### 3.1211 $$\int \frac{(a+b x+c x^2)^{3/2}}{(b d+2 c d x)^5} \, dx$$

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

[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)

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Rubi [A]  time = 0.0731791, antiderivative size = 123, normalized size of antiderivative = 1., 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} $\frac{3 \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{128 c^{5/2} d^5 \sqrt{b^2-4 a c}}-\frac{3 \sqrt{a+b x+c x^2}}{64 c^2 d^5 (b+2 c x)^2}-\frac{\left (a+b x+c x^2\right )^{3/2}}{8 c d^5 (b+2 c x)^4}$

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 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]

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]

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*}

Mathematica [A]  time = 0.245584, size = 162, normalized size = 1.32 $\frac{-2 c \left (8 a^2 c+a \left (3 b^2+28 b c x+28 c^2 x^2\right )+x \left (23 b^2 c x+3 b^3+40 b c^2 x^2+20 c^3 x^3\right )\right )-3 (b+2 c x)^4 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \tanh ^{-1}\left (2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right )}{128 c^3 d^5 (b+2 c x)^4 \sqrt{a+x (b+c x)}}$

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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Maple [B]  time = 0.194, size = 622, normalized size = 5.1 \begin{align*} -{\frac{1}{32\,{c}^{4}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) } \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-4}}-{\frac{1}{16\,{c}^{2}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) ^{2}} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-2}}+{\frac{1}{16\,{d}^{5}c \left ( 4\,ac-{b}^{2} \right ) ^{2}} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,a}{32\,{d}^{5}c \left ( 4\,ac-{b}^{2} \right ) ^{2}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}-{\frac{3\,{b}^{2}}{128\,{c}^{2}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}-{\frac{3\,{a}^{2}}{8\,{d}^{5}c \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ({ \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}+{\frac{3\,{b}^{2}a}{16\,{c}^{2}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ({ \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}-{\frac{3\,{b}^{4}}{128\,{c}^{3}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ({ \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

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Fricas [B]  time = 12.4489, size = 1320, normalized size = 10.73 \begin{align*} \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{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}

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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Giac [B]  time = 1.79571, size = 911, normalized size = 7.41 \begin{align*} \text{result too large to display} \end{align*}

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]

1/96*(3*sqrt(-b^2*c + 4*a*c^2)*log(abs(c))*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d)/(b^6*c^7*d^8*abs(c) - 12*a*b^4
*c^8*d^8*abs(c) + 48*a^2*b^2*c^9*d^8*abs(c) - 64*a^3*c^10*d^8*abs(c)) + 2*sqrt(-b^2*c*d^2/(2*c*d*x + b*d)^2 +
4*a*c^2*d^2/(2*c*d*x + b*d)^2 + c)*(5*(b^4*c^5*d^10*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d) - 8*a*b^2*c^6*d^10*sg
n(1/(2*c*d*x + b*d))*sgn(c)*sgn(d) + 16*a^2*c^7*d^10*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d))/(b^8*c^12*d^16 - 16
*a*b^6*c^13*d^16 + 96*a^2*b^4*c^14*d^16 - 256*a^3*b^2*c^15*d^16 + 256*a^4*c^16*d^16) - 2*(b^6*c^7*d^14*sgn(1/(
2*c*d*x + b*d))*sgn(c)*sgn(d) - 12*a*b^4*c^8*d^14*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d) + 48*a^2*b^2*c^9*d^14*s
gn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d) - 64*a^3*c^10*d^14*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d))/((b^8*c^12*d^16 -
16*a*b^6*c^13*d^16 + 96*a^2*b^4*c^14*d^16 - 256*a^3*b^2*c^15*d^16 + 256*a^4*c^16*d^16)*(2*c*d*x + b*d)^2*c^2*
d^2))/((2*c*d*x + b*d)*c*d) - 6*sqrt(-b^2*c + 4*a*c^2)*log(abs(sqrt(-b^2*c*d^2/(2*c*d*x + b*d)^2 + 4*a*c^2*d^2
/(2*c*d*x + b*d)^2 + c) + sqrt(-b^2*c^3*d^4 + 4*a*c^4*d^4)/((2*c*d*x + b*d)*c*d)))*sgn(1/(2*c*d*x + b*d))*sgn(
c)*sgn(d)/((b^6*c^7 - 12*a*b^4*c^8 + 48*a^2*b^2*c^9 - 64*a^3*c^10)*d^8*abs(c)))*d^2*abs(c)