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

Optimal. Leaf size=123 \[ \frac {3 \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{128 c^{5/2} d^5 \sqrt {b^2-4 a c}}-\frac {3 \sqrt {a+b x+c x^2}}{64 c^2 d^5 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{8 c d^5 (b+2 c x)^4} \]

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Rubi [A]  time = 0.07, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {684, 688, 205} \begin {gather*} \frac {3 \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{128 c^{5/2} d^5 \sqrt {b^2-4 a c}}-\frac {3 \sqrt {a+b x+c x^2}}{64 c^2 d^5 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{8 c d^5 (b+2 c x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[a + b*x + c*x^2])/(64*c^2*d^5*(b + 2*c*x)^2) - (a + b*x + c*x^2)^(3/2)/(8*c*d^5*(b + 2*c*x)^4) + (3*A
rcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(128*c^(5/2)*Sqrt[b^2 - 4*a*c]*d^5)

Rule 205

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

Rule 684

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 1)), x] - Dist[(b*p)/(d*e*(m + 1)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1
), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] &&
GtQ[p, 0] && LtQ[m, -1] &&  !(IntegerQ[m/2] && LtQ[m + 2*p + 3, 0]) && IntegerQ[2*p]

Rule 688

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[4*c, Subst[Int[1/(b^2*e
- 4*a*c*e + 4*c*e*x^2), x], x, Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^5} \, dx &=-\frac {\left (a+b x+c x^2\right )^{3/2}}{8 c d^5 (b+2 c x)^4}+\frac {3 \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^3} \, dx}{16 c d^2}\\ &=-\frac {3 \sqrt {a+b x+c x^2}}{64 c^2 d^5 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{8 c d^5 (b+2 c x)^4}+\frac {3 \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx}{128 c^2 d^4}\\ &=-\frac {3 \sqrt {a+b x+c x^2}}{64 c^2 d^5 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{8 c d^5 (b+2 c x)^4}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt {a+b x+c x^2}\right )}{32 c d^4}\\ &=-\frac {3 \sqrt {a+b x+c x^2}}{64 c^2 d^5 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{8 c d^5 (b+2 c x)^4}+\frac {3 \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{128 c^{5/2} \sqrt {b^2-4 a c} d^5}\\ \end {align*}

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Mathematica [A]  time = 0.25, size = 162, normalized size = 1.32 \begin {gather*} \frac {-2 c \left (8 a^2 c+a \left (3 b^2+28 b c x+28 c^2 x^2\right )+x \left (3 b^3+23 b^2 c x+40 b c^2 x^2+20 c^3 x^3\right )\right )-3 (b+2 c x)^4 \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}} \tanh ^{-1}\left (2 \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}}\right )}{128 c^3 d^5 (b+2 c x)^4 \sqrt {a+x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c*(8*a^2*c + a*(3*b^2 + 28*b*c*x + 28*c^2*x^2) + x*(3*b^3 + 23*b^2*c*x + 40*b*c^2*x^2 + 20*c^3*x^3)) - 3*(
b + 2*c*x)^4*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*ArcTanh[2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]])
/(128*c^3*d^5*(b + 2*c*x)^4*Sqrt[a + x*(b + c*x)])

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IntegrateAlgebraic [B]  time = 5.97, size = 1233, normalized size = 10.02 \begin {gather*} \frac {-2560 c^7 x^8+2560 c^{13/2} \sqrt {c x^2+b x+a} x^7-3 b^7 x-3 a b^6+c \left (-119 x^2 b^6-196 a x b^5-80 a^2 b^4\right )+c^{5/2} \sqrt {c x^2+b x+a} \left (2736 x^3 b^4+3968 a x^2 b^3+1728 a^2 x b^2+256 a^3 b\right )+c^2 \left (-1256 x^3 b^5-2340 a x^2 b^4-1360 a^2 x b^3-240 a^3 b^2\right )+c^{7/2} \sqrt {c x^2+b x+a} \left (8160 b^3 x^4+10752 a b^2 x^3+4224 a^2 b x^2+512 a^3 x\right )+c^3 \left (-128 a^4-1472 b x a^3-6288 b^2 x^2 a^2-10432 b^3 x^3 a-5748 b^4 x^4\right )+c^{9/2} \sqrt {c x^2+b x+a} \left (12224 b^2 x^5+12160 a b x^4+2816 a^2 x^3\right )+c^4 \left (-13312 b^3 x^5-20576 a b^2 x^4-9856 a^2 b x^3-1472 a^3 x^2\right )+c^{11/2} \sqrt {c x^2+b x+a} \left (8960 b x^6+4864 a x^5\right )+c^5 \left (-16384 b^2 x^6-18432 a b x^5-4928 a^2 x^4\right )+c^6 \left (-10240 b x^7-6144 a x^6\right )+\sqrt {c} \left (24 x b^6+24 a b^5\right ) \sqrt {c x^2+b x+a}+c^{3/2} \left (424 x^2 b^5+560 a x b^4+160 a^2 b^3\right ) \sqrt {c x^2+b x+a}}{-131072 d^5 x^9 c^{21/2}+131072 d^5 x^8 \sqrt {c x^2+b x+a} c^{10}+64 d^5 \left (-9216 b x^8-3072 a x^7\right ) c^{19/2}+64 d^5 \sqrt {c x^2+b x+a} \left (8192 b x^7+2048 a x^6\right ) c^9+64 d^5 \left (-17664 b^2 x^7-10752 a b x^6-1024 a^2 x^5\right ) c^{17/2}+64 d^5 \sqrt {c x^2+b x+a} \left (13824 b^2 x^6+6144 a b x^5+256 a^2 x^4\right ) c^8+64 d^5 \left (-18816 b^3 x^6-15616 a b^2 x^5-2560 a^2 b x^4\right ) c^{15/2}+64 d^5 \sqrt {c x^2+b x+a} \left (12800 b^3 x^5+7552 a b^2 x^4+512 a^2 b x^3\right ) c^7+64 d^5 \left (-12160 b^4 x^5-12160 a b^3 x^4-2560 a^2 b^2 x^3\right ) c^{13/2}+64 d^5 \sqrt {c x^2+b x+a} \left (7056 b^4 x^4+4864 a b^3 x^3+384 a^2 b^2 x^2\right ) c^6+64 d^5 \left (-4864 x^4 b^5-5440 a x^3 b^4-1280 a^2 x^2 b^3\right ) c^{11/2}+64 d^5 \sqrt {c x^2+b x+a} \left (2336 x^3 b^5+1728 a x^2 b^4+128 a^2 x b^3\right ) c^5+64 d^5 \left (-1168 x^3 b^6-1376 a x^2 b^5-320 a^2 x b^4\right ) c^{9/2}+64 d^5 \left (440 x^2 b^6+320 a x b^5+16 a^2 b^4\right ) \sqrt {c x^2+b x+a} c^4+64 d^5 \left (-152 x^2 b^7-176 a x b^6-32 a^2 b^5\right ) c^{7/2}+64 d^5 \left (40 x b^7+24 a b^6\right ) \sqrt {c x^2+b x+a} c^3+64 d^5 \left (-8 x b^8-8 a b^7\right ) c^{5/2}+64 b^8 d^5 \sqrt {c x^2+b x+a} c^2}-\frac {3 \tan ^{-1}\left (\frac {b}{\sqrt {b^2-4 a c}}+\frac {2 c x}{\sqrt {b^2-4 a c}}-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}\right )}{64 c^{5/2} \sqrt {b^2-4 a c} d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*b^6 - 3*b^7*x - 2560*c^7*x^8 + c*(-80*a^2*b^4 - 196*a*b^5*x - 119*b^6*x^2) + 2560*c^(13/2)*x^7*Sqrt[a +
b*x + c*x^2] + Sqrt[c]*(24*a*b^5 + 24*b^6*x)*Sqrt[a + b*x + c*x^2] + c^(3/2)*(160*a^2*b^3 + 560*a*b^4*x + 424*
b^5*x^2)*Sqrt[a + b*x + c*x^2] + c^(5/2)*Sqrt[a + b*x + c*x^2]*(256*a^3*b + 1728*a^2*b^2*x + 3968*a*b^3*x^2 +
2736*b^4*x^3) + c^2*(-240*a^3*b^2 - 1360*a^2*b^3*x - 2340*a*b^4*x^2 - 1256*b^5*x^3) + c^(7/2)*Sqrt[a + b*x + c
*x^2]*(512*a^3*x + 4224*a^2*b*x^2 + 10752*a*b^2*x^3 + 8160*b^3*x^4) + c^3*(-128*a^4 - 1472*a^3*b*x - 6288*a^2*
b^2*x^2 - 10432*a*b^3*x^3 - 5748*b^4*x^4) + c^(9/2)*Sqrt[a + b*x + c*x^2]*(2816*a^2*x^3 + 12160*a*b*x^4 + 1222
4*b^2*x^5) + c^4*(-1472*a^3*x^2 - 9856*a^2*b*x^3 - 20576*a*b^2*x^4 - 13312*b^3*x^5) + c^(11/2)*Sqrt[a + b*x +
c*x^2]*(4864*a*x^5 + 8960*b*x^6) + c^5*(-4928*a^2*x^4 - 18432*a*b*x^5 - 16384*b^2*x^6) + c^6*(-6144*a*x^6 - 10
240*b*x^7))/(-131072*c^(21/2)*d^5*x^9 + 64*c^(5/2)*d^5*(-8*a*b^7 - 8*b^8*x) + 64*c^(7/2)*d^5*(-32*a^2*b^5 - 17
6*a*b^6*x - 152*b^7*x^2) + 64*b^8*c^2*d^5*Sqrt[a + b*x + c*x^2] + 131072*c^10*d^5*x^8*Sqrt[a + b*x + c*x^2] +
64*c^3*d^5*(24*a*b^6 + 40*b^7*x)*Sqrt[a + b*x + c*x^2] + 64*c^4*d^5*(16*a^2*b^4 + 320*a*b^5*x + 440*b^6*x^2)*S
qrt[a + b*x + c*x^2] + 64*c^5*d^5*Sqrt[a + b*x + c*x^2]*(128*a^2*b^3*x + 1728*a*b^4*x^2 + 2336*b^5*x^3) + 64*c
^(9/2)*d^5*(-320*a^2*b^4*x - 1376*a*b^5*x^2 - 1168*b^6*x^3) + 64*c^6*d^5*Sqrt[a + b*x + c*x^2]*(384*a^2*b^2*x^
2 + 4864*a*b^3*x^3 + 7056*b^4*x^4) + 64*c^(11/2)*d^5*(-1280*a^2*b^3*x^2 - 5440*a*b^4*x^3 - 4864*b^5*x^4) + 64*
c^7*d^5*Sqrt[a + b*x + c*x^2]*(512*a^2*b*x^3 + 7552*a*b^2*x^4 + 12800*b^3*x^5) + 64*c^(13/2)*d^5*(-2560*a^2*b^
2*x^3 - 12160*a*b^3*x^4 - 12160*b^4*x^5) + 64*c^8*d^5*Sqrt[a + b*x + c*x^2]*(256*a^2*x^4 + 6144*a*b*x^5 + 1382
4*b^2*x^6) + 64*c^(15/2)*d^5*(-2560*a^2*b*x^4 - 15616*a*b^2*x^5 - 18816*b^3*x^6) + 64*c^9*d^5*Sqrt[a + b*x + c
*x^2]*(2048*a*x^6 + 8192*b*x^7) + 64*c^(17/2)*d^5*(-1024*a^2*x^5 - 10752*a*b*x^6 - 17664*b^2*x^7) + 64*c^(19/2
)*d^5*(-3072*a*x^7 - 9216*b*x^8)) - (3*ArcTan[b/Sqrt[b^2 - 4*a*c] + (2*c*x)/Sqrt[b^2 - 4*a*c] - (2*Sqrt[c]*Sqr
t[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(64*c^(5/2)*Sqrt[b^2 - 4*a*c]*d^5)

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fricas [B]  time = 1.16, size = 622, normalized size = 5.06 \begin {gather*} \left [-\frac {3 \, {\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \sqrt {-b^{2} c + 4 \, a c^{2}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt {-b^{2} c + 4 \, a c^{2}} \sqrt {c x^{2} + b x + a}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + 4 \, {\left (3 \, b^{4} c - 4 \, a b^{2} c^{2} - 32 \, a^{2} c^{3} + 20 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + 20 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{256 \, {\left (16 \, {\left (b^{2} c^{7} - 4 \, a c^{8}\right )} d^{5} x^{4} + 32 \, {\left (b^{3} c^{6} - 4 \, a b c^{7}\right )} d^{5} x^{3} + 24 \, {\left (b^{4} c^{5} - 4 \, a b^{2} c^{6}\right )} d^{5} x^{2} + 8 \, {\left (b^{5} c^{4} - 4 \, a b^{3} c^{5}\right )} d^{5} x + {\left (b^{6} c^{3} - 4 \, a b^{4} c^{4}\right )} d^{5}\right )}}, -\frac {3 \, {\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \sqrt {b^{2} c - 4 \, a c^{2}} \arctan \left (\frac {\sqrt {b^{2} c - 4 \, a c^{2}} \sqrt {c x^{2} + b x + a}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (3 \, b^{4} c - 4 \, a b^{2} c^{2} - 32 \, a^{2} c^{3} + 20 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + 20 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{128 \, {\left (16 \, {\left (b^{2} c^{7} - 4 \, a c^{8}\right )} d^{5} x^{4} + 32 \, {\left (b^{3} c^{6} - 4 \, a b c^{7}\right )} d^{5} x^{3} + 24 \, {\left (b^{4} c^{5} - 4 \, a b^{2} c^{6}\right )} d^{5} x^{2} + 8 \, {\left (b^{5} c^{4} - 4 \, a b^{3} c^{5}\right )} d^{5} x + {\left (b^{6} c^{3} - 4 \, a b^{4} c^{4}\right )} d^{5}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/256*(3*(16*c^4*x^4 + 32*b*c^3*x^3 + 24*b^2*c^2*x^2 + 8*b^3*c*x + b^4)*sqrt(-b^2*c + 4*a*c^2)*log(-(4*c^2*x
^2 + 4*b*c*x - b^2 + 8*a*c - 4*sqrt(-b^2*c + 4*a*c^2)*sqrt(c*x^2 + b*x + a))/(4*c^2*x^2 + 4*b*c*x + b^2)) + 4*
(3*b^4*c - 4*a*b^2*c^2 - 32*a^2*c^3 + 20*(b^2*c^3 - 4*a*c^4)*x^2 + 20*(b^3*c^2 - 4*a*b*c^3)*x)*sqrt(c*x^2 + b*
x + a))/(16*(b^2*c^7 - 4*a*c^8)*d^5*x^4 + 32*(b^3*c^6 - 4*a*b*c^7)*d^5*x^3 + 24*(b^4*c^5 - 4*a*b^2*c^6)*d^5*x^
2 + 8*(b^5*c^4 - 4*a*b^3*c^5)*d^5*x + (b^6*c^3 - 4*a*b^4*c^4)*d^5), -1/128*(3*(16*c^4*x^4 + 32*b*c^3*x^3 + 24*
b^2*c^2*x^2 + 8*b^3*c*x + b^4)*sqrt(b^2*c - 4*a*c^2)*arctan(1/2*sqrt(b^2*c - 4*a*c^2)*sqrt(c*x^2 + b*x + a)/(c
^2*x^2 + b*c*x + a*c)) + 2*(3*b^4*c - 4*a*b^2*c^2 - 32*a^2*c^3 + 20*(b^2*c^3 - 4*a*c^4)*x^2 + 20*(b^3*c^2 - 4*
a*b*c^3)*x)*sqrt(c*x^2 + b*x + a))/(16*(b^2*c^7 - 4*a*c^8)*d^5*x^4 + 32*(b^3*c^6 - 4*a*b*c^7)*d^5*x^3 + 24*(b^
4*c^5 - 4*a*b^2*c^6)*d^5*x^2 + 8*(b^5*c^4 - 4*a*b^3*c^5)*d^5*x + (b^6*c^3 - 4*a*b^4*c^4)*d^5)]

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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+b*x+a)^(3/2)/(2*c*d*x+b*d)^5,x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.06, size = 622, normalized size = 5.06 \begin {gather*} -\frac {3 a^{2} \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{8 \left (4 a c -b^{2}\right )^{2} \sqrt {\frac {4 a c -b^{2}}{c}}\, c \,d^{5}}+\frac {3 a \,b^{2} \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{16 \left (4 a c -b^{2}\right )^{2} \sqrt {\frac {4 a c -b^{2}}{c}}\, c^{2} d^{5}}-\frac {3 b^{4} \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{128 \left (4 a c -b^{2}\right )^{2} \sqrt {\frac {4 a c -b^{2}}{c}}\, c^{3} d^{5}}+\frac {3 \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}\, a}{32 \left (4 a c -b^{2}\right )^{2} c \,d^{5}}-\frac {3 \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}\, b^{2}}{128 \left (4 a c -b^{2}\right )^{2} c^{2} d^{5}}+\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{16 \left (4 a c -b^{2}\right )^{2} c \,d^{5}}-\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{16 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right )^{2} c^{2} d^{5}}-\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{32 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{4} c^{4} d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^5,x)

[Out]

-1/32/d^5/c^4/(4*a*c-b^2)/(x+1/2*b/c)^4*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)-1/16/d^5/c^2/(4*a*c-b^2)^2/(
x+1/2*b/c)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)+1/16/d^5/c/(4*a*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^
2)/c)^(3/2)+3/32/d^5/c/(4*a*c-b^2)^2*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*a-3/128/d^5/c^2/(4*a*c-b^2)^2*(4*
(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*b^2-3/8/d^5/c/(4*a*c-b^2)^2/((4*a*c-b^2)/c)^(1/2)*ln((1/2*(4*a*c-b^2)/c+1
/2*((4*a*c-b^2)/c)^(1/2)*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2))/(x+1/2*b/c))*a^2+3/16/d^5/c^2/(4*a*c-b^2)^2/
((4*a*c-b^2)/c)^(1/2)*ln((1/2*(4*a*c-b^2)/c+1/2*((4*a*c-b^2)/c)^(1/2)*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2))
/(x+1/2*b/c))*a*b^2-3/128/d^5/c^3/(4*a*c-b^2)^2/((4*a*c-b^2)/c)^(1/2)*ln((1/2*(4*a*c-b^2)/c+1/2*((4*a*c-b^2)/c
)^(1/2)*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2))/(x+1/2*b/c))*b^4

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (b\,d+2\,c\,d\,x\right )}^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^5,x)

[Out]

int((a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^5, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a \sqrt {a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx + \int \frac {b x \sqrt {a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx + \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx}{d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a*sqrt(a + b*x + c*x**2)/(b**5 + 10*b**4*c*x + 40*b**3*c**2*x**2 + 80*b**2*c**3*x**3 + 80*b*c**4*x**
4 + 32*c**5*x**5), x) + Integral(b*x*sqrt(a + b*x + c*x**2)/(b**5 + 10*b**4*c*x + 40*b**3*c**2*x**2 + 80*b**2*
c**3*x**3 + 80*b*c**4*x**4 + 32*c**5*x**5), x) + Integral(c*x**2*sqrt(a + b*x + c*x**2)/(b**5 + 10*b**4*c*x +
40*b**3*c**2*x**2 + 80*b**2*c**3*x**3 + 80*b*c**4*x**4 + 32*c**5*x**5), x))/d**5

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