Optimal. Leaf size=285 \[ -\frac {2 x \left (x \left (32 a^2 B c^2+16 a A b c^2-32 a b^2 B c-2 A b^3 c+5 b^4 B\right )+a \left (24 a A c^2-28 a b B c-2 A b^2 c+5 b^3 B\right )\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {\sqrt {a+b x+c x^2} \left (128 a^2 B c^2+40 a A b c^2-100 a b^2 B c-6 A b^3 c+15 b^4 B\right )}{3 c^3 \left (b^2-4 a c\right )^2}-\frac {2 x^3 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {(5 b B-2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{7/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.27, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {818, 640, 621, 206} \begin {gather*} -\frac {2 x \left (x \left (32 a^2 B c^2+16 a A b c^2-32 a b^2 B c-2 A b^3 c+5 b^4 B\right )+a \left (24 a A c^2-28 a b B c-2 A b^2 c+5 b^3 B\right )\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {\sqrt {a+b x+c x^2} \left (128 a^2 B c^2+40 a A b c^2-100 a b^2 B c-6 A b^3 c+15 b^4 B\right )}{3 c^3 \left (b^2-4 a c\right )^2}-\frac {2 x^3 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {(5 b B-2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 621
Rule 640
Rule 818
Rubi steps
\begin {align*} \int \frac {x^4 (A+B x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 x^3 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \int \frac {x^2 \left (3 a (b B-2 A c)+\frac {1}{2} \left (5 b^2 B-2 A b c-16 a B c\right ) x\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 c \left (b^2-4 a c\right )}\\ &=-\frac {2 x^3 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 x \left (a \left (5 b^3 B-2 A b^2 c-28 a b B c+24 a A c^2\right )+\left (5 b^4 B-2 A b^3 c-32 a b^2 B c+16 a A b c^2+32 a^2 B c^2\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {4 \int \frac {\frac {1}{2} a \left (5 b^3 B-2 A b^2 c-28 a b B c+24 a A c^2\right )+\frac {1}{4} \left (15 b^4 B-6 A b^3 c-100 a b^2 B c+40 a A b c^2+128 a^2 B c^2\right ) x}{\sqrt {a+b x+c x^2}} \, dx}{3 c^2 \left (b^2-4 a c\right )^2}\\ &=-\frac {2 x^3 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 x \left (a \left (5 b^3 B-2 A b^2 c-28 a b B c+24 a A c^2\right )+\left (5 b^4 B-2 A b^3 c-32 a b^2 B c+16 a A b c^2+32 a^2 B c^2\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {\left (15 b^4 B-6 A b^3 c-100 a b^2 B c+40 a A b c^2+128 a^2 B c^2\right ) \sqrt {a+b x+c x^2}}{3 c^3 \left (b^2-4 a c\right )^2}-\frac {(5 b B-2 A c) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 c^3}\\ &=-\frac {2 x^3 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 x \left (a \left (5 b^3 B-2 A b^2 c-28 a b B c+24 a A c^2\right )+\left (5 b^4 B-2 A b^3 c-32 a b^2 B c+16 a A b c^2+32 a^2 B c^2\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {\left (15 b^4 B-6 A b^3 c-100 a b^2 B c+40 a A b c^2+128 a^2 B c^2\right ) \sqrt {a+b x+c x^2}}{3 c^3 \left (b^2-4 a c\right )^2}-\frac {(5 b B-2 A c) \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 )}{c^3}\\ &=-\frac {2 x^3 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 x \left (a \left (5 b^3 B-2 A b^2 c-28 a b B c+24 a A c^2\right )+\left (5 b^4 B-2 A b^3 c-32 a b^2 B c+16 a A b c^2+32 a^2 B c^2\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {\left (15 b^4 B-6 A b^3 c-100 a b^2 B c+40 a A b c^2+128 a^2 B c^2\right ) \sqrt {a+b x+c x^2}}{3 c^3 \left (b^2-4 a c\right )^2}-\frac {(5 b B-2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.79, size = 414, normalized size = 1.45 \begin {gather*} \frac {2 \left (\frac {x^5 \left (A \left (16 a^2 c^2-28 a b^2 c-20 a b c^2 x+7 b^4+7 b^3 c x\right )+2 a B \left (16 a b c+12 a c^2 x-5 b^3-5 b^2 c x\right )\right )}{a \left (4 a c-b^2\right ) \sqrt {a+x (b+c x)}}+\frac {3 a^2 \left (b^2-4 a c\right )^2 (5 b B-2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-2 \sqrt {c} \sqrt {a+x (b+c x)} \left (128 a^4 B c^2-4 a^3 c \left (-2 b c (5 A+7 B x)+4 c^2 x (3 A+4 B x)+25 b^2 B\right )+a^2 \left (-2 b^3 c (3 A+5 B x)+4 b^2 c^2 x (A+2 B x)+16 b c^3 x^2 (A+B x)+16 c^4 x^3 (2 A+3 B x)+15 b^4 B\right )-4 a b c^3 x^3 (4 A b+10 A c x+5 b B x)+14 A b^3 c^3 x^4\right )}{4 a c^{7/2} \left (4 a c-b^2\right )}+\frac {x^5 \left (A \left (-2 a c+b^2+b c x\right )-a B (b+2 c x)\right )}{(a+x (b+c x))^{3/2}}\right )}{3 a \left (b^2-4 a c\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 2.35, size = 371, normalized size = 1.30 \begin {gather*} \frac {128 a^4 B c^2+40 a^3 A b c^2-48 a^3 A c^3 x-100 a^3 b^2 B c+312 a^3 b B c^2 x+192 a^3 B c^3 x^2-6 a^2 A b^3 c+84 a^2 A b^2 c^2 x-64 a^2 A c^4 x^3+15 a^2 b^4 B-210 a^2 b^3 B c x+48 a^2 b^2 B c^2 x^2+256 a^2 b B c^3 x^3+48 a^2 B c^4 x^4-12 a A b^4 c x+36 a A b^3 c^2 x^2+56 a A b^2 c^3 x^3+30 a b^5 B x-90 a b^4 B c x^2-148 a b^3 B c^2 x^3-24 a b^2 B c^3 x^4-6 A b^5 c x^2-8 A b^4 c^2 x^3+15 b^6 B x^2+20 b^5 B c x^3+3 b^4 B c^2 x^4}{3 c^3 \left (4 a c-b^2\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {(5 b B-2 A c) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{2 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.48, size = 1601, normalized size = 5.62
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.29, size = 458, normalized size = 1.61 \begin {gather*} \frac {{\left ({\left ({\left (\frac {3 \, {\left (B b^{4} c^{2} - 8 \, B a b^{2} c^{3} + 16 \, B a^{2} c^{4}\right )} x}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}} + \frac {4 \, {\left (5 \, B b^{5} c - 37 \, B a b^{3} c^{2} - 2 \, A b^{4} c^{2} + 64 \, B a^{2} b c^{3} + 14 \, A a b^{2} c^{3} - 16 \, A a^{2} c^{4}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x + \frac {3 \, {\left (5 \, B b^{6} - 30 \, B a b^{4} c - 2 \, A b^{5} c + 16 \, B a^{2} b^{2} c^{2} + 12 \, A a b^{3} c^{2} + 64 \, B a^{3} c^{3}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x + \frac {6 \, {\left (5 \, B a b^{5} - 35 \, B a^{2} b^{3} c - 2 \, A a b^{4} c + 52 \, B a^{3} b c^{2} + 14 \, A a^{2} b^{2} c^{2} - 8 \, A a^{3} c^{3}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x + \frac {15 \, B a^{2} b^{4} - 100 \, B a^{3} b^{2} c - 6 \, A a^{2} b^{3} c + 128 \, B a^{4} c^{2} + 40 \, A a^{3} b c^{2}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} + \frac {{\left (5 \, B b - 2 \, A c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{2 \, c^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 1262, normalized size = 4.43
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4\,\left (A+B\,x\right )}{{\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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________