Optimal. Leaf size=381 \[ -\frac {5 \left (-4 a A c-4 a b B+7 A b^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{9/2}}-\frac {2 \left (-A \left (40 a^2 c^2-42 a b^2 c+7 b^4\right )-c x \left (32 a^2 B c-36 a A b c-4 a b^2 B+7 A b^3\right )+4 a b B \left (b^2-6 a c\right )\right )}{3 a^2 x^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {\sqrt {a+b x+c x^2} \left (4 a B \left (128 a^2 c^2-100 a b^2 c+15 b^4\right )-A \left (1296 a^2 b c^2-760 a b^3 c+105 b^5\right )\right )}{12 a^4 x \left (b^2-4 a c\right )^2}+\frac {\sqrt {a+b x+c x^2} \left (4 a b B \left (5 b^2-28 a c\right )-A \left (240 a^2 c^2-216 a b^2 c+35 b^4\right )\right )}{6 a^3 x^2 \left (b^2-4 a c\right )^2}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{3 a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.46, antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {822, 834, 806, 724, 206} \[ \frac {\sqrt {a+b x+c x^2} \left (4 a b B \left (5 b^2-28 a c\right )-A \left (240 a^2 c^2-216 a b^2 c+35 b^4\right )\right )}{6 a^3 x^2 \left (b^2-4 a c\right )^2}-\frac {\sqrt {a+b x+c x^2} \left (4 a B \left (128 a^2 c^2-100 a b^2 c+15 b^4\right )-A \left (1296 a^2 b c^2-760 a b^3 c+105 b^5\right )\right )}{12 a^4 x \left (b^2-4 a c\right )^2}-\frac {2 \left (-c x \left (32 a^2 B c-36 a A b c-4 a b^2 B+7 A b^3\right )-A \left (40 a^2 c^2-42 a b^2 c+7 b^4\right )+4 a b B \left (b^2-6 a c\right )\right )}{3 a^2 x^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {5 \left (-4 a A c-4 a b B+7 A b^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{9/2}}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{3 a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 724
Rule 806
Rule 822
Rule 834
Rubi steps
\begin {align*} \int \frac {A+B x}{x^3 \left (a+b x+c x^2\right )^{5/2}} \, dx &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{3 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (-7 A b^2+4 a b B+20 a A c\right )-4 (A b-2 a B) c x}{x^3 \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 a \left (b^2-4 a c\right )}\\ &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{3 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (4 a b B \left (b^2-6 a c\right )-A \left (7 b^4-42 a b^2 c+40 a^2 c^2\right )-c \left (7 A b^3-4 a b^2 B-36 a A b c+32 a^2 B c\right ) x\right )}{3 a^2 \left (b^2-4 a c\right )^2 x^2 \sqrt {a+b x+c x^2}}+\frac {4 \int \frac {\frac {1}{4} \left (-4 a b B \left (5 b^2-28 a c\right )+4 A \left (\frac {35 b^4}{4}-54 a b^2 c+60 a^2 c^2\right )\right )-c \left (4 a B \left (b^2-8 a c\right )-A \left (7 b^3-36 a b c\right )\right ) x}{x^3 \sqrt {a+b x+c x^2}} \, dx}{3 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{3 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (4 a b B \left (b^2-6 a c\right )-A \left (7 b^4-42 a b^2 c+40 a^2 c^2\right )-c \left (7 A b^3-4 a b^2 B-36 a A b c+32 a^2 B c\right ) x\right )}{3 a^2 \left (b^2-4 a c\right )^2 x^2 \sqrt {a+b x+c x^2}}+\frac {\left (4 a b B \left (5 b^2-28 a c\right )-A \left (35 b^4-216 a b^2 c+240 a^2 c^2\right )\right ) \sqrt {a+b x+c x^2}}{6 a^3 \left (b^2-4 a c\right )^2 x^2}-\frac {2 \int \frac {\frac {1}{8} \left (105 A b^5-60 a b^4 B-760 a A b^3 c+400 a^2 b^2 B c+1296 a^2 A b c^2-512 a^3 B c^2\right )-\frac {1}{4} c \left (4 a b B \left (5 b^2-28 a c\right )-A \left (35 b^4-216 a b^2 c+240 a^2 c^2\right )\right ) x}{x^2 \sqrt {a+b x+c x^2}} \, dx}{3 a^3 \left (b^2-4 a c\right )^2}\\ &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{3 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (4 a b B \left (b^2-6 a c\right )-A \left (7 b^4-42 a b^2 c+40 a^2 c^2\right )-c \left (7 A b^3-4 a b^2 B-36 a A b c+32 a^2 B c\right ) x\right )}{3 a^2 \left (b^2-4 a c\right )^2 x^2 \sqrt {a+b x+c x^2}}+\frac {\left (4 a b B \left (5 b^2-28 a c\right )-A \left (35 b^4-216 a b^2 c+240 a^2 c^2\right )\right ) \sqrt {a+b x+c x^2}}{6 a^3 \left (b^2-4 a c\right )^2 x^2}-\frac {\left (4 a B \left (15 b^4-100 a b^2 c+128 a^2 c^2\right )-A \left (105 b^5-760 a b^3 c+1296 a^2 b c^2\right )\right ) \sqrt {a+b x+c x^2}}{12 a^4 \left (b^2-4 a c\right )^2 x}+\frac {\left (5 \left (7 A b^2-4 a b B-4 a A c\right )\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{8 a^4}\\ &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{3 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (4 a b B \left (b^2-6 a c\right )-A \left (7 b^4-42 a b^2 c+40 a^2 c^2\right )-c \left (7 A b^3-4 a b^2 B-36 a A b c+32 a^2 B c\right ) x\right )}{3 a^2 \left (b^2-4 a c\right )^2 x^2 \sqrt {a+b x+c x^2}}+\frac {\left (4 a b B \left (5 b^2-28 a c\right )-A \left (35 b^4-216 a b^2 c+240 a^2 c^2\right )\right ) \sqrt {a+b x+c x^2}}{6 a^3 \left (b^2-4 a c\right )^2 x^2}-\frac {\left (4 a B \left (15 b^4-100 a b^2 c+128 a^2 c^2\right )-A \left (105 b^5-760 a b^3 c+1296 a^2 b c^2\right )\right ) \sqrt {a+b x+c x^2}}{12 a^4 \left (b^2-4 a c\right )^2 x}-\frac {\left (5 \left (7 A b^2-4 a b B-4 a A c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{4 a^4}\\ &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{3 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (4 a b B \left (b^2-6 a c\right )-A \left (7 b^4-42 a b^2 c+40 a^2 c^2\right )-c \left (7 A b^3-4 a b^2 B-36 a A b c+32 a^2 B c\right ) x\right )}{3 a^2 \left (b^2-4 a c\right )^2 x^2 \sqrt {a+b x+c x^2}}+\frac {\left (4 a b B \left (5 b^2-28 a c\right )-A \left (35 b^4-216 a b^2 c+240 a^2 c^2\right )\right ) \sqrt {a+b x+c x^2}}{6 a^3 \left (b^2-4 a c\right )^2 x^2}-\frac {\left (4 a B \left (15 b^4-100 a b^2 c+128 a^2 c^2\right )-A \left (105 b^5-760 a b^3 c+1296 a^2 b c^2\right )\right ) \sqrt {a+b x+c x^2}}{12 a^4 \left (b^2-4 a c\right )^2 x}-\frac {5 \left (7 A b^2-4 a b B-4 a A c\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{9/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.73, size = 343, normalized size = 0.90 \[ \frac {-\frac {8 \left (A \left (40 a^2 c^2-42 a b^2 c-36 a b c^2 x+7 b^4+7 b^3 c x\right )+4 a B \left (6 a b c+8 a c^2 x-b^3-b^2 c x\right )\right )}{a \left (4 a c-b^2\right ) \sqrt {a+x (b+c x)}}+\frac {2 \sqrt {a} \sqrt {a+x (b+c x)} \left (32 a^3 c (15 A c+7 b B+16 B c x)-8 a^2 b \left (54 A b c+162 A c^2 x+5 b^2 B+50 b B c x\right )+10 a b^3 (7 A b+76 A c x+6 b B x)-105 A b^5 x\right )+15 x^2 \left (b^2-4 a c\right )^2 \left (-4 a A c-4 a b B+7 A b^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )}{2 a^{7/2} \left (4 a c-b^2\right )}+\frac {8 A \left (-2 a c+b^2+b c x\right )-8 a B (b+2 c x)}{(a+x (b+c x))^{3/2}}}{12 a x^2 \left (b^2-4 a c\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 21.40, size = 2057, normalized size = 5.40 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.28, size = 765, normalized size = 2.01 \[ -\frac {2 \, {\left ({\left ({\left (\frac {{\left (6 \, B a^{12} b^{4} c^{2} - 9 \, A a^{11} b^{5} c^{2} - 38 \, B a^{13} b^{2} c^{3} + 62 \, A a^{12} b^{3} c^{3} + 40 \, B a^{14} c^{4} - 96 \, A a^{13} b c^{4}\right )} x}{a^{15} b^{4} - 8 \, a^{16} b^{2} c + 16 \, a^{17} c^{2}} + \frac {3 \, {\left (4 \, B a^{12} b^{5} c - 6 \, A a^{11} b^{6} c - 27 \, B a^{13} b^{3} c^{2} + 44 \, A a^{12} b^{4} c^{2} + 36 \, B a^{14} b c^{3} - 80 \, A a^{13} b^{2} c^{3} + 16 \, A a^{14} c^{4}\right )}}{a^{15} b^{4} - 8 \, a^{16} b^{2} c + 16 \, a^{17} c^{2}}\right )} x + \frac {3 \, {\left (2 \, B a^{12} b^{6} - 3 \, A a^{11} b^{7} - 12 \, B a^{13} b^{4} c + 20 \, A a^{12} b^{5} c + 8 \, B a^{14} b^{2} c^{2} - 25 \, A a^{13} b^{3} c^{2} + 16 \, B a^{15} c^{3} - 20 \, A a^{14} b c^{3}\right )}}{a^{15} b^{4} - 8 \, a^{16} b^{2} c + 16 \, a^{17} c^{2}}\right )} x + \frac {7 \, B a^{13} b^{5} - 10 \, A a^{12} b^{6} - 50 \, B a^{14} b^{3} c + 78 \, A a^{13} b^{4} c + 80 \, B a^{15} b c^{2} - 162 \, A a^{14} b^{2} c^{2} + 56 \, A a^{15} c^{3}}{a^{15} b^{4} - 8 \, a^{16} b^{2} c + 16 \, a^{17} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} - \frac {5 \, {\left (4 \, B a b - 7 \, A b^{2} + 4 \, A a c\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{4}} + \frac {4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a b - 11 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A b^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a c + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{2} \sqrt {c} - 16 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a b \sqrt {c} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{2} b + 13 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a b^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} c - 8 \, B a^{3} \sqrt {c} + 24 \, A a^{2} b \sqrt {c}}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{2} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.08, size = 1051, normalized size = 2.76 \[ \frac {88 A b \,c^{3} x}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}\, a^{2}}-\frac {70 A \,b^{3} c^{2} x}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}\, a^{3}}-\frac {128 B \,c^{3} x}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}\, a}+\frac {40 B \,b^{2} c^{2} x}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}\, a^{2}}+\frac {44 A \,b^{2} c^{2}}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}\, a^{2}}+\frac {11 A b \,c^{2} x}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2}}-\frac {35 A \,b^{4} c}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}\, a^{3}}-\frac {35 A \,b^{3} c x}{12 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{3}}-\frac {64 B b \,c^{2}}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}\, a}-\frac {16 B \,c^{2} x}{3 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a}+\frac {20 B \,b^{3} c}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}\, a^{2}}+\frac {5 B \,b^{2} c x}{3 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2}}+\frac {11 A \,b^{2} c}{2 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2}}-\frac {35 A \,b^{4}}{24 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{3}}+\frac {5 A b \,c^{2} x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{3}}-\frac {35 A \,b^{3} c x}{4 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{4}}-\frac {8 B b c}{3 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a}+\frac {5 B \,b^{3}}{6 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2}}+\frac {5 B \,b^{2} c x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{3}}+\frac {5 A \,b^{2} c}{2 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{3}}-\frac {35 A \,b^{4}}{8 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{4}}+\frac {5 B \,b^{3}}{2 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{3}}-\frac {5 A c}{6 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2}}+\frac {35 A \,b^{2}}{24 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{3}}-\frac {5 B b}{6 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2}}+\frac {5 A c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {7}{2}}}-\frac {35 A \,b^{2} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{8 a^{\frac {9}{2}}}+\frac {5 B b \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {7}{2}}}+\frac {7 A b}{4 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2} x}-\frac {5 A c}{2 \sqrt {c \,x^{2}+b x +a}\, a^{3}}+\frac {35 A \,b^{2}}{8 \sqrt {c \,x^{2}+b x +a}\, a^{4}}-\frac {B}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a x}-\frac {5 B b}{2 \sqrt {c \,x^{2}+b x +a}\, a^{3}}-\frac {A}{2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {A+B\,x}{x^3\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________