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

Optimal. Leaf size=136 \[ 40 c^{3/2} d^6 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )+80 c^2 d^6 (b+2 c x) \sqrt {a+b x+c x^2}-\frac {40 c d^6 (b+2 c x)^3}{3 \sqrt {a+b x+c x^2}}-\frac {2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}} \]

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Rubi [A]  time = 0.08, antiderivative size = 136, 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 = {686, 692, 621, 206} \begin {gather*} 40 c^{3/2} d^6 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )+80 c^2 d^6 (b+2 c x) \sqrt {a+b x+c x^2}-\frac {40 c d^6 (b+2 c x)^3}{3 \sqrt {a+b x+c x^2}}-\frac {2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d^6*(b + 2*c*x)^5)/(3*(a + b*x + c*x^2)^(3/2)) - (40*c*d^6*(b + 2*c*x)^3)/(3*Sqrt[a + b*x + c*x^2]) + 80*c
^2*d^6*(b + 2*c*x)*Sqrt[a + b*x + c*x^2] + 40*c^(3/2)*(b^2 - 4*a*c)*d^6*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a
+ b*x + c*x^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 621

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 686

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*(d + e*x)^(m - 1)*
(a + b*x + c*x^2)^(p + 1))/(b*(p + 1)), x] - Dist[(d*e*(m - 1))/(b*(p + 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] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]

Rule 692

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

Rubi steps

\begin {align*} \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}}+\frac {1}{3} \left (20 c d^2\right ) \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac {2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {40 c d^6 (b+2 c x)^3}{3 \sqrt {a+b x+c x^2}}+\left (80 c^2 d^4\right ) \int \frac {(b d+2 c d x)^2}{\sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {40 c d^6 (b+2 c x)^3}{3 \sqrt {a+b x+c x^2}}+80 c^2 d^6 (b+2 c x) \sqrt {a+b x+c x^2}+\left (40 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {40 c d^6 (b+2 c x)^3}{3 \sqrt {a+b x+c x^2}}+80 c^2 d^6 (b+2 c x) \sqrt {a+b x+c x^2}+\left (80 c^2 \left (b^2-4 a c\right ) d^6\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )\\ &=-\frac {2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {40 c d^6 (b+2 c x)^3}{3 \sqrt {a+b x+c x^2}}+80 c^2 d^6 (b+2 c x) \sqrt {a+b x+c x^2}+40 c^{3/2} \left (b^2-4 a c\right ) d^6 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )\\ \end {align*}

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Mathematica [A]  time = 1.47, size = 204, normalized size = 1.50 \begin {gather*} \frac {d^6 \left (3 (b+2 c x)^5-\frac {5 \left (b^2-4 a c\right ) \left (\sqrt {4 a-\frac {b^2}{c}} (b+2 c x) \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}} \left (4 c \left (3 a+4 c x^2\right )+b^2+16 b c x\right )-24 c^{3/2} (a+x (b+c x))^2 \sinh ^{-1}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {4 a-\frac {b^2}{c}}}\right )\right )}{\sqrt {4 a-\frac {b^2}{c}} \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}}}\right )}{3 (a+x (b+c x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^6*(3*(b + 2*c*x)^5 - (5*(b^2 - 4*a*c)*(Sqrt[4*a - b^2/c]*(b + 2*c*x)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a
*c)]*(b^2 + 16*b*c*x + 4*c*(3*a + 4*c*x^2)) - 24*c^(3/2)*(a + x*(b + c*x))^2*ArcSinh[(b + 2*c*x)/(Sqrt[4*a - b
^2/c]*Sqrt[c])]))/(Sqrt[4*a - b^2/c]*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])))/(3*(a + x*(b + c*x))^(3/2))

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IntegrateAlgebraic [A]  time = 1.04, size = 214, normalized size = 1.57 \begin {gather*} -\frac {2 \left (-120 a^2 b c^2 d^6-240 a^2 c^3 d^6 x+20 a b^3 c d^6-120 a b^2 c^2 d^6 x-480 a b c^3 d^6 x^2-320 a c^4 d^6 x^3+b^5 d^6+30 b^4 c d^6 x+60 b^3 c^2 d^6 x^2-40 b^2 c^3 d^6 x^3-120 b c^4 d^6 x^4-48 c^5 d^6 x^5\right )}{3 \left (a+b x+c x^2\right )^{3/2}}-40 \left (b^2 c^{3/2} d^6-4 a c^{5/2} d^6\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b^5*d^6 + 20*a*b^3*c*d^6 - 120*a^2*b*c^2*d^6 + 30*b^4*c*d^6*x - 120*a*b^2*c^2*d^6*x - 240*a^2*c^3*d^6*x +
60*b^3*c^2*d^6*x^2 - 480*a*b*c^3*d^6*x^2 - 40*b^2*c^3*d^6*x^3 - 320*a*c^4*d^6*x^3 - 120*b*c^4*d^6*x^4 - 48*c^
5*d^6*x^5))/(3*(a + b*x + c*x^2)^(3/2)) - 40*(b^2*c^(3/2)*d^6 - 4*a*c^(5/2)*d^6)*Log[b + 2*c*x - 2*Sqrt[c]*Sqr
t[a + b*x + c*x^2]]

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fricas [B]  time = 1.12, size = 693, normalized size = 5.10 \begin {gather*} \left [-\frac {2 \, {\left (30 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{4} + 2 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{6} x^{3} + {\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x^{2} + 2 \, {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{6} x + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} d^{6}\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 ) - {\left (48 \, c^{5} d^{6} x^{5} + 120 \, b c^{4} d^{6} x^{4} + 40 \, {\left (b^{2} c^{3} + 8 \, a c^{4}\right )} d^{6} x^{3} - 60 \, {\left (b^{3} c^{2} - 8 \, a b c^{3}\right )} d^{6} x^{2} - 30 \, {\left (b^{4} c - 4 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x - {\left (b^{5} + 20 \, a b^{3} c - 120 \, a^{2} b c^{2}\right )} d^{6}\right )} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}, -\frac {2 \, {\left (60 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{4} + 2 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{6} x^{3} + {\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x^{2} + 2 \, {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{6} x + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} d^{6}\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 ) - {\left (48 \, c^{5} d^{6} x^{5} + 120 \, b c^{4} d^{6} x^{4} + 40 \, {\left (b^{2} c^{3} + 8 \, a c^{4}\right )} d^{6} x^{3} - 60 \, {\left (b^{3} c^{2} - 8 \, a b c^{3}\right )} d^{6} x^{2} - 30 \, {\left (b^{4} c - 4 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x - {\left (b^{5} + 20 \, a b^{3} c - 120 \, a^{2} b c^{2}\right )} d^{6}\right )} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-2/3*(30*((b^2*c^3 - 4*a*c^4)*d^6*x^4 + 2*(b^3*c^2 - 4*a*b*c^3)*d^6*x^3 + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d
^6*x^2 + 2*(a*b^3*c - 4*a^2*b*c^2)*d^6*x + (a^2*b^2*c - 4*a^3*c^2)*d^6)*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) - (48*c^5*d^6*x^5 + 120*b*c^4*d^6*x^4 + 40*(b^2*c^3 +
8*a*c^4)*d^6*x^3 - 60*(b^3*c^2 - 8*a*b*c^3)*d^6*x^2 - 30*(b^4*c - 4*a*b^2*c^2 - 8*a^2*c^3)*d^6*x - (b^5 + 20*a
*b^3*c - 120*a^2*b*c^2)*d^6)*sqrt(c*x^2 + b*x + a))/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2),
-2/3*(60*((b^2*c^3 - 4*a*c^4)*d^6*x^4 + 2*(b^3*c^2 - 4*a*b*c^3)*d^6*x^3 + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d
^6*x^2 + 2*(a*b^3*c - 4*a^2*b*c^2)*d^6*x + (a^2*b^2*c - 4*a^3*c^2)*d^6)*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)) - (48*c^5*d^6*x^5 + 120*b*c^4*d^6*x^4 + 40*(b^2*c^3 + 8*a*c^
4)*d^6*x^3 - 60*(b^3*c^2 - 8*a*b*c^3)*d^6*x^2 - 30*(b^4*c - 4*a*b^2*c^2 - 8*a^2*c^3)*d^6*x - (b^5 + 20*a*b^3*c
- 120*a^2*b*c^2)*d^6)*sqrt(c*x^2 + b*x + a))/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)]

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giac [B]  time = 0.31, size = 528, normalized size = 3.88 \begin {gather*} -\frac {40 \, {\left (b^{2} c^{2} d^{6} - 4 \, a c^{3} d^{6}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{\sqrt {c}} + \frac {2 \, {\left (2 \, {\left (2 \, {\left (2 \, {\left (3 \, {\left (\frac {2 \, {\left (b^{4} c^{8} d^{6} - 8 \, a b^{2} c^{9} d^{6} + 16 \, a^{2} c^{10} d^{6}\right )} x}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}} + \frac {5 \, {\left (b^{5} c^{7} d^{6} - 8 \, a b^{3} c^{8} d^{6} + 16 \, a^{2} b c^{9} d^{6}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x + \frac {5 \, {\left (b^{6} c^{6} d^{6} - 48 \, a^{2} b^{2} c^{8} d^{6} + 128 \, a^{3} c^{9} d^{6}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac {15 \, {\left (b^{7} c^{5} d^{6} - 16 \, a b^{5} c^{6} d^{6} + 80 \, a^{2} b^{3} c^{7} d^{6} - 128 \, a^{3} b c^{8} d^{6}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac {15 \, {\left (b^{8} c^{4} d^{6} - 12 \, a b^{6} c^{5} d^{6} + 40 \, a^{2} b^{4} c^{6} d^{6} - 128 \, a^{4} c^{8} d^{6}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac {b^{9} c^{3} d^{6} + 12 \, a b^{7} c^{4} d^{6} - 264 \, a^{2} b^{5} c^{5} d^{6} + 1280 \, a^{3} b^{3} c^{6} d^{6} - 1920 \, a^{4} b c^{7} d^{6}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-40*(b^2*c^2*d^6 - 4*a*c^3*d^6)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/sqrt(c) + 2/3*(2*
(2*(2*(3*(2*(b^4*c^8*d^6 - 8*a*b^2*c^9*d^6 + 16*a^2*c^10*d^6)*x/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5) + 5*(b^5*
c^7*d^6 - 8*a*b^3*c^8*d^6 + 16*a^2*b*c^9*d^6)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x + 5*(b^6*c^6*d^6 - 48*a^
2*b^2*c^8*d^6 + 128*a^3*c^9*d^6)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x - 15*(b^7*c^5*d^6 - 16*a*b^5*c^6*d^6
+ 80*a^2*b^3*c^7*d^6 - 128*a^3*b*c^8*d^6)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x - 15*(b^8*c^4*d^6 - 12*a*b^6
*c^5*d^6 + 40*a^2*b^4*c^6*d^6 - 128*a^4*c^8*d^6)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x - (b^9*c^3*d^6 + 12*a
*b^7*c^4*d^6 - 264*a^2*b^5*c^5*d^6 + 1280*a^3*b^3*c^6*d^6 - 1920*a^4*b*c^7*d^6)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^
2*c^5))/(c*x^2 + b*x + a)^(3/2)

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maple [B]  time = 0.07, size = 997, normalized size = 7.33 \begin {gather*} \frac {32 c^{5} d^{6} x^{5}}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {1920 a^{2} b^{2} c^{4} d^{6} x}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}-\frac {960 a \,b^{4} c^{3} d^{6} x}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}+\frac {120 b^{6} c^{2} d^{6} x}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}+\frac {80 b \,c^{4} d^{6} x^{4}}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {960 a^{2} b^{3} c^{3} d^{6}}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}+\frac {240 a^{2} b^{2} c^{3} d^{6} x}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {480 a \,b^{5} c^{2} d^{6}}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}-\frac {120 a \,b^{4} c^{2} d^{6} x}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {160 a \,c^{4} d^{6} x^{3}}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {60 b^{7} c \,d^{6}}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}+\frac {15 b^{6} c \,d^{6} x}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {40 b^{2} c^{3} d^{6} x^{3}}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {120 a^{2} b^{3} c^{2} d^{6}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {60 a \,b^{5} c \,d^{6}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {160 a \,b^{2} c^{3} d^{6} x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {240 a b \,c^{3} d^{6} x^{2}}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {15 b^{7} d^{6}}{2 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {40 b^{4} c^{2} d^{6} x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {140 b^{3} c^{2} d^{6} x^{2}}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {80 a \,b^{3} c^{2} d^{6}}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {60 a \,b^{2} c^{2} d^{6} x}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {20 b^{5} c \,d^{6}}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {65 b^{4} c \,d^{6} x}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {160 a^{2} b \,c^{2} d^{6}}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {310 a \,b^{3} c \,d^{6}}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {160 a \,c^{3} d^{6} x}{\sqrt {c \,x^{2}+b x +a}}+\frac {41 b^{5} d^{6}}{6 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {40 b^{2} c^{2} d^{6} x}{\sqrt {c \,x^{2}+b x +a}}-160 a \,c^{\frac {5}{2}} d^{6} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )+40 b^{2} c^{\frac {3}{2}} d^{6} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )-\frac {80 a b \,c^{2} d^{6}}{\sqrt {c \,x^{2}+b x +a}}+\frac {20 b^{3} c \,d^{6}}{\sqrt {c \,x^{2}+b x +a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15*d^6*c*b^6/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-480*d^6*c^2*b^5*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+60*d^6*c^2*
b^2*a*x/(c*x^2+b*x+a)^(3/2)+120*d^6*c^2*b^3*a^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+120*d^6*c^2*b^6/(4*a*c-b^2)^2/
(c*x^2+b*x+a)^(1/2)*x-60*d^6*c*b^5*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+960*d^6*c^3*b^3*a^2/(4*a*c-b^2)^2/(c*x^2+
b*x+a)^(1/2)+40*d^6*c^2*b^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-80*d^6*c^2*a*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)
+240*d^6*c^3*b*a*x^2/(c*x^2+b*x+a)^(3/2)-40*d^6*c^2*b^2*x/(c*x^2+b*x+a)^(1/2)+32*d^6*c^5*x^5/(c*x^2+b*x+a)^(3/
2)-160*d^6*c^(5/2)*a*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+40*d^6*c^(3/2)*b^2*ln((c*x+1/2*b)/c^(1/2)+(c*
x^2+b*x+a)^(1/2))+15/2*d^6*b^7/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+20*d^6*c*b^3/(c*x^2+b*x+a)^(1/2)-120*d^6*c^2*b^
4*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-160*d^6*c^3*a*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+240*d^6*c^3*b^2*a^2/
(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+1920*d^6*c^4*b^2*a^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-960*d^6*c^3*b^4*a/(
4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-40/3*d^6*c^3*b^2*x^3/(c*x^2+b*x+a)^(3/2)+160/3*d^6*c^4*a*x^3/(c*x^2+b*x+a)^
(3/2)+160*d^6*c^3*a*x/(c*x^2+b*x+a)^(1/2)-80*d^6*c^2*a*b/(c*x^2+b*x+a)^(1/2)+160*d^6*c^2*b*a^2/(c*x^2+b*x+a)^(
3/2)+80*d^6*c^4*b*x^4/(c*x^2+b*x+a)^(3/2)+20*d^6*c*b^5/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-140*d^6*c^2*b^3*x^2/(c*
x^2+b*x+a)^(3/2)-65*d^6*c*b^4*x/(c*x^2+b*x+a)^(3/2)+41/6*d^6*b^5/(c*x^2+b*x+a)^(3/2)+60*d^6*c*b^7/(4*a*c-b^2)^
2/(c*x^2+b*x+a)^(1/2)-310/3*d^6*c*b^3*a/(c*x^2+b*x+a)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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