3.13.27 \(\int \frac {(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^6} \, dx\) [1227]

Optimal. Leaf size=139 \[ -\frac {\sqrt {a+b x+c x^2}}{32 c^3 d^6 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}+\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 c^{7/2} d^6} \]

[Out]

-1/24*(c*x^2+b*x+a)^(3/2)/c^2/d^6/(2*c*x+b)^3-1/10*(c*x^2+b*x+a)^(5/2)/c/d^6/(2*c*x+b)^5+1/64*arctanh(1/2*(2*c
*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(7/2)/d^6-1/32*(c*x^2+b*x+a)^(1/2)/c^3/d^6/(2*c*x+b)

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Rubi [A]
time = 0.05, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {698, 635, 212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 c^{7/2} d^6}-\frac {\sqrt {a+b x+c x^2}}{32 c^3 d^6 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^6,x]

[Out]

-1/32*Sqrt[a + b*x + c*x^2]/(c^3*d^6*(b + 2*c*x)) - (a + b*x + c*x^2)^(3/2)/(24*c^2*d^6*(b + 2*c*x)^3) - (a +
b*x + c*x^2)^(5/2)/(10*c*d^6*(b + 2*c*x)^5) + ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]/(64*c^(7/
2)*d^6)

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 635

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

Rule 698

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]

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^6} \, dx &=-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}+\frac {\int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^4} \, dx}{4 c d^2}\\ &=-\frac {\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}+\frac {\int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^2} \, dx}{16 c^2 d^4}\\ &=-\frac {\sqrt {a+b x+c x^2}}{32 c^3 d^6 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}+\frac {\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{64 c^3 d^6}\\ &=-\frac {\sqrt {a+b x+c x^2}}{32 c^3 d^6 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}+\frac {\text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{32 c^3 d^6}\\ &=-\frac {\sqrt {a+b x+c x^2}}{32 c^3 d^6 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}+\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 c^{7/2} d^6}\\ \end {align*}

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Mathematica [A]
time = 0.74, size = 145, normalized size = 1.04 \begin {gather*} \frac {-\frac {\sqrt {a+x (b+c x)} \left (15 b^4+140 b^3 c x+16 b c^2 x \left (11 a+46 c x^2\right )+4 b^2 c \left (5 a+127 c x^2\right )+16 c^2 \left (3 a^2+11 a c x^2+23 c^2 x^4\right )\right )}{480 c^3 (b+2 c x)^5}-\frac {\log \left (c^3 \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{64 c^{7/2}}}{d^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^6,x]

[Out]

(-1/480*(Sqrt[a + x*(b + c*x)]*(15*b^4 + 140*b^3*c*x + 16*b*c^2*x*(11*a + 46*c*x^2) + 4*b^2*c*(5*a + 127*c*x^2
) + 16*c^2*(3*a^2 + 11*a*c*x^2 + 23*c^2*x^4)))/(c^3*(b + 2*c*x)^5) - Log[c^3*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x
*(b + c*x)])]/(64*c^(7/2)))/d^6

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(439\) vs. \(2(117)=234\).
time = 0.71, size = 440, normalized size = 3.17

method	result	size
default	\(\frac {-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{5 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{5}}+\frac {8 c^{2} \left (-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{3}}+\frac {16 c^{2} \left (-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )}+\frac {24 c^{2} \left (\frac {\left (x +\frac {b}{2 c}\right ) \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{6}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (x +\frac {b}{2 c}\right ) \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{4}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (x +\frac {b}{2 c}\right ) \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}{2}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\sqrt {c}\, \left (x +\frac {b}{2 c}\right )+\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{4 a c -b^{2}}\right )}{3 \left (4 a c -b^{2}\right )}\right )}{5 \left (4 a c -b^{2}\right )}}{64 d^{6} c^{6}}\)	\(440\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^6,x,method=_RETURNVERBOSE)

[Out]

1/64/d^6/c^6*(-4/5/(4*a*c-b^2)*c/(x+1/2*b/c)^5*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)+8/5*c^2/(4*a*c-b^2)*(
-4/3/(4*a*c-b^2)*c/(x+1/2*b/c)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)+16/3*c^2/(4*a*c-b^2)*(-4/(4*a*c-b^2
)*c/(x+1/2*b/c)*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)+24*c^2/(4*a*c-b^2)*(1/6*(x+1/2*b/c)*((x+1/2*b/c)^2*c
+1/4*(4*a*c-b^2)/c)^(5/2)+5/24*(4*a*c-b^2)/c*(1/4*(x+1/2*b/c)*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)+3/16*(
4*a*c-b^2)/c*(1/2*(x+1/2*b/c)*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln(c^(1/2)*(x+
1/2*b/c)+((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2))))))))

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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)^(5/2)/(2*c*d*x+b*d)^6,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 deta

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (117) = 234\).
time = 4.44, size = 549, normalized size = 3.95 \begin {gather*} \left [\frac {15 \, {\left (32 \, c^{5} x^{5} + 80 \, b c^{4} x^{4} + 80 \, b^{2} c^{3} x^{3} + 40 \, b^{3} c^{2} x^{2} + 10 \, b^{4} c x + b^{5}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (368 \, c^{5} x^{4} + 736 \, b c^{4} x^{3} + 15 \, b^{4} c + 20 \, a b^{2} c^{2} + 48 \, a^{2} c^{3} + 4 \, {\left (127 \, b^{2} c^{3} + 44 \, a c^{4}\right )} x^{2} + 4 \, {\left (35 \, b^{3} c^{2} + 44 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{1920 \, {\left (32 \, c^{9} d^{6} x^{5} + 80 \, b c^{8} d^{6} x^{4} + 80 \, b^{2} c^{7} d^{6} x^{3} + 40 \, b^{3} c^{6} d^{6} x^{2} + 10 \, b^{4} c^{5} d^{6} x + b^{5} c^{4} d^{6}\right )}}, -\frac {15 \, {\left (32 \, c^{5} x^{5} + 80 \, b c^{4} x^{4} + 80 \, b^{2} c^{3} x^{3} + 40 \, b^{3} c^{2} x^{2} + 10 \, b^{4} c x + b^{5}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (368 \, c^{5} x^{4} + 736 \, b c^{4} x^{3} + 15 \, b^{4} c + 20 \, a b^{2} c^{2} + 48 \, a^{2} c^{3} + 4 \, {\left (127 \, b^{2} c^{3} + 44 \, a c^{4}\right )} x^{2} + 4 \, {\left (35 \, b^{3} c^{2} + 44 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{960 \, {\left (32 \, c^{9} d^{6} x^{5} + 80 \, b c^{8} d^{6} x^{4} + 80 \, b^{2} c^{7} d^{6} x^{3} + 40 \, b^{3} c^{6} d^{6} x^{2} + 10 \, b^{4} c^{5} d^{6} x + b^{5} c^{4} d^{6}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^6,x, algorithm="fricas")

[Out]

[1/1920*(15*(32*c^5*x^5 + 80*b*c^4*x^4 + 80*b^2*c^3*x^3 + 40*b^3*c^2*x^2 + 10*b^4*c*x + b^5)*sqrt(c)*log(-8*c^
2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(368*c^5*x^4 + 736*b*c^4*x^3
+ 15*b^4*c + 20*a*b^2*c^2 + 48*a^2*c^3 + 4*(127*b^2*c^3 + 44*a*c^4)*x^2 + 4*(35*b^3*c^2 + 44*a*b*c^3)*x)*sqrt(
c*x^2 + b*x + a))/(32*c^9*d^6*x^5 + 80*b*c^8*d^6*x^4 + 80*b^2*c^7*d^6*x^3 + 40*b^3*c^6*d^6*x^2 + 10*b^4*c^5*d^
6*x + b^5*c^4*d^6), -1/960*(15*(32*c^5*x^5 + 80*b*c^4*x^4 + 80*b^2*c^3*x^3 + 40*b^3*c^2*x^2 + 10*b^4*c*x + b^5
)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(368*c^5*x^4 + 7
36*b*c^4*x^3 + 15*b^4*c + 20*a*b^2*c^2 + 48*a^2*c^3 + 4*(127*b^2*c^3 + 44*a*c^4)*x^2 + 4*(35*b^3*c^2 + 44*a*b*
c^3)*x)*sqrt(c*x^2 + b*x + a))/(32*c^9*d^6*x^5 + 80*b*c^8*d^6*x^4 + 80*b^2*c^7*d^6*x^3 + 40*b^3*c^6*d^6*x^2 +
10*b^4*c^5*d^6*x + b^5*c^4*d^6)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a^{2} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx}{d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**6,x)

[Out]

(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*
c**4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integral(b**2*x**2*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*
x + 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integr
al(c**2*x**4*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c*
*4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integral(2*a*b*x*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x +
60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integral(2
*a*c*x**2*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*
x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integral(2*b*c*x**3*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x +
60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x))/d**6

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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((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^6,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{%%%{64,[6]%%%},[12,6,12,0]%%%}+%%%{%%%{-1536,[7]%%%},[12
,6,10,1]%%%

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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 )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^6,x)

[Out]

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^6, x)

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