Optimal. Leaf size=225 \[ -\frac {16 (b+2 c x) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )}{15 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}+\frac {2 (b+2 c x) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}} \]
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Rubi [A] time = 0.18, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {777, 614, 613} \begin {gather*} -\frac {16 (b+2 c x) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )}{15 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}+\frac {2 (b+2 c x) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 613
Rule 614
Rule 777
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx &=\frac {2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {\left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right ) \int \frac {1}{\left (a+b x+c x^2\right )^{5/2}} \, dx}{5 c \left (b^2-4 a c\right )}\\ &=\frac {2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 \left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right ) (b+2 c x)}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {\left (8 \left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right )\right ) \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{15 \left (b^2-4 a c\right )^2}\\ &=\frac {2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 \left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right ) (b+2 c x)}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {16 \left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right ) (b+2 c x)}{15 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 200, normalized size = 0.89 \begin {gather*} \frac {2 \left (-3 \left (b^2-4 a c\right )^2 (A c (-2 a e+b (d-e x)+2 c d x)+B (a b e-2 a c (d+e x)+b x (b e-c d)))+\left (b^2-4 a c\right ) (b+2 c x) (a+x (b+c x)) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )-8 c (b+2 c x) (a+x (b+c x))^2 \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )\right )}{15 c \left (b^2-4 a c\right )^3 (a+x (b+c x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 6.96, size = 587, normalized size = 2.61 \begin {gather*} -\frac {2 \left (-96 a^3 A c^2 e+96 a^3 b B c e-96 a^3 B c^2 d-48 a^2 A b^2 c e+240 a^2 A b c^2 d-240 a^2 A b c^2 e x+480 a^2 A c^3 d x+8 a^2 b^3 B e-48 a^2 b^2 B c d+240 a^2 b^2 B c e x-240 a^2 b B c^2 d x+240 a^2 b B c^2 e x^2+160 a^2 B c^3 e x^3+2 a A b^4 e-40 a A b^3 c d-120 a A b^3 c e x+240 a A b^2 c^2 d x-480 a A b^2 c^2 e x^2+960 a A b c^3 d x^2-320 a A b c^3 e x^3+640 a A c^4 d x^3+2 a b^4 B d+20 a b^4 B e x-120 a b^3 B c d x+200 a b^3 B c e x^2-480 a b^2 B c^2 d x^2+240 a b^2 B c^2 e x^3-320 a b B c^3 d x^3+160 a b B c^3 e x^4+64 a B c^4 e x^5+3 A b^5 d+5 A b^5 e x-10 A b^4 c d x-40 A b^4 c e x^2+80 A b^3 c^2 d x^2-240 A b^3 c^2 e x^3+480 A b^2 c^3 d x^3-320 A b^2 c^3 e x^4+640 A b c^4 d x^4-128 A b c^4 e x^5+256 A c^5 d x^5+5 b^5 B d x+15 b^5 B e x^2-40 b^4 B c d x^2+90 b^4 B c e x^3-240 b^3 B c^2 d x^3+120 b^3 B c^2 e x^4-320 b^2 B c^3 d x^4+48 b^2 B c^3 e x^5-128 b B c^4 d x^5\right )}{15 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 127.21, size = 805, normalized size = 3.58 \begin {gather*} \frac {2 \, {\left (16 \, {\left (8 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d - {\left (3 \, B b^{2} c^{3} + 4 \, {\left (B a - 2 \, A b\right )} c^{4}\right )} e\right )} x^{5} + 40 \, {\left (8 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d - {\left (3 \, B b^{3} c^{2} + 4 \, {\left (B a b - 2 \, A b^{2}\right )} c^{3}\right )} e\right )} x^{4} + 10 \, {\left (8 \, {\left (3 \, B b^{3} c^{2} - 8 \, A a c^{4} + 2 \, {\left (2 \, B a b - 3 \, A b^{2}\right )} c^{3}\right )} d - {\left (9 \, B b^{4} c + 16 \, {\left (B a^{2} - 2 \, A a b\right )} c^{3} + 24 \, {\left (B a b^{2} - A b^{3}\right )} c^{2}\right )} e\right )} x^{3} + 5 \, {\left (8 \, {\left (B b^{4} c - 24 \, A a b c^{3} + 2 \, {\left (6 \, B a b^{2} - A b^{3}\right )} c^{2}\right )} d - {\left (3 \, B b^{5} + 48 \, {\left (B a^{2} b - 2 \, A a b^{2}\right )} c^{2} + 8 \, {\left (5 \, B a b^{3} - A b^{4}\right )} c\right )} e\right )} x^{2} - {\left (2 \, B a b^{4} + 3 \, A b^{5} - 48 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} c^{2} - 8 \, {\left (6 \, B a^{2} b^{2} + 5 \, A a b^{3}\right )} c\right )} d - 2 \, {\left (4 \, B a^{2} b^{3} + A a b^{4} - 48 \, A a^{3} c^{2} + 24 \, {\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} c\right )} e - 5 \, {\left ({\left (B b^{5} + 96 \, A a^{2} c^{3} - 48 \, {\left (B a^{2} b - A a b^{2}\right )} c^{2} - 2 \, {\left (12 \, B a b^{3} + A b^{4}\right )} c\right )} d + {\left (4 \, B a b^{4} + A b^{5} - 48 \, A a^{2} b c^{2} + 24 \, {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} c\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{15 \, {\left (a^{3} b^{6} - 12 \, a^{4} b^{4} c + 48 \, a^{5} b^{2} c^{2} - 64 \, a^{6} c^{3} + {\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} x^{6} + 3 \, {\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} x^{5} + 3 \, {\left (b^{8} c - 11 \, a b^{6} c^{2} + 36 \, a^{2} b^{4} c^{3} - 16 \, a^{3} b^{2} c^{4} - 64 \, a^{4} c^{5}\right )} x^{4} + {\left (b^{9} - 6 \, a b^{7} c - 24 \, a^{2} b^{5} c^{2} + 224 \, a^{3} b^{3} c^{3} - 384 \, a^{4} b c^{4}\right )} x^{3} + 3 \, {\left (a b^{8} - 11 \, a^{2} b^{6} c + 36 \, a^{3} b^{4} c^{2} - 16 \, a^{4} b^{2} c^{3} - 64 \, a^{5} c^{4}\right )} x^{2} + 3 \, {\left (a^{2} b^{7} - 12 \, a^{3} b^{5} c + 48 \, a^{4} b^{3} c^{2} - 64 \, a^{5} b c^{3}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 727, normalized size = 3.23 \begin {gather*} \frac {2 \, {\left ({\left ({\left (2 \, {\left (4 \, {\left (\frac {2 \, {\left (8 \, B b c^{4} d - 16 \, A c^{5} d - 3 \, B b^{2} c^{3} e - 4 \, B a c^{4} e + 8 \, A b c^{4} e\right )} x}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}} + \frac {5 \, {\left (8 \, B b^{2} c^{3} d - 16 \, A b c^{4} d - 3 \, B b^{3} c^{2} e - 4 \, B a b c^{3} e + 8 \, A b^{2} c^{3} e\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x + \frac {5 \, {\left (24 \, B b^{3} c^{2} d + 32 \, B a b c^{3} d - 48 \, A b^{2} c^{3} d - 64 \, A a c^{4} d - 9 \, B b^{4} c e - 24 \, B a b^{2} c^{2} e + 24 \, A b^{3} c^{2} e - 16 \, B a^{2} c^{3} e + 32 \, A a b c^{3} e\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x + \frac {5 \, {\left (8 \, B b^{4} c d + 96 \, B a b^{2} c^{2} d - 16 \, A b^{3} c^{2} d - 192 \, A a b c^{3} d - 3 \, B b^{5} e - 40 \, B a b^{3} c e + 8 \, A b^{4} c e - 48 \, B a^{2} b c^{2} e + 96 \, A a b^{2} c^{2} e\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x - \frac {5 \, {\left (B b^{5} d - 24 \, B a b^{3} c d - 2 \, A b^{4} c d - 48 \, B a^{2} b c^{2} d + 48 \, A a b^{2} c^{2} d + 96 \, A a^{2} c^{3} d + 4 \, B a b^{4} e + A b^{5} e + 48 \, B a^{2} b^{2} c e - 24 \, A a b^{3} c e - 48 \, A a^{2} b c^{2} e\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x - \frac {2 \, B a b^{4} d + 3 \, A b^{5} d - 48 \, B a^{2} b^{2} c d - 40 \, A a b^{3} c d - 96 \, B a^{3} c^{2} d + 240 \, A a^{2} b c^{2} d + 8 \, B a^{2} b^{3} e + 2 \, A a b^{4} e + 96 \, B a^{3} b c e - 48 \, A a^{2} b^{2} c e - 96 \, A a^{3} c^{2} e}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )}}{15 \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 608, normalized size = 2.70 \begin {gather*} -\frac {2 \left (128 A b \,c^{4} e \,x^{5}-256 A \,c^{5} d \,x^{5}-64 B a \,c^{4} e \,x^{5}-48 B \,b^{2} c^{3} e \,x^{5}+128 B b \,c^{4} d \,x^{5}+320 A \,b^{2} c^{3} e \,x^{4}-640 A b \,c^{4} d \,x^{4}-160 B a b \,c^{3} e \,x^{4}-120 B \,b^{3} c^{2} e \,x^{4}+320 B \,b^{2} c^{3} d \,x^{4}+320 A a b \,c^{3} e \,x^{3}-640 A a \,c^{4} d \,x^{3}+240 A \,b^{3} c^{2} e \,x^{3}-480 A \,b^{2} c^{3} d \,x^{3}-160 B \,a^{2} c^{3} e \,x^{3}-240 B a \,b^{2} c^{2} e \,x^{3}+320 B a b \,c^{3} d \,x^{3}-90 B \,b^{4} c e \,x^{3}+240 B \,b^{3} c^{2} d \,x^{3}+480 A a \,b^{2} c^{2} e \,x^{2}-960 A a b \,c^{3} d \,x^{2}+40 A \,b^{4} c e \,x^{2}-80 A \,b^{3} c^{2} d \,x^{2}-240 B \,a^{2} b \,c^{2} e \,x^{2}-200 B a \,b^{3} c e \,x^{2}+480 B a \,b^{2} c^{2} d \,x^{2}-15 B \,b^{5} e \,x^{2}+40 B \,b^{4} c d \,x^{2}+240 A \,a^{2} b \,c^{2} e x -480 A \,a^{2} c^{3} d x +120 A a \,b^{3} c e x -240 A a \,b^{2} c^{2} d x -5 A \,b^{5} e x +10 A \,b^{4} c d x -240 B \,a^{2} b^{2} c e x +240 B \,a^{2} b \,c^{2} d x -20 B a \,b^{4} e x +120 B a \,b^{3} c d x -5 B \,b^{5} d x +96 A \,a^{3} c^{2} e +48 A \,a^{2} b^{2} c e -240 A \,a^{2} b \,c^{2} d -2 A a \,b^{4} e +40 A a \,b^{3} c d -3 A \,b^{5} d -96 B \,a^{3} b c e +96 B \,a^{3} c^{2} d -8 B \,a^{2} b^{3} e +48 B \,a^{2} b^{2} c d -2 B a \,b^{4} d \right )}{15 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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mupad [B] time = 3.26, size = 892, normalized size = 3.96 \begin {gather*} \frac {\frac {16\,B\,c^2\,e\,x}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}+\frac {8\,B\,b\,c\,e}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}}{\sqrt {c\,x^2+b\,x+a}}-\frac {x\,\left (\frac {b\,\left (\frac {2\,c^2\,\left (\frac {2\,A\,e}{5}+\frac {2\,B\,d}{5}\right )}{4\,a\,c^2-b^2\,c}-\frac {2\,B\,b\,c\,e}{5\,\left (4\,a\,c^2-b^2\,c\right )}\right )}{c}-\frac {b\,c\,\left (\frac {2\,A\,e}{5}+\frac {2\,B\,d}{5}\right )}{4\,a\,c^2-b^2\,c}-\frac {4\,A\,c^2\,d}{5\,\left (4\,a\,c^2-b^2\,c\right )}+\frac {4\,B\,a\,c\,e}{5\,\left (4\,a\,c^2-b^2\,c\right )}\right )+\frac {a\,\left (\frac {2\,c^2\,\left (\frac {2\,A\,e}{5}+\frac {2\,B\,d}{5}\right )}{4\,a\,c^2-b^2\,c}-\frac {2\,B\,b\,c\,e}{5\,\left (4\,a\,c^2-b^2\,c\right )}\right )}{c}-\frac {2\,A\,b\,c\,d}{5\,\left (4\,a\,c^2-b^2\,c\right )}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}}+\frac {x\,\left (\frac {b\,\left (\frac {16\,c^3\,\left (A\,e+B\,d\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}-\frac {8\,B\,b\,c^2\,e}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}\right )}{c}+\frac {2\,c\,\left (32\,A\,c^2\,d+8\,B\,b^2\,e-20\,A\,b\,c\,e+8\,B\,a\,c\,e-20\,B\,b\,c\,d\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}-\frac {8\,b\,c^2\,\left (A\,e+B\,d\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}+\frac {16\,B\,a\,c^2\,e}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}\right )+\frac {a\,\left (\frac {16\,c^3\,\left (A\,e+B\,d\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}-\frac {8\,B\,b\,c^2\,e}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}\right )}{c}+\frac {b\,\left (32\,A\,c^2\,d+8\,B\,b^2\,e-20\,A\,b\,c\,e+8\,B\,a\,c\,e-20\,B\,b\,c\,d\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}}+\frac {\frac {b\,c\,\left (256\,A\,c^2\,d+56\,B\,b^2\,e-128\,A\,b\,c\,e+32\,B\,a\,c\,e-128\,B\,b\,c\,d\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}+\frac {2\,c^2\,x\,\left (256\,A\,c^2\,d+56\,B\,b^2\,e-128\,A\,b\,c\,e+32\,B\,a\,c\,e-128\,B\,b\,c\,d\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}}{\sqrt {c\,x^2+b\,x+a}}-\frac {\frac {4\,A\,c\,e-2\,B\,b\,e+4\,B\,c\,d}{15\,c\,\left (4\,a\,c-b^2\right )}+\frac {4\,B\,e\,x}{15\,\left (4\,a\,c-b^2\right )}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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