Optimal. Leaf size=423 \[ \frac {\left (b x+c x^2\right )^{7/2} \left (14 c e x (18 A c e-11 b B e+4 B c d)+18 A c e (32 c d-9 b e)+B \left (99 b^2 e^2-324 b c d e+64 c^2 d^2\right )\right )}{2016 c^3}-\frac {5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{12288 c^5}+\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{768 c^4}-\frac {5 b^6 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{32768 c^{13/2}}+\frac {5 b^4 (b+2 c x) \sqrt {b x+c x^2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{32768 c^6}+\frac {B \left (b x+c x^2\right )^{7/2} (d+e x)^2}{9 c} \]
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Rubi [A] time = 0.43, antiderivative size = 423, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {832, 779, 612, 620, 206} \begin {gather*} \frac {5 b^4 (b+2 c x) \sqrt {b x+c x^2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{32768 c^6}-\frac {5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{12288 c^5}+\frac {\left (b x+c x^2\right )^{7/2} \left (14 c e x (18 A c e-11 b B e+4 B c d)+18 A c e (32 c d-9 b e)+B \left (99 b^2 e^2-324 b c d e+64 c^2 d^2\right )\right )}{2016 c^3}+\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{768 c^4}-\frac {5 b^6 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{32768 c^{13/2}}+\frac {B \left (b x+c x^2\right )^{7/2} (d+e x)^2}{9 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rule 779
Rule 832
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^{5/2} \, dx &=\frac {B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {\int (d+e x) \left (-\frac {1}{2} (7 b B-18 A c) d+\frac {1}{2} (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{5/2} \, dx}{9 c}\\ &=\frac {B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (18 A c e (32 c d-9 b e)+B \left (64 c^2 d^2-324 b c d e+99 b^2 e^2\right )+14 c e (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}+\frac {\left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) \int \left (b x+c x^2\right )^{5/2} \, dx}{64 c^3}\\ &=\frac {\left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (18 A c e (32 c d-9 b e)+B \left (64 c^2 d^2-324 b c d e+99 b^2 e^2\right )+14 c e (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {\left (5 b^2 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right )\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{1536 c^4}\\ &=-\frac {5 b^2 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{12288 c^5}+\frac {\left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (18 A c e (32 c d-9 b e)+B \left (64 c^2 d^2-324 b c d e+99 b^2 e^2\right )+14 c e (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}+\frac {\left (5 b^4 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right )\right ) \int \sqrt {b x+c x^2} \, dx}{8192 c^5}\\ &=\frac {5 b^4 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{32768 c^6}-\frac {5 b^2 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{12288 c^5}+\frac {\left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (18 A c e (32 c d-9 b e)+B \left (64 c^2 d^2-324 b c d e+99 b^2 e^2\right )+14 c e (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {\left (5 b^6 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{65536 c^6}\\ &=\frac {5 b^4 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{32768 c^6}-\frac {5 b^2 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{12288 c^5}+\frac {\left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (18 A c e (32 c d-9 b e)+B \left (64 c^2 d^2-324 b c d e+99 b^2 e^2\right )+14 c e (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {\left (5 b^6 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{32768 c^6}\\ &=\frac {5 b^4 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{32768 c^6}-\frac {5 b^2 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{12288 c^5}+\frac {\left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (18 A c e (32 c d-9 b e)+B \left (64 c^2 d^2-324 b c d e+99 b^2 e^2\right )+14 c e (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {5 b^6 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{32768 c^{13/2}}\\ \end {align*}
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Mathematica [A] time = 1.45, size = 417, normalized size = 0.99 \begin {gather*} \frac {(x (b+c x))^{7/2} \left (\frac {1323 A \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right ) \left (\sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \left (15 b^5-10 b^4 c x+8 b^3 c^2 x^2+432 b^2 c^3 x^3+640 b c^4 x^4+256 c^5 x^5\right )-15 b^{11/2} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )\right )}{1024 c^{9/2} x^{7/2} \sqrt {\frac {c x}{b}+1}}+\frac {5103 A e (b+c x)^3 (2 c d-b e)}{c}+7938 A e (b+c x)^3 (d+e x)+\frac {189 B \left (11 b^2 e^2-36 b c d e+32 c^2 d^2\right ) \left (105 b^{13/2} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )+\sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \left (-105 b^6+70 b^5 c x-56 b^4 c^2 x^2+48 b^3 c^3 x^3+4736 b^2 c^4 x^4+7424 b c^5 x^5+3072 c^6 x^6\right )\right )}{2048 c^{11/2} x^{7/2} \sqrt {\frac {c x}{b}+1}}+\frac {441 B e x (b+c x)^3 (20 c d-11 b e)}{c}+7056 B e x (b+c x)^3 (d+e x)\right )}{63504 c (b+c x)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.28, size = 713, normalized size = 1.69 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (5670 A b^7 c e^2-20160 A b^6 c^2 d e-3780 A b^6 c^2 e^2 x+20160 A b^5 c^3 d^2+13440 A b^5 c^3 d e x+3024 A b^5 c^3 e^2 x^2-13440 A b^4 c^4 d^2 x-10752 A b^4 c^4 d e x^2-2592 A b^4 c^4 e^2 x^3+10752 A b^3 c^5 d^2 x^2+9216 A b^3 c^5 d e x^3+2304 A b^3 c^5 e^2 x^4+580608 A b^2 c^6 d^2 x^3+909312 A b^2 c^6 d e x^4+373248 A b^2 c^6 e^2 x^5+860160 A b c^7 d^2 x^4+1425408 A b c^7 d e x^5+608256 A b c^7 e^2 x^6+344064 A c^8 d^2 x^5+589824 A c^8 d e x^6+258048 A c^8 e^2 x^7-3465 b^8 B e^2+11340 b^7 B c d e+2310 b^7 B c e^2 x-10080 b^6 B c^2 d^2-7560 b^6 B c^2 d e x-1848 b^6 B c^2 e^2 x^2+6720 b^5 B c^3 d^2 x+6048 b^5 B c^3 d e x^2+1584 b^5 B c^3 e^2 x^3-5376 b^4 B c^4 d^2 x^2-5184 b^4 B c^4 d e x^3-1408 b^4 B c^4 e^2 x^4+4608 b^3 B c^5 d^2 x^3+4608 b^3 B c^5 d e x^4+1280 b^3 B c^5 e^2 x^5+454656 b^2 B c^6 d^2 x^4+746496 b^2 B c^6 d e x^5+316416 b^2 B c^6 e^2 x^6+712704 b B c^7 d^2 x^5+1216512 b B c^7 d e x^6+530432 b B c^7 e^2 x^7+294912 B c^8 d^2 x^6+516096 B c^8 d e x^7+229376 B c^8 e^2 x^8\right )}{2064384 c^6}-\frac {5 \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right ) \left (-18 A b^8 c e^2+64 A b^7 c^2 d e-64 A b^6 c^3 d^2+11 b^9 B e^2-36 b^8 B c d e+32 b^7 B c^2 d^2\right )}{65536 c^{13/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 1292, normalized size = 3.05
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 681, normalized size = 1.61 \begin {gather*} \frac {1}{2064384} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (14 \, {\left (16 \, B c^{2} x e^{2} + \frac {36 \, B c^{10} d e + 37 \, B b c^{9} e^{2} + 18 \, A c^{10} e^{2}}{c^{8}}\right )} x + \frac {3 \, {\left (96 \, B c^{10} d^{2} + 396 \, B b c^{9} d e + 192 \, A c^{10} d e + 103 \, B b^{2} c^{8} e^{2} + 198 \, A b c^{9} e^{2}\right )}}{c^{8}}\right )} x + \frac {2784 \, B b c^{9} d^{2} + 1344 \, A c^{10} d^{2} + 2916 \, B b^{2} c^{8} d e + 5568 \, A b c^{9} d e + 5 \, B b^{3} c^{7} e^{2} + 1458 \, A b^{2} c^{8} e^{2}}{c^{8}}\right )} x + \frac {3552 \, B b^{2} c^{8} d^{2} + 6720 \, A b c^{9} d^{2} + 36 \, B b^{3} c^{7} d e + 7104 \, A b^{2} c^{8} d e - 11 \, B b^{4} c^{6} e^{2} + 18 \, A b^{3} c^{7} e^{2}}{c^{8}}\right )} x + \frac {9 \, {\left (32 \, B b^{3} c^{7} d^{2} + 4032 \, A b^{2} c^{8} d^{2} - 36 \, B b^{4} c^{6} d e + 64 \, A b^{3} c^{7} d e + 11 \, B b^{5} c^{5} e^{2} - 18 \, A b^{4} c^{6} e^{2}\right )}}{c^{8}}\right )} x - \frac {21 \, {\left (32 \, B b^{4} c^{6} d^{2} - 64 \, A b^{3} c^{7} d^{2} - 36 \, B b^{5} c^{5} d e + 64 \, A b^{4} c^{6} d e + 11 \, B b^{6} c^{4} e^{2} - 18 \, A b^{5} c^{5} e^{2}\right )}}{c^{8}}\right )} x + \frac {105 \, {\left (32 \, B b^{5} c^{5} d^{2} - 64 \, A b^{4} c^{6} d^{2} - 36 \, B b^{6} c^{4} d e + 64 \, A b^{5} c^{5} d e + 11 \, B b^{7} c^{3} e^{2} - 18 \, A b^{6} c^{4} e^{2}\right )}}{c^{8}}\right )} x - \frac {315 \, {\left (32 \, B b^{6} c^{4} d^{2} - 64 \, A b^{5} c^{5} d^{2} - 36 \, B b^{7} c^{3} d e + 64 \, A b^{6} c^{4} d e + 11 \, B b^{8} c^{2} e^{2} - 18 \, A b^{7} c^{3} e^{2}\right )}}{c^{8}}\right )} - \frac {5 \, {\left (32 \, B b^{7} c^{2} d^{2} - 64 \, A b^{6} c^{3} d^{2} - 36 \, B b^{8} c d e + 64 \, A b^{7} c^{2} d e + 11 \, B b^{9} e^{2} - 18 \, A b^{8} c e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{65536 \, c^{\frac {13}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 1227, normalized size = 2.90
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 944, normalized size = 2.23
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right )\,{\left (d+e\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right ) \left (d + e x\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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