3.13.23 \(\int \frac {(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^2} \, dx\) [1223]

Optimal. Leaf size=153 \[ -\frac {15 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{256 c^3 d^2}+\frac {5 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{32 c^2 d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 c d^2 (b+2 c x)}+\frac {15 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{512 c^{7/2} d^2} \]

[Out]

5/32*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^2/d^2-1/2*(c*x^2+b*x+a)^(5/2)/c/d^2/(2*c*x+b)+15/512*(-4*a*c+b^2)^2*arcta
nh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(7/2)/d^2-15/256*(-4*a*c+b^2)*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^
3/d^2

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Rubi [A]
time = 0.04, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {698, 626, 635, 212} \begin {gather*} \frac {15 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{512 c^{7/2} d^2}-\frac {15 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{256 c^3 d^2}+\frac {5 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{32 c^2 d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 c d^2 (b+2 c x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*(b^2 - 4*a*c)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(256*c^3*d^2) + (5*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/
(32*c^2*d^2) - (a + b*x + c*x^2)^(5/2)/(2*c*d^2*(b + 2*c*x)) + (15*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt
[c]*Sqrt[a + b*x + c*x^2])])/(512*c^(7/2)*d^2)

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 626

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

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)^2} \, dx &=-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 c d^2 (b+2 c x)}+\frac {5 \int \left (a+b x+c x^2\right )^{3/2} \, dx}{4 c d^2}\\ &=\frac {5 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{32 c^2 d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 c d^2 (b+2 c x)}-\frac {\left (15 \left (b^2-4 a c\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{64 c^2 d^2}\\ &=-\frac {15 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{256 c^3 d^2}+\frac {5 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{32 c^2 d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 c d^2 (b+2 c x)}+\frac {\left (15 \left (b^2-4 a c\right )^2\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{512 c^3 d^2}\\ &=-\frac {15 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{256 c^3 d^2}+\frac {5 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{32 c^2 d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 c d^2 (b+2 c x)}+\frac {\left (15 \left (b^2-4 a c\right )^2\right ) \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 )}{256 c^3 d^2}\\ &=-\frac {15 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{256 c^3 d^2}+\frac {5 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{32 c^2 d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 c d^2 (b+2 c x)}+\frac {15 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{512 c^{7/2} d^2}\\ \end {align*}

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Mathematica [A]
time = 0.64, size = 152, normalized size = 0.99 \begin {gather*} \frac {\frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (-15 b^4-20 b^3 c x+4 b^2 c \left (25 a+3 c x^2\right )+16 b c^2 x \left (9 a+4 c x^2\right )+16 c^2 \left (-8 a^2+9 a c x^2+2 c^2 x^4\right )\right )}{b+2 c x}-15 \left (b^2-4 a c\right )^2 \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{512 c^{7/2} d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-15*b^4 - 20*b^3*c*x + 4*b^2*c*(25*a + 3*c*x^2) + 16*b*c^2*x*(9*a + 4*c*x^2
) + 16*c^2*(-8*a^2 + 9*a*c*x^2 + 2*c^2*x^4)))/(b + 2*c*x) - 15*(b^2 - 4*a*c)^2*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[
a + x*(b + c*x)]])/(512*c^(7/2)*d^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(293\) vs. \(2(131)=262\).
time = 0.87, size = 294, normalized size = 1.92

method	result	size
default	\(\frac {-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )}+\frac {24 c^{2} \left (\frac {\left (x +\frac {b}{2 c}\right ) \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{6}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (x +\frac {b}{2 c}\right ) \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{4}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (x +\frac {b}{2 c}\right ) \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}{2}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\sqrt {c}\, \left (x +\frac {b}{2 c}\right )+\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{4 a c -b^{2}}}{4 d^{2} c^{2}}\)	\(294\)
risch	\(\frac {\left (16 c^{3} x^{3}+24 b \,c^{2} x^{2}+72 a \,c^{2} x -6 b^{2} c x +36 a b c -7 b^{3}\right ) \sqrt {c \,x^{2}+b x +a}}{256 c^{3} d^{2}}+\frac {\frac {15 a^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32 c^{\frac {3}{2}}}-\frac {15 a \,b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64 c^{\frac {5}{2}}}+\frac {15 b^{4} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{512 c^{\frac {7}{2}}}-\frac {\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, a^{3}}{c \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )}+\frac {3 \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, a^{2} b^{2}}{4 c^{2} \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )}-\frac {3 \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, a \,b^{4}}{16 c^{3} \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )}+\frac {\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, b^{6}}{64 c^{4} \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )}}{d^{2}}\)	\(413\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

1/4/d^2/c^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)^2,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 [A]
time = 2.29, size = 427, normalized size = 2.79 \begin {gather*} \left [\frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2} + 2 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x\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 (32 \, c^{5} x^{4} + 64 \, b c^{4} x^{3} - 15 \, b^{4} c + 100 \, a b^{2} c^{2} - 128 \, a^{2} c^{3} + 12 \, {\left (b^{2} c^{3} + 12 \, a c^{4}\right )} x^{2} - 4 \, {\left (5 \, b^{3} c^{2} - 36 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{1024 \, {\left (2 \, c^{5} d^{2} x + b c^{4} d^{2}\right )}}, -\frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2} + 2 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x\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 (32 \, c^{5} x^{4} + 64 \, b c^{4} x^{3} - 15 \, b^{4} c + 100 \, a b^{2} c^{2} - 128 \, a^{2} c^{3} + 12 \, {\left (b^{2} c^{3} + 12 \, a c^{4}\right )} x^{2} - 4 \, {\left (5 \, b^{3} c^{2} - 36 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{512 \, {\left (2 \, c^{5} d^{2} x + b c^{4} d^{2}\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)^2,x, algorithm="fricas")

[Out]

[1/1024*(15*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2 + 2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*x)*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*(32*c^5*x^4 + 64*b*c^4*x^3 - 15*b^4*
c + 100*a*b^2*c^2 - 128*a^2*c^3 + 12*(b^2*c^3 + 12*a*c^4)*x^2 - 4*(5*b^3*c^2 - 36*a*b*c^3)*x)*sqrt(c*x^2 + b*x
+ a))/(2*c^5*d^2*x + b*c^4*d^2), -1/512*(15*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2 + 2*(b^4*c - 8*a*b^2*c^2 + 16*a^2
*c^3)*x)*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*(32*c^5*x
^4 + 64*b*c^4*x^3 - 15*b^4*c + 100*a*b^2*c^2 - 128*a^2*c^3 + 12*(b^2*c^3 + 12*a*c^4)*x^2 - 4*(5*b^3*c^2 - 36*a
*b*c^3)*x)*sqrt(c*x^2 + b*x + a))/(2*c^5*d^2*x + b*c^4*d^2)]

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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^{2} + 4 b c x + 4 c^{2} x^{2}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{2} + 4 b c x + 4 c^{2} x^{2}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{2} + 4 b c x + 4 c^{2} x^{2}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{2} + 4 b c x + 4 c^{2} x^{2}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{2} + 4 b c x + 4 c^{2} x^{2}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{2} + 4 b c x + 4 c^{2} x^{2}}\, dx}{d^{2}} \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)**2,x)

[Out]

(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**2 + 4*b*c*x + 4*c**2*x**2), x) + Integral(b**2*x**2*sqrt(a + b*x + c
*x**2)/(b**2 + 4*b*c*x + 4*c**2*x**2), x) + Integral(c**2*x**4*sqrt(a + b*x + c*x**2)/(b**2 + 4*b*c*x + 4*c**2
*x**2), x) + Integral(2*a*b*x*sqrt(a + b*x + c*x**2)/(b**2 + 4*b*c*x + 4*c**2*x**2), x) + Integral(2*a*c*x**2*
sqrt(a + b*x + c*x**2)/(b**2 + 4*b*c*x + 4*c**2*x**2), x) + Integral(2*b*c*x**3*sqrt(a + b*x + c*x**2)/(b**2 +
4*b*c*x + 4*c**2*x**2), x))/d**2

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 814 vs. \(2 (131) = 262\).
time = 3.17, size = 814, normalized size = 5.32 \begin {gather*} -\frac {1}{512} \, d^{2} {\left (\frac {15 \, {\left (b^{4} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) - 8 \, a b^{2} c \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) + 16 \, a^{2} c^{2} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )\right )} \arctan \left (\frac {\sqrt {-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c^{3} d^{4} {\left | c \right |}} + \frac {8 \, {\left (\sqrt {-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c} b^{4} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) - 8 \, \sqrt {-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c} a b^{2} c \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) + 16 \, \sqrt {-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c} a^{2} c^{2} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )\right )}}{c^{4} d^{4} {\left | c \right |}} - \frac {9 \, {\left (-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c\right )}^{\frac {3}{2}} b^{4} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) - 72 \, {\left (-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c\right )}^{\frac {3}{2}} a b^{2} c \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) - 7 \, \sqrt {-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c} b^{4} c \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) + 144 \, {\left (-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c\right )}^{\frac {3}{2}} a^{2} c^{2} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) + 56 \, \sqrt {-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c} a b^{2} c^{2} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) - 112 \, \sqrt {-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c} a^{2} c^{3} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )}{{\left (\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}}\right )}^{2} c^{3} d^{4} {\left | c \right |}}\right )} {\left | c \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)^2,x, algorithm="giac")

[Out]

-1/512*d^2*(15*(b^4*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d) - 8*a*b^2*c*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d) + 16
*a^2*c^2*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d))*arctan(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(-c))/(sqrt(-c)*c^3*d^4*abs(c)) + 8*(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)*b^4*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d) - 8*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)*a*b^2*c*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d) + 16*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)*a^2*c^2*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d))/(c^4*d^4*abs(c)) - (9*(-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)^(3/2)*b^4*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d) - 72
*(-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)^(3/2)*a*b^2*c*sgn(1/(2*c*d*x + b*d))*sgn(c
)*sgn(d) - 7*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)*b^4*c*sgn(1/(2*c*d*x + b*d
))*sgn(c)*sgn(d) + 144*(-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)^(3/2)*a^2*c^2*sgn(1/
(2*c*d*x + b*d))*sgn(c)*sgn(d) + 56*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)*a*b
^2*c^2*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d) - 112*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)*a^2*c^3*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d))/((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)^2*c^3*d^4*abs(c)))*abs(c)

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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 )}^2} \,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)^2,x)

[Out]

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

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