^5*e^6*x^2) + 48*c^(3/2)*e^6*(d + e*x)^6*(18*a^5*d^2
*e^4*x + 38*a^5*e^6*x^3) + 48*c^2*e^6*(d + e*x)^6*Sqrt[a + c*x^2]*(-3*a^4*d^4*e^2 - 54*a^4*d^2*e^4*x^2 - 48*a^
4*e^6*x^4) + 48*c^(5/2)*e^6*(d + e*x)^6*(18*a^4*d^4*e^2*x + 114*a^4*d^2*e^4*x^3 + 64*a^4*e^6*x^5) + 48*c^5*e^6
*(d + e*x)^6*Sqrt[a + c*x^2]*(-48*a*d^6*x^4 - 96*a*d^4*e^2*x^6) + 48*c^4*e^6*(d + e*x)^6*Sqrt[a + c*x^2]*(-18*
a^2*d^6*x^2 - 144*a^2*d^4*e^2*x^4 - 96*a^2*d^2*e^4*x^6) + 48*c^3*e^6*(d + e*x)^6*Sqrt[a + c*x^2]*(-(a^3*d^6) -
54*a^3*d^4*e^2*x^2 - 144*a^3*d^2*e^4*x^4 - 32*a^3*e^6*x^6) + 48*c^(11/2)*e^6*(d + e*x)^6*(64*a*d^6*x^5 + 96*a
*d^4*e^2*x^7) + 48*c^(9/2)*e^6*(d + e*x)^6*(38*a^2*d^6*x^3 + 192*a^2*d^4*e^2*x^5 + 96*a^2*d^2*e^4*x^7) + 48*c^
(7/2)*e^6*(d + e*x)^6*(6*a^3*d^6*x + 114*a^3*d^4*e^2*x^3 + 192*a^3*d^2*e^4*x^5 + 32*a^3*e^6*x^7)) + (5*a^3*c^3
*ArcTan[(Sqrt[c]*d)/Sqrt[-(c*d^2) - a*e^2] + (Sqrt[c]*e*x)/Sqrt[-(c*d^2) - a*e^2] - (e*Sqrt[a + c*x^2])/Sqrt[-
(c*d^2) - a*e^2]])/(8*(-(c*d^2) - a*e^2)^(7/2))
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fricas [B] time = 13.69, size = 1929, normalized size = 9.50
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+a)^(5/2)/(e*x+d)^7,x, algorithm="fricas")
[Out]
[1/96*(15*(a^3*c^3*e^6*x^6 + 6*a^3*c^3*d*e^5*x^5 + 15*a^3*c^3*d^2*e^4*x^4 + 20*a^3*c^3*d^3*e^3*x^3 + 15*a^3*c^
3*d^4*e^2*x^2 + 6*a^3*c^3*d^5*e*x + a^3*c^3*d^6)*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*(33*a^3*c^3*d^6*e + 59*a^4*c^2*d^4*e^3 + 34*a^5*c*d^2*e^5 + 8*a^6*e^7 - (8*c^6*d^7 + 34*a*c^5*d^5*e^2 + 59*a
^2*c^4*d^3*e^4 + 33*a^3*c^3*d*e^6)*x^5 - 3*(4*a*c^5*d^6*e + 22*a^2*c^4*d^4*e^3 + 7*a^3*c^3*d^2*e^5 - 11*a^4*c^
2*e^7)*x^4 - 2*(13*a*c^5*d^7 + 74*a^2*c^4*d^5*e^2 + 34*a^3*c^3*d^3*e^4 - 27*a^4*c^2*d*e^6)*x^3 - 2*(27*a^2*c^4
*d^6*e - 34*a^3*c^3*d^4*e^3 - 74*a^4*c^2*d^2*e^5 - 13*a^5*c*e^7)*x^2 - 3*(11*a^2*c^4*d^7 - 7*a^3*c^3*d^5*e^2 -
22*a^4*c^2*d^3*e^4 - 4*a^5*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^4*d^14 + 4*a*c^3*d^12*e^2 + 6*a^2*c^2*d^10*e^4 + 4
*a^3*c*d^8*e^6 + a^4*d^6*e^8 + (c^4*d^8*e^6 + 4*a*c^3*d^6*e^8 + 6*a^2*c^2*d^4*e^10 + 4*a^3*c*d^2*e^12 + a^4*e^
14)*x^6 + 6*(c^4*d^9*e^5 + 4*a*c^3*d^7*e^7 + 6*a^2*c^2*d^5*e^9 + 4*a^3*c*d^3*e^11 + a^4*d*e^13)*x^5 + 15*(c^4*
d^10*e^4 + 4*a*c^3*d^8*e^6 + 6*a^2*c^2*d^6*e^8 + 4*a^3*c*d^4*e^10 + a^4*d^2*e^12)*x^4 + 20*(c^4*d^11*e^3 + 4*a
*c^3*d^9*e^5 + 6*a^2*c^2*d^7*e^7 + 4*a^3*c*d^5*e^9 + a^4*d^3*e^11)*x^3 + 15*(c^4*d^12*e^2 + 4*a*c^3*d^10*e^4 +
6*a^2*c^2*d^8*e^6 + 4*a^3*c*d^6*e^8 + a^4*d^4*e^10)*x^2 + 6*(c^4*d^13*e + 4*a*c^3*d^11*e^3 + 6*a^2*c^2*d^9*e^
5 + 4*a^3*c*d^7*e^7 + a^4*d^5*e^9)*x), -1/48*(15*(a^3*c^3*e^6*x^6 + 6*a^3*c^3*d*e^5*x^5 + 15*a^3*c^3*d^2*e^4*x
^4 + 20*a^3*c^3*d^3*e^3*x^3 + 15*a^3*c^3*d^4*e^2*x^2 + 6*a^3*c^3*d^5*e*x + a^3*c^3*d^6)*sqrt(-c*d^2 - a*e^2)*a
rctan(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)) + (33*
a^3*c^3*d^6*e + 59*a^4*c^2*d^4*e^3 + 34*a^5*c*d^2*e^5 + 8*a^6*e^7 - (8*c^6*d^7 + 34*a*c^5*d^5*e^2 + 59*a^2*c^4
*d^3*e^4 + 33*a^3*c^3*d*e^6)*x^5 - 3*(4*a*c^5*d^6*e + 22*a^2*c^4*d^4*e^3 + 7*a^3*c^3*d^2*e^5 - 11*a^4*c^2*e^7)
*x^4 - 2*(13*a*c^5*d^7 + 74*a^2*c^4*d^5*e^2 + 34*a^3*c^3*d^3*e^4 - 27*a^4*c^2*d*e^6)*x^3 - 2*(27*a^2*c^4*d^6*e
- 34*a^3*c^3*d^4*e^3 - 74*a^4*c^2*d^2*e^5 - 13*a^5*c*e^7)*x^2 - 3*(11*a^2*c^4*d^7 - 7*a^3*c^3*d^5*e^2 - 22*a^
4*c^2*d^3*e^4 - 4*a^5*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^4*d^14 + 4*a*c^3*d^12*e^2 + 6*a^2*c^2*d^10*e^4 + 4*a^3*c
*d^8*e^6 + a^4*d^6*e^8 + (c^4*d^8*e^6 + 4*a*c^3*d^6*e^8 + 6*a^2*c^2*d^4*e^10 + 4*a^3*c*d^2*e^12 + a^4*e^14)*x^
6 + 6*(c^4*d^9*e^5 + 4*a*c^3*d^7*e^7 + 6*a^2*c^2*d^5*e^9 + 4*a^3*c*d^3*e^11 + a^4*d*e^13)*x^5 + 15*(c^4*d^10*e
^4 + 4*a*c^3*d^8*e^6 + 6*a^2*c^2*d^6*e^8 + 4*a^3*c*d^4*e^10 + a^4*d^2*e^12)*x^4 + 20*(c^4*d^11*e^3 + 4*a*c^3*d
^9*e^5 + 6*a^2*c^2*d^7*e^7 + 4*a^3*c*d^5*e^9 + a^4*d^3*e^11)*x^3 + 15*(c^4*d^12*e^2 + 4*a*c^3*d^10*e^4 + 6*a^2
*c^2*d^8*e^6 + 4*a^3*c*d^6*e^8 + a^4*d^4*e^10)*x^2 + 6*(c^4*d^13*e + 4*a*c^3*d^11*e^3 + 6*a^2*c^2*d^9*e^5 + 4*
a^3*c*d^7*e^7 + a^4*d^5*e^9)*x)]
________________________________________________________________________________________
giac [B] time = 0.56, size = 1895, normalized size = 9.33
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+a)^(5/2)/(e*x+d)^7,x, algorithm="giac")
[Out]
5/8*a^3*c^3*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c^3*d^6 + 3*a*c^2*d^
4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*sqrt(-c*d^2 - a*e^2)) + 1/24*(768*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^8*d^10*
e + 256*(sqrt(c)*x - sqrt(c*x^2 + a))^6*c^(17/2)*d^11 + 960*(sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(15/2)*d^9*e^2 +
640*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^7*d^8*e^3 - 768*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^8*d^10*e + 240*(sqr
t(c)*x - sqrt(c*x^2 + a))^10*c^(13/2)*d^7*e^4 - 1088*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(15/2)*d^9*e^2 + 48*(
sqrt(c)*x - sqrt(c*x^2 + a))^11*c^6*d^6*e^5 + 576*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c^7*d^8*e^3 + 2160*(sqrt(c
)*x - sqrt(c*x^2 + a))^8*a*c^(13/2)*d^7*e^4 + 960*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(15/2)*d^9*e^2 + 1840*
(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^6*d^6*e^5 - 576*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c^7*d^8*e^3 + 720*(sqr
t(c)*x - sqrt(c*x^2 + a))^10*a*c^(11/2)*d^5*e^6 - 3744*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(13/2)*d^7*e^4 +
144*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a*c^5*d^4*e^7 - 2592*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^6*d^6*e^5 - 64
0*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^3*c^7*d^8*e^3 + 720*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(11/2)*d^5*e^6 +
2160*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^3*c^(13/2)*d^7*e^4 + 1680*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^5*d^4*
e^7 + 2592*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^6*d^6*e^5 + 720*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^2*c^(9/2)*
d^3*e^8 - 3320*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(11/2)*d^5*e^6 + 240*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^4*
c^(13/2)*d^7*e^4 + 144*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^2*c^4*d^2*e^9 - 5640*(sqrt(c)*x - sqrt(c*x^2 + a))^7
*a^3*c^5*d^4*e^7 - 1840*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^4*c^6*d^6*e^5 - 2910*(sqrt(c)*x - sqrt(c*x^2 + a))^8
*a^3*c^(9/2)*d^3*e^8 + 1080*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^4*c^(11/2)*d^5*e^6 - 340*(sqrt(c)*x - sqrt(c*x^2
+ a))^9*a^3*c^4*d^2*e^9 + 7080*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^5*d^4*e^7 - 48*(sqrt(c)*x - sqrt(c*x^2 +
a))*a^5*c^6*d^6*e^5 + 75*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^3*c^(7/2)*d*e^10 + 5680*(sqrt(c)*x - sqrt(c*x^2 +
a))^6*a^4*c^(9/2)*d^3*e^8 + 792*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^5*c^(11/2)*d^5*e^6 + 33*(sqrt(c)*x - sqrt(c
*x^2 + a))^11*a^3*c^3*e^11 + 1800*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^4*c^4*d^2*e^9 - 2040*(sqrt(c)*x - sqrt(c*x
^2 + a))^3*a^5*c^5*d^4*e^7 + 45*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^4*c^(7/2)*d*e^10 - 4620*(sqrt(c)*x - sqrt(c*
x^2 + a))^4*a^5*c^(9/2)*d^3*e^8 + 8*a^6*c^(11/2)*d^5*e^6 + 5*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^4*c^3*e^11 - 21
60*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^5*c^4*d^2*e^9 - 168*(sqrt(c)*x - sqrt(c*x^2 + a))*a^6*c^5*d^4*e^7 - 330*(
sqrt(c)*x - sqrt(c*x^2 + a))^6*a^5*c^(7/2)*d*e^10 + 1104*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^6*c^(9/2)*d^3*e^8 +
90*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^5*c^3*e^11 + 1640*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^6*c^4*d^2*e^9 + 450*
(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^6*c^(7/2)*d*e^10 + 26*a^7*c^(9/2)*d^3*e^8 + 90*(sqrt(c)*x - sqrt(c*x^2 + a))
^5*a^6*c^3*e^11 - 252*(sqrt(c)*x - sqrt(c*x^2 + a))*a^7*c^4*d^2*e^9 - 273*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^7*
c^(7/2)*d*e^10 + 5*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^7*c^3*e^11 + 33*a^8*c^(7/2)*d*e^10 + 33*(sqrt(c)*x - sqrt
(c*x^2 + a))*a^8*c^3*e^11)/((c^3*d^6*e^6 + 3*a*c^2*d^4*e^8 + 3*a^2*c*d^2*e^10 + a^3*e^12)*((sqrt(c)*x - sqrt(c
*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^6)
________________________________________________________________________________________
maple [B] time = 0.09, size = 7616, normalized size = 37.52 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c*x^2+a)^(5/2)/(e*x+d)^7,x)
[Out]
result too large to display
________________________________________________________________________________________
maxima [B] time = 5.61, size = 4635, normalized size = 22.83
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+a)^(5/2)/(e*x+d)^7,x, algorithm="maxima")
[Out]
-5/32*c^8*d^9*arcsinh(c*x/sqrt(a*c))/(c^(11/2)*d^10*e^6 + 5*a*c^(9/2)*d^8*e^8 + 10*a^2*c^(7/2)*d^6*e^10 + 10*a
^3*c^(5/2)*d^4*e^12 + 5*a^4*c^(3/2)*d^2*e^14 + a^5*sqrt(c)*e^16) - 5/32*a*c^7*d^7*arcsinh(c*x/sqrt(a*c))/(c^(1
1/2)*d^10*e^4 + 5*a*c^(9/2)*d^8*e^6 + 10*a^2*c^(7/2)*d^6*e^8 + 10*a^3*c^(5/2)*d^4*e^10 + 5*a^4*c^(3/2)*d^2*e^1
2 + a^5*sqrt(c)*e^14) + 5/32*sqrt(c*x^2 + a)*c^7*d^7*x/(c^5*d^10*e^4 + 5*a*c^4*d^8*e^6 + 10*a^2*c^3*d^6*e^8 +
10*a^3*c^2*d^4*e^10 + 5*a^4*c*d^2*e^12 + a^5*e^14) + 15/16*c^7*d^7*arcsinh(c*x/sqrt(a*c))/(c^(9/2)*d^8*e^6 + 4
*a*c^(7/2)*d^6*e^8 + 6*a^2*c^(5/2)*d^4*e^10 + 4*a^3*c^(3/2)*d^2*e^12 + a^4*sqrt(c)*e^14) - 5/48*(c*x^2 + a)^(3
/2)*c^6*d^6/(c^5*d^10*e^3 + 5*a*c^4*d^8*e^5 + 10*a^2*c^3*d^6*e^7 + 10*a^3*c^2*d^4*e^9 + 5*a^4*c*d^2*e^11 + a^5
*e^13) + 5/48*(c*x^2 + a)^(3/2)*c^6*d^5*x/(c^5*d^10*e^2 + 5*a*c^4*d^8*e^4 + 10*a^2*c^3*d^6*e^6 + 10*a^3*c^2*d^
4*e^8 + 5*a^4*c*d^2*e^10 + a^5*e^12) + 5/32*sqrt(c*x^2 + a)*a*c^6*d^5*x/(c^5*d^10*e^2 + 5*a*c^4*d^8*e^4 + 10*a
^2*c^3*d^6*e^6 + 10*a^3*c^2*d^4*e^8 + 5*a^4*c*d^2*e^10 + a^5*e^12) + 25/32*a*c^6*d^5*arcsinh(c*x/sqrt(a*c))/(c
^(9/2)*d^8*e^4 + 4*a*c^(7/2)*d^6*e^6 + 6*a^2*c^(5/2)*d^4*e^8 + 4*a^3*c^(3/2)*d^2*e^10 + a^4*sqrt(c)*e^12) - 1/
16*(c*x^2 + a)^(5/2)*c^5*d^5/(c^5*d^10*e^2*x + 5*a*c^4*d^8*e^4*x + 10*a^2*c^3*d^6*e^6*x + 10*a^3*c^2*d^4*e^8*x
+ 5*a^4*c*d^2*e^10*x + a^5*e^12*x + c^5*d^11*e + 5*a*c^4*d^9*e^3 + 10*a^2*c^3*d^7*e^5 + 10*a^3*c^2*d^5*e^7 +
5*a^4*c*d^3*e^9 + a^5*d*e^11) - 5/16*sqrt(c*x^2 + a)*c^6*d^6/(c^4*d^8*e^5 + 4*a*c^3*d^6*e^7 + 6*a^2*c^2*d^4*e^
9 + 4*a^3*c*d^2*e^11 + a^4*e^13) - 15/32*sqrt(c*x^2 + a)*c^6*d^5*x/(c^4*d^8*e^4 + 4*a*c^3*d^6*e^6 + 6*a^2*c^2*
d^4*e^8 + 4*a^3*c*d^2*e^10 + a^4*e^12) - 15/8*c^6*d^5*arcsinh(c*x/sqrt(a*c))/(c^(7/2)*d^6*e^6 + 3*a*c^(5/2)*d^
4*e^8 + 3*a^2*c^(3/2)*d^2*e^10 + a^3*sqrt(c)*e^12) + 1/48*(c*x^2 + a)^(7/2)*c^4*d^4/(c^5*d^10*e*x^2 + 5*a*c^4*
d^8*e^3*x^2 + 10*a^2*c^3*d^6*e^5*x^2 + 10*a^3*c^2*d^4*e^7*x^2 + 5*a^4*c*d^2*e^9*x^2 + a^5*e^11*x^2 + 2*c^5*d^1
1*x + 10*a*c^4*d^9*e^2*x + 20*a^2*c^3*d^7*e^4*x + 20*a^3*c^2*d^5*e^6*x + 10*a^4*c*d^3*e^8*x + 2*a^5*d*e^10*x +
c^5*d^12/e + 5*a*c^4*d^10*e + 10*a^2*c^3*d^8*e^3 + 10*a^3*c^2*d^6*e^5 + 5*a^4*c*d^4*e^7 + a^5*d^2*e^9) - 1/48
*(c*x^2 + a)^(5/2)*c^5*d^4/(c^5*d^10*e + 5*a*c^4*d^8*e^3 + 10*a^2*c^3*d^6*e^5 + 10*a^3*c^2*d^4*e^7 + 5*a^4*c*d
^2*e^9 + a^5*e^11) + 5/16*(c*x^2 + a)^(3/2)*c^5*d^4/(c^4*d^8*e^3 + 4*a*c^3*d^6*e^5 + 6*a^2*c^2*d^4*e^7 + 4*a^3
*c*d^2*e^9 + a^4*e^11) - 5/12*(c*x^2 + a)^(3/2)*c^5*d^3*x/(c^4*d^8*e^2 + 4*a*c^3*d^6*e^4 + 6*a^2*c^2*d^4*e^6 +
4*a^3*c*d^2*e^8 + a^4*e^10) - 5/8*sqrt(c*x^2 + a)*a*c^5*d^3*x/(c^4*d^8*e^2 + 4*a*c^3*d^6*e^4 + 6*a^2*c^2*d^4*
e^6 + 4*a^3*c*d^2*e^8 + a^4*e^10) - 35/32*a*c^5*d^3*arcsinh(c*x/sqrt(a*c))/(c^(7/2)*d^6*e^4 + 3*a*c^(5/2)*d^4*
e^6 + 3*a^2*c^(3/2)*d^2*e^8 + a^3*sqrt(c)*e^10) - 1/24*(c*x^2 + a)^(7/2)*c^3*d^3/(c^4*d^8*e^2*x^3 + 4*a*c^3*d^
6*e^4*x^3 + 6*a^2*c^2*d^4*e^6*x^3 + 4*a^3*c*d^2*e^8*x^3 + a^4*e^10*x^3 + 3*c^4*d^9*e*x^2 + 12*a*c^3*d^7*e^3*x^
2 + 18*a^2*c^2*d^5*e^5*x^2 + 12*a^3*c*d^3*e^7*x^2 + 3*a^4*d*e^9*x^2 + 3*c^4*d^10*x + 12*a*c^3*d^8*e^2*x + 18*a
^2*c^2*d^6*e^4*x + 12*a^3*c*d^4*e^6*x + 3*a^4*d^2*e^8*x + c^4*d^11/e + 4*a*c^3*d^9*e + 6*a^2*c^2*d^7*e^3 + 4*a
^3*c*d^5*e^5 + a^4*d^3*e^7) + 5/24*(c*x^2 + a)^(5/2)*c^4*d^3/(c^4*d^8*e^2*x + 4*a*c^3*d^6*e^4*x + 6*a^2*c^2*d^
4*e^6*x + 4*a^3*c*d^2*e^8*x + a^4*e^10*x + c^4*d^9*e + 4*a*c^3*d^7*e^3 + 6*a^2*c^2*d^5*e^5 + 4*a^3*c*d^3*e^7 +
a^4*d*e^9) + 15/16*sqrt(c*x^2 + a)*c^5*d^4/(c^3*d^6*e^5 + 3*a*c^2*d^4*e^7 + 3*a^2*c*d^2*e^9 + a^3*e^11) + 15/
32*sqrt(c*x^2 + a)*c^5*d^3*x/(c^3*d^6*e^4 + 3*a*c^2*d^4*e^6 + 3*a^2*c*d^2*e^8 + a^3*e^10) + 25/16*c^5*d^3*arcs
inh(c*x/sqrt(a*c))/(c^(5/2)*d^4*e^6 + 2*a*c^(3/2)*d^2*e^8 + a^2*sqrt(c)*e^10) - 1/8*(c*x^2 + a)^(7/2)*c^3*d^2/
(c^4*d^8*e*x^2 + 4*a*c^3*d^6*e^3*x^2 + 6*a^2*c^2*d^4*e^5*x^2 + 4*a^3*c*d^2*e^7*x^2 + a^4*e^9*x^2 + 2*c^4*d^9*x
+ 8*a*c^3*d^7*e^2*x + 12*a^2*c^2*d^5*e^4*x + 8*a^3*c*d^3*e^6*x + 2*a^4*d*e^8*x + c^4*d^10/e + 4*a*c^3*d^8*e +
6*a^2*c^2*d^6*e^3 + 4*a^3*c*d^4*e^5 + a^4*d^2*e^7) + 1/8*(c*x^2 + a)^(5/2)*c^4*d^2/(c^4*d^8*e + 4*a*c^3*d^6*e
^3 + 6*a^2*c^2*d^4*e^5 + 4*a^3*c*d^2*e^7 + a^4*e^9) - 1/8*(c*x^2 + a)^(7/2)*c^2*d^2/(c^3*d^6*e^3*x^4 + 3*a*c^2
*d^4*e^5*x^4 + 3*a^2*c*d^2*e^7*x^4 + a^3*e^9*x^4 + 4*c^3*d^7*e^2*x^3 + 12*a*c^2*d^5*e^4*x^3 + 12*a^2*c*d^3*e^6
*x^3 + 4*a^3*d*e^8*x^3 + 6*c^3*d^8*e*x^2 + 18*a*c^2*d^6*e^3*x^2 + 18*a^2*c*d^4*e^5*x^2 + 6*a^3*d^2*e^7*x^2 + 4
*c^3*d^9*x + 12*a*c^2*d^7*e^2*x + 12*a^2*c*d^5*e^4*x + 4*a^3*d^3*e^6*x + c^3*d^10/e + 3*a*c^2*d^8*e + 3*a^2*c*
d^6*e^3 + a^3*d^4*e^5) - 5/16*(c*x^2 + a)^(3/2)*c^4*d^2/(c^3*d^6*e^3 + 3*a*c^2*d^4*e^5 + 3*a^2*c*d^2*e^7 + a^3
*e^9) + 5/16*(c*x^2 + a)^(3/2)*c^4*d*x/(c^3*d^6*e^2 + 3*a*c^2*d^4*e^4 + 3*a^2*c*d^2*e^6 + a^3*e^8) + 15/32*sqr
t(c*x^2 + a)*a*c^4*d*x/(c^3*d^6*e^2 + 3*a*c^2*d^4*e^4 + 3*a^2*c*d^2*e^6 + a^3*e^8) + 15/32*a*c^4*d*arcsinh(c*x
/sqrt(a*c))/(c^(5/2)*d^4*e^4 + 2*a*c^(3/2)*d^2*e^6 + a^2*sqrt(c)*e^8) - 1/8*(c*x^2 + a)^(7/2)*c^2*d/(c^3*d^6*e
^2*x^3 + 3*a*c^2*d^4*e^4*x^3 + 3*a^2*c*d^2*e^6*x^3 + a^3*e^8*x^3 + 3*c^3*d^7*e*x^2 + 9*a*c^2*d^5*e^3*x^2 + 9*a
^2*c*d^3*e^5*x^2 + 3*a^3*d*e^7*x^2 + 3*c^3*d^8*x + 9*a*c^2*d^6*e^2*x + 9*a^2*c*d^4*e^4*x + 3*a^3*d^2*e^6*x + c
^3*d^9/e + 3*a*c^2*d^7*e + 3*a^2*c*d^5*e^3 + a^3*d^3*e^5) - 5/16*(c*x^2 + a)^(5/2)*c^3*d/(c^3*d^6*e^2*x + 3*a*
c^2*d^4*e^4*x + 3*a^2*c*d^2*e^6*x + a^3*e^8*x + c^3*d^7*e + 3*a*c^2*d^5*e^3 + 3*a^2*c*d^3*e^5 + a^3*d*e^7) - 1
5/16*sqrt(c*x^2 + a)*c^4*d^2/(c^2*d^4*e^5 + 2*a*c*d^2*e^7 + a^2*e^9) - 5/32*sqrt(c*x^2 + a)*c^4*d*x/(c^2*d^4*e
^4 + 2*a*c*d^2*e^6 + a^2*e^8) - 15/32*c^4*d*arcsinh(c*x/sqrt(a*c))/(c^(3/2)*d^2*e^6 + a*sqrt(c)*e^8) - 1/16*(c
*x^2 + a)^(7/2)*c^2/(c^3*d^6*e*x^2 + 3*a*c^2*d^4*e^3*x^2 + 3*a^2*c*d^2*e^5*x^2 + a^3*e^7*x^2 + 2*c^3*d^7*x + 6
*a*c^2*d^5*e^2*x + 6*a^2*c*d^3*e^4*x + 2*a^3*d*e^6*x + c^3*d^8/e + 3*a*c^2*d^6*e + 3*a^2*c*d^4*e^3 + a^3*d^2*e
^5) + 1/16*(c*x^2 + a)^(5/2)*c^3/(c^3*d^6*e + 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 + a^3*e^7) - 1/6*(c*x^2 + a)^(
7/2)*c*d/(c^2*d^4*e^4*x^5 + 2*a*c*d^2*e^6*x^5 + a^2*e^8*x^5 + 5*c^2*d^5*e^3*x^4 + 10*a*c*d^3*e^5*x^4 + 5*a^2*d
*e^7*x^4 + 10*c^2*d^6*e^2*x^3 + 20*a*c*d^4*e^4*x^3 + 10*a^2*d^2*e^6*x^3 + 10*c^2*d^7*e*x^2 + 20*a*c*d^5*e^3*x^
2 + 10*a^2*d^3*e^5*x^2 + 5*c^2*d^8*x + 10*a*c*d^6*e^2*x + 5*a^2*d^4*e^4*x + c^2*d^9/e + 2*a*c*d^7*e + a^2*d^5*
e^3) - 1/24*(c*x^2 + a)^(7/2)*c/(c^2*d^4*e^3*x^4 + 2*a*c*d^2*e^5*x^4 + a^2*e^7*x^4 + 4*c^2*d^5*e^2*x^3 + 8*a*c
*d^3*e^4*x^3 + 4*a^2*d*e^6*x^3 + 6*c^2*d^6*e*x^2 + 12*a*c*d^4*e^3*x^2 + 6*a^2*d^2*e^5*x^2 + 4*c^2*d^7*x + 8*a*
c*d^5*e^2*x + 4*a^2*d^3*e^4*x + c^2*d^8/e + 2*a*c*d^6*e + a^2*d^4*e^3) + 5/48*(c*x^2 + a)^(3/2)*c^3/(c^2*d^4*e
^3 + 2*a*c*d^2*e^5 + a^2*e^7) - 1/6*(c*x^2 + a)^(7/2)/(c*d^2*e^5*x^6 + a*e^7*x^6 + 6*c*d^3*e^4*x^5 + 6*a*d*e^6
*x^5 + 15*c*d^4*e^3*x^4 + 15*a*d^2*e^5*x^4 + 20*c*d^5*e^2*x^3 + 20*a*d^3*e^4*x^3 + 15*c*d^6*e*x^2 + 15*a*d^4*e
^3*x^2 + 6*c*d^7*x + 6*a*d^5*e^2*x + c*d^8/e + a*d^6*e) + 5/16*sqrt(c*x^2 + a)*c^3/(c*d^2*e^5 + a*e^7) - 5/16*
c^6*d^6*arcsinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(e*x + d)))/((a + c*d^2/e^2)^(7/2)*e^13) +
15/16*c^5*d^4*arcsinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(e*x + d)))/((a + c*d^2/e^2)^(5/2)*e^
11) - 15/16*c^4*d^2*arcsinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(e*x + d)))/((a + c*d^2/e^2)^(3
/2)*e^9) + 5/16*c^3*arcsinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(e*x + d)))/(sqrt(a + c*d^2/e^2
)*e^7)
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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 )}^7} \,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)^7,x)
[Out]
int((a + c*x^2)^(5/2)/(d + e*x)^7, 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 {5}{2}}}{\left (d + e x\right )^{7}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x**2+a)**(5/2)/(e*x+d)**7,x)
[Out]
Integral((a + c*x**2)**(5/2)/(d + e*x)**7, x)
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