3.5.62 \(\int \frac {(a+c x^2)^{3/2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=153 \[ -\frac {3 a^2 c^2 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{8 \left (a e^2+c d^2\right )^{5/2}}-\frac {3 a c \sqrt {a+c x^2} (a e-c d x)}{8 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac {\left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )} \]

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Rubi [A]  time = 0.06, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {3 a^2 c^2 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{8 \left (a e^2+c d^2\right )^{5/2}}-\frac {3 a c \sqrt {a+c x^2} (a e-c d x)}{8 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac {\left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^(3/2)/(d + e*x)^5,x]

[Out]

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

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

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Mathematica [A]  time = 0.28, size = 198, normalized size = 1.29 \begin {gather*} \frac {1}{8} \left (-\frac {3 a^2 c^2 \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 )^{5/2}}+\frac {3 a^2 c^2 \log (d+e x)}{\left (a e^2+c d^2\right )^{5/2}}+\frac {\sqrt {a+c x^2} \left (-2 a^3 e^3-a^2 c e \left (5 d^2+4 d e x+5 e^2 x^2\right )+a c^2 d x \left (5 d^2+4 d e x+5 e^2 x^2\right )+2 c^3 d^3 x^3\right )}{(d+e x)^4 \left (a e^2+c d^2\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^(3/2)/(d + e*x)^5,x]

[Out]

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

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IntegrateAlgebraic [B]  time = 6.18, size = 1331, normalized size = 8.70 \begin {gather*} \frac {2 a^6 e^7-8 a^5 \sqrt {c} x \sqrt {c x^2+a} e^7+c \left (23 a^5 x^2 e^7+4 a^5 d x e^6+5 a^5 d^2 e^5\right )+c^{3/2} \sqrt {c x^2+a} \left (-44 a^4 x^3 e^7-16 a^4 d x^2 e^6-20 a^4 d^2 x e^5\right )+c^2 \left (77 a^4 x^4 e^7+31 a^4 d x^3 e^6+41 a^4 d^2 x^2 e^5-5 a^4 d^3 x e^4\right )+c^{5/2} \sqrt {c x^2+a} \left (-76 a^3 x^5 e^7-33 a^3 d x^4 e^6-64 a^3 d^2 x^3 e^5-10 a^3 d^3 x^2 e^4-20 a^3 d^4 x e^3-5 a^3 d^5 e^2\right )+c^3 \left (96 a^3 x^6 e^7+39 a^3 d x^5 e^6+124 a^3 d^2 x^4 e^5+73 a^3 d^3 x^3 e^4+80 a^3 d^4 x^2 e^3+20 a^3 d^5 x e^2\right )+c^{11/2} \sqrt {c x^2+a} \left (-16 x^4 d^7-64 e x^5 d^6-96 e^2 x^6 d^5-64 e^3 x^7 d^4\right )+c^{9/2} \sqrt {c x^2+a} \left (-16 a x^2 d^7-64 a e x^3 d^6-136 a e^2 x^4 d^5-224 a e^3 x^5 d^4-192 a e^4 x^6 d^3-128 a e^5 x^7 d^2\right )+c^{7/2} \sqrt {c x^2+a} \left (-2 a^2 d^7-8 a^2 e x d^6-52 a^2 e^2 x^2 d^5-168 a^2 e^3 x^3 d^4-174 a^2 e^4 x^4 d^3-152 a^2 e^5 x^5 d^2-12 a^2 e^6 x^6 d-40 a^2 e^7 x^7\right )+c^6 \left (64 d^4 e^3 x^8+96 d^5 e^2 x^7+64 d^6 e x^6+16 d^7 x^5\right )+c^5 \left (128 a d^2 e^5 x^8+192 a d^3 e^4 x^7+256 a d^4 e^3 x^6+184 a d^5 e^2 x^5+96 a d^6 e x^4+24 a d^7 x^3\right )+c^4 \left (40 a^2 e^7 x^8+12 a^2 d e^6 x^7+216 a^2 d^2 e^5 x^6+270 a^2 d^3 e^4 x^5+272 a^2 d^4 e^3 x^4+108 a^2 d^5 e^2 x^3+32 a^2 d^6 e x^2+8 a^2 d^7 x\right )}{32 a^4 \sqrt {c} x (d+e x)^4 e^8-8 a^4 (d+e x)^4 \sqrt {c x^2+a} e^8+64 c^{9/2} d^4 x^5 (d+e x)^4 e^4+8 c (d+e x)^4 \sqrt {c x^2+a} \left (-8 a^3 x^2 e^4-2 a^3 d^2 e^2\right ) e^4+8 c^{3/2} (d+e x)^4 \left (12 a^3 x^3 e^4+8 a^3 d^2 x e^2\right ) e^4+8 c^3 (d+e x)^4 \sqrt {c x^2+a} \left (-8 a x^2 d^4-16 a e^2 x^4 d^2\right ) e^4+8 c^2 (d+e x)^4 \sqrt {c x^2+a} \left (-a^2 d^4-16 a^2 e^2 x^2 d^2-8 a^2 e^4 x^4\right ) e^4+8 c^{7/2} (d+e x)^4 \left (16 a d^2 e^2 x^5+12 a d^4 x^3\right ) e^4+8 c^{5/2} (d+e x)^4 \left (8 a^2 e^4 x^5+24 a^2 d^2 e^2 x^3+4 a^2 d^4 x\right ) e^4-64 c^4 d^4 x^4 (d+e x)^4 \sqrt {c x^2+a} e^4}-\frac {3 a^2 c^2 \tan ^{-1}\left (\frac {\sqrt {c} d}{\sqrt {-c d^2-a e^2}}+\frac {\sqrt {c} e x}{\sqrt {-c d^2-a e^2}}-\frac {e \sqrt {c x^2+a}}{\sqrt {-c d^2-a e^2}}\right )}{4 \left (-c d^2-a e^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + c*x^2)^(3/2)/(d + e*x)^5,x]

[Out]

(2*a^6*e^7 - 8*a^5*Sqrt[c]*e^7*x*Sqrt[a + c*x^2] + c*(5*a^5*d^2*e^5 + 4*a^5*d*e^6*x + 23*a^5*e^7*x^2) + c^(3/2
)*Sqrt[a + c*x^2]*(-20*a^4*d^2*e^5*x - 16*a^4*d*e^6*x^2 - 44*a^4*e^7*x^3) + c^2*(-5*a^4*d^3*e^4*x + 41*a^4*d^2
*e^5*x^2 + 31*a^4*d*e^6*x^3 + 77*a^4*e^7*x^4) + c^(5/2)*Sqrt[a + c*x^2]*(-5*a^3*d^5*e^2 - 20*a^3*d^4*e^3*x - 1
0*a^3*d^3*e^4*x^2 - 64*a^3*d^2*e^5*x^3 - 33*a^3*d*e^6*x^4 - 76*a^3*e^7*x^5) + c^3*(20*a^3*d^5*e^2*x + 80*a^3*d
^4*e^3*x^2 + 73*a^3*d^3*e^4*x^3 + 124*a^3*d^2*e^5*x^4 + 39*a^3*d*e^6*x^5 + 96*a^3*e^7*x^6) + c^(11/2)*Sqrt[a +
c*x^2]*(-16*d^7*x^4 - 64*d^6*e*x^5 - 96*d^5*e^2*x^6 - 64*d^4*e^3*x^7) + c^(9/2)*Sqrt[a + c*x^2]*(-16*a*d^7*x^
2 - 64*a*d^6*e*x^3 - 136*a*d^5*e^2*x^4 - 224*a*d^4*e^3*x^5 - 192*a*d^3*e^4*x^6 - 128*a*d^2*e^5*x^7) + c^(7/2)*
Sqrt[a + c*x^2]*(-2*a^2*d^7 - 8*a^2*d^6*e*x - 52*a^2*d^5*e^2*x^2 - 168*a^2*d^4*e^3*x^3 - 174*a^2*d^3*e^4*x^4 -
152*a^2*d^2*e^5*x^5 - 12*a^2*d*e^6*x^6 - 40*a^2*e^7*x^7) + c^6*(16*d^7*x^5 + 64*d^6*e*x^6 + 96*d^5*e^2*x^7 +
64*d^4*e^3*x^8) + c^5*(24*a*d^7*x^3 + 96*a*d^6*e*x^4 + 184*a*d^5*e^2*x^5 + 256*a*d^4*e^3*x^6 + 192*a*d^3*e^4*x
^7 + 128*a*d^2*e^5*x^8) + c^4*(8*a^2*d^7*x + 32*a^2*d^6*e*x^2 + 108*a^2*d^5*e^2*x^3 + 272*a^2*d^4*e^3*x^4 + 27
0*a^2*d^3*e^4*x^5 + 216*a^2*d^2*e^5*x^6 + 12*a^2*d*e^6*x^7 + 40*a^2*e^7*x^8))/(32*a^4*Sqrt[c]*e^8*x*(d + e*x)^
4 + 64*c^(9/2)*d^4*e^4*x^5*(d + e*x)^4 - 8*a^4*e^8*(d + e*x)^4*Sqrt[a + c*x^2] - 64*c^4*d^4*e^4*x^4*(d + e*x)^
4*Sqrt[a + c*x^2] + 8*c*e^4*(d + e*x)^4*Sqrt[a + c*x^2]*(-2*a^3*d^2*e^2 - 8*a^3*e^4*x^2) + 8*c^(3/2)*e^4*(d +
e*x)^4*(8*a^3*d^2*e^2*x + 12*a^3*e^4*x^3) + 8*c^3*e^4*(d + e*x)^4*Sqrt[a + c*x^2]*(-8*a*d^4*x^2 - 16*a*d^2*e^2
*x^4) + 8*c^2*e^4*(d + e*x)^4*Sqrt[a + c*x^2]*(-(a^2*d^4) - 16*a^2*d^2*e^2*x^2 - 8*a^2*e^4*x^4) + 8*c^(7/2)*e^
4*(d + e*x)^4*(12*a*d^4*x^3 + 16*a*d^2*e^2*x^5) + 8*c^(5/2)*e^4*(d + e*x)^4*(4*a^2*d^4*x + 24*a^2*d^2*e^2*x^3
+ 8*a^2*e^4*x^5)) - (3*a^2*c^2*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]])/(4*(-(c*d^2) - a*e^2)^(5/2))

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fricas [B]  time = 2.23, size = 1123, normalized size = 7.34 \begin {gather*} \left [\frac {3 \, {\left (a^{2} c^{2} e^{4} x^{4} + 4 \, a^{2} c^{2} d e^{3} x^{3} + 6 \, a^{2} c^{2} d^{2} e^{2} x^{2} + 4 \, a^{2} c^{2} d^{3} e x + a^{2} c^{2} d^{4}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (5 \, a^{2} c^{2} d^{4} e + 7 \, a^{3} c d^{2} e^{3} + 2 \, a^{4} e^{5} - {\left (2 \, c^{4} d^{5} + 7 \, a c^{3} d^{3} e^{2} + 5 \, a^{2} c^{2} d e^{4}\right )} x^{3} - {\left (4 \, a c^{3} d^{4} e - a^{2} c^{2} d^{2} e^{3} - 5 \, a^{3} c e^{5}\right )} x^{2} - {\left (5 \, a c^{3} d^{5} + a^{2} c^{2} d^{3} e^{2} - 4 \, a^{3} c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{16 \, {\left (c^{3} d^{10} + 3 \, a c^{2} d^{8} e^{2} + 3 \, a^{2} c d^{6} e^{4} + a^{3} d^{4} e^{6} + {\left (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}\right )} x^{4} + 4 \, {\left (c^{3} d^{7} e^{3} + 3 \, a c^{2} d^{5} e^{5} + 3 \, a^{2} c d^{3} e^{7} + a^{3} d e^{9}\right )} x^{3} + 6 \, {\left (c^{3} d^{8} e^{2} + 3 \, a c^{2} d^{6} e^{4} + 3 \, a^{2} c d^{4} e^{6} + a^{3} d^{2} e^{8}\right )} x^{2} + 4 \, {\left (c^{3} d^{9} e + 3 \, a c^{2} d^{7} e^{3} + 3 \, a^{2} c d^{5} e^{5} + a^{3} d^{3} e^{7}\right )} x\right )}}, -\frac {3 \, {\left (a^{2} c^{2} e^{4} x^{4} + 4 \, a^{2} c^{2} d e^{3} x^{3} + 6 \, a^{2} c^{2} d^{2} e^{2} x^{2} + 4 \, a^{2} c^{2} d^{3} e x + a^{2} c^{2} d^{4}\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}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) + {\left (5 \, a^{2} c^{2} d^{4} e + 7 \, a^{3} c d^{2} e^{3} + 2 \, a^{4} e^{5} - {\left (2 \, c^{4} d^{5} + 7 \, a c^{3} d^{3} e^{2} + 5 \, a^{2} c^{2} d e^{4}\right )} x^{3} - {\left (4 \, a c^{3} d^{4} e - a^{2} c^{2} d^{2} e^{3} - 5 \, a^{3} c e^{5}\right )} x^{2} - {\left (5 \, a c^{3} d^{5} + a^{2} c^{2} d^{3} e^{2} - 4 \, a^{3} c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{8 \, {\left (c^{3} d^{10} + 3 \, a c^{2} d^{8} e^{2} + 3 \, a^{2} c d^{6} e^{4} + a^{3} d^{4} e^{6} + {\left (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}\right )} x^{4} + 4 \, {\left (c^{3} d^{7} e^{3} + 3 \, a c^{2} d^{5} e^{5} + 3 \, a^{2} c d^{3} e^{7} + a^{3} d e^{9}\right )} x^{3} + 6 \, {\left (c^{3} d^{8} e^{2} + 3 \, a c^{2} d^{6} e^{4} + 3 \, a^{2} c d^{4} e^{6} + a^{3} d^{2} e^{8}\right )} x^{2} + 4 \, {\left (c^{3} d^{9} e + 3 \, a c^{2} d^{7} e^{3} + 3 \, a^{2} c d^{5} e^{5} + a^{3} d^{3} e^{7}\right )} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^5,x, algorithm="fricas")

[Out]

[1/16*(3*(a^2*c^2*e^4*x^4 + 4*a^2*c^2*d*e^3*x^3 + 6*a^2*c^2*d^2*e^2*x^2 + 4*a^2*c^2*d^3*e*x + a^2*c^2*d^4)*sqr
t(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*(5*a^2*c^2*d^4*e + 7*a^3*c*d^2*e^3 + 2*a^4*e^5 - (
2*c^4*d^5 + 7*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4)*x^3 - (4*a*c^3*d^4*e - a^2*c^2*d^2*e^3 - 5*a^3*c*e^5)*x^2 - (5*
a*c^3*d^5 + a^2*c^2*d^3*e^2 - 4*a^3*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c^3*d^10 + 3*a*c^2*d^8*e^2 + 3*a^2*c*d^6*e^4
+ a^3*d^4*e^6 + (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)*x^4 + 4*(c^3*d^7*e^3 + 3*a*c^2*d
^5*e^5 + 3*a^2*c*d^3*e^7 + a^3*d*e^9)*x^3 + 6*(c^3*d^8*e^2 + 3*a*c^2*d^6*e^4 + 3*a^2*c*d^4*e^6 + a^3*d^2*e^8)*
x^2 + 4*(c^3*d^9*e + 3*a*c^2*d^7*e^3 + 3*a^2*c*d^5*e^5 + a^3*d^3*e^7)*x), -1/8*(3*(a^2*c^2*e^4*x^4 + 4*a^2*c^2
*d*e^3*x^3 + 6*a^2*c^2*d^2*e^2*x^2 + 4*a^2*c^2*d^3*e*x + a^2*c^2*d^4)*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)) + (5*a^2*c^2*d^4*e + 7*a
^3*c*d^2*e^3 + 2*a^4*e^5 - (2*c^4*d^5 + 7*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4)*x^3 - (4*a*c^3*d^4*e - a^2*c^2*d^2*
e^3 - 5*a^3*c*e^5)*x^2 - (5*a*c^3*d^5 + a^2*c^2*d^3*e^2 - 4*a^3*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c^3*d^10 + 3*a*c
^2*d^8*e^2 + 3*a^2*c*d^6*e^4 + a^3*d^4*e^6 + (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)*x^4
+ 4*(c^3*d^7*e^3 + 3*a*c^2*d^5*e^5 + 3*a^2*c*d^3*e^7 + a^3*d*e^9)*x^3 + 6*(c^3*d^8*e^2 + 3*a*c^2*d^6*e^4 + 3*a
^2*c*d^4*e^6 + a^3*d^2*e^8)*x^2 + 4*(c^3*d^9*e + 3*a*c^2*d^7*e^3 + 3*a^2*c*d^5*e^5 + a^3*d^3*e^7)*x)]

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giac [F(-1)]  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)^(3/2)/(e*x+d)^5,x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.07, size = 3528, normalized size = 23.06 \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)^(3/2)/(e*x+d)^5,x)

[Out]

9/8/e^2*c^(7/2)*d^3/(a*e^2+c*d^2)^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))*a-9/16/e^2/(a*e^2+c*d^2)^2*c^(5/2)*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-3/16/e^2/(a*e^2+c*d^2)^2*c^3*d*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/
2)*x-3/8/e/(a*e^2+c*d^2)^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^2-3/8/e^5/(a*e^2+c*d^2)^2*c
^4/((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^4-1/4/e^2*c*d/(a*e^2+c*d^2)^2/(x+d/e)^3*(-2*(x+d/e)*c*d
/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(5/2)-1/8/e*c^2*d^2/(a*e^2+c*d^2)^3/(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)+1/8/e/(a*e^2+c*d^2)^2*c^2*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)-1/4/
e^3/(a*e^2+c*d^2)/(x+d/e)^4*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(5/2)-3/8*c^2*d/(a*e^2+c*d^2)^3/(
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)-3/8/e^4/(a*e^2+c*d^2)^2*c^(7/2)*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))-1/8*c^4*d^3/(a*e^2+c*d^2)^4*(-2*(x+d
/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)*x+9/16*c^(5/2)*d/(a*e^2+c*d^2)^3*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))-3/16*c^(7/2)*d^3/(a*e^2+c*d^2)^4*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))+3/8*c^3*d/(a*e^2+c*d^2)^3*(-2*(x+d/e)
*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)*x+3/8/e^3*c^5*d^6/(a*e^2+c*d^2)^4*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a
*e^2+c*d^2)/e^2)^(1/2)-1/4/e*c^3*d^2/(a*e^2+c*d^2)^3*(-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^3*c^4*d^4/(a*e^2+c*d^2)^3*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)+3/8/e/(a*e^2+c*d^2)^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)*a+3/8/e^3/(a*e^2+c*d^2)^2*c^3*(-2*(x+d/e)*c*d/e+(x+d/e
)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)*d^2-3/16/e^2*c^5*d^5/(a*e^2+c*d^2)^4*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2
)/e^2)^(1/2)*x+3/8/e*c^4*d^4/(a*e^2+c*d^2)^4*(-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^4*d^4/(a*e^2+c*d^2)^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))*a-3/8/e*c^4*d^4/(a*e^2+c*d^2)^4/((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+3/4/e*c^3*d^2/(a*e^2+c*d^2)^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))*a^2-3/4/e^3*c^5*d^6/(a*e^2+c*d^2)^4/((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/
4/e^3/(a*e^2+c*d^2)^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))*a*d^2+1/8*c^3*d^3/(a*e^2+c*d^2)^4/
(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)+1/8/e*c^4*d^4/(a*e^2+c*d^2)^4*(-2*(x+d/e)*c*d/e
+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)-3/8/e^4*c^(11/2)*d^7/(a*e^2+c*d^2)^4*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))+3/4/e^4*c^(9/2)*d^5/(a*e^2+c*d^2)^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))-1/8/e/(a*e^2+c*d^2)^2*c/(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)+9/16*c^3*d/(a*e^2+c*d^2)^3*a*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*
d^2)/e^2)^(1/2)*x-3/16*c^4*d^3/(a*e^2+c*d^2)^4*a*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)*x-3/8/
e^5*c^6*d^8/(a*e^2+c*d^2)^4/((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))+3/8/e^2*c^4*d^3/(a*e^2+c*d^2)^3*
(-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/e*c^3*d^2/(a*e^2+c*d^2)^3*(-2*(x+d/e)*c*d/e+(x+d/
e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)*a+3/4/e^5*c^5*d^6/(a*e^2+c*d^2)^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))-9/16/e^2*c^(9/2)*d^5/(a*e^2+c*d^2)^4*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

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maxima [B]  time = 2.42, size = 1223, normalized size = 7.99

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)^5,x, algorithm="maxima")

[Out]

3/8*sqrt(c*x^2 + a)*c^4*d^4/(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) - 3/8*sqrt(c*x^2 + a)*
c^4*d^3*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) + 1/8*(c*x^2 + a)^(3/2)*c^3*d^3/(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) - 1/8*(c*x^2 + a)^(5/2)*c^2*d^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) + 1/8*(c*x^2 + a)^(3/2)*c^3*d^2/(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) - 3/4*sqrt(c*x^2 + a)*c^3*d^2/(c^2*d^4*e^3 + 2*a*c*d^2*e^5 + a^2*e^7) + 3/8*sqrt(c*x^2 + a)*c^3*d*x/(c^2
*d^4*e^2 + 2*a*c*d^2*e^4 + a^2*e^6) - 1/4*(c*x^2 + a)^(5/2)*c*d/(c^2*d^4*e^2*x^3 + 2*a*c*d^2*e^4*x^3 + a^2*e^6
*x^3 + 3*c^2*d^5*e*x^2 + 6*a*c*d^3*e^3*x^2 + 3*a^2*d*e^5*x^2 + 3*c^2*d^6*x + 6*a*c*d^4*e^2*x + 3*a^2*d^2*e^4*x
+ c^2*d^7/e + 2*a*c*d^5*e + a^2*d^3*e^3) - 3/8*(c*x^2 + a)^(3/2)*c^2*d/(c^2*d^4*e^2*x + 2*a*c*d^2*e^4*x + a^2
*e^6*x + c^2*d^5*e + 2*a*c*d^3*e^3 + a^2*d*e^5) - 1/8*(c*x^2 + a)^(5/2)*c/(c^2*d^4*e*x^2 + 2*a*c*d^2*e^3*x^2 +
a^2*e^5*x^2 + 2*c^2*d^5*x + 4*a*c*d^3*e^2*x + 2*a^2*d*e^4*x + c^2*d^6/e + 2*a*c*d^4*e + a^2*d^2*e^3) + 1/8*(c
*x^2 + a)^(3/2)*c^2/(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5) - 1/4*(c*x^2 + a)^(5/2)/(c*d^2*e^3*x^4 + a*e^5*x^4 +
4*c*d^3*e^2*x^3 + 4*a*d*e^4*x^3 + 6*c*d^4*e*x^2 + 6*a*d^2*e^3*x^2 + 4*c*d^5*x + 4*a*d^3*e^2*x + c*d^6/e + a*d
^4*e) + 3/8*sqrt(c*x^2 + a)*c^2/(c*d^2*e^3 + a*e^5) + 3/8*c^4*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^9) - 3/4*c^3*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^7) + 3/8*c^2*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^5)

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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 )}^5} \,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)^5,x)

[Out]

int((a + c*x^2)^(3/2)/(d + e*x)^5, 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 )^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(3/2)/(e*x+d)**5,x)

[Out]

Integral((a + c*x**2)**(3/2)/(d + e*x)**5, x)

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