Optimal. Leaf size=198 \[ \frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (-3 a B e+A b e+2 b B d)}{8 b^4}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e) (-3 a B e+2 A b e+b B d)}{7 b^4}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)^2}{6 b^4}+\frac {B e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^4} \]
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Rubi [A] time = 0.34, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {770, 77} \begin {gather*} \frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (-3 a B e+A b e+2 b B d)}{8 b^4}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e) (-3 a B e+2 A b e+b B d)}{7 b^4}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)^2}{6 b^4}+\frac {B e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^5 (A+B x) (d+e x)^2 \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {(A b-a B) (b d-a e)^2 \left (a b+b^2 x\right )^5}{b^3}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) \left (a b+b^2 x\right )^6}{b^4}+\frac {e (2 b B d+A b e-3 a B e) \left (a b+b^2 x\right )^7}{b^5}+\frac {B e^2 \left (a b+b^2 x\right )^8}{b^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {(A b-a B) (b d-a e)^2 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^4}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^4}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^4}+\frac {B e^2 (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^4}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 347, normalized size = 1.75 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (42 a^5 \left (4 A \left (3 d^2+3 d e x+e^2 x^2\right )+B x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )+42 a^4 b x \left (5 A \left (6 d^2+8 d e x+3 e^2 x^2\right )+2 B x \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )+84 a^3 b^2 x^2 \left (2 A \left (10 d^2+15 d e x+6 e^2 x^2\right )+B x \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )+12 a^2 b^3 x^3 \left (7 A \left (15 d^2+24 d e x+10 e^2 x^2\right )+4 B x \left (21 d^2+35 d e x+15 e^2 x^2\right )\right )+3 a b^4 x^4 \left (8 A \left (21 d^2+35 d e x+15 e^2 x^2\right )+5 B x \left (28 d^2+48 d e x+21 e^2 x^2\right )\right )+b^5 x^5 \left (3 A \left (28 d^2+48 d e x+21 e^2 x^2\right )+2 B x \left (36 d^2+63 d e x+28 e^2 x^2\right )\right )\right )}{504 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 4.57, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.41, size = 384, normalized size = 1.94 \begin {gather*} \frac {1}{9} \, B b^{5} e^{2} x^{9} + A a^{5} d^{2} x + \frac {1}{8} \, {\left (2 \, B b^{5} d e + {\left (5 \, B a b^{4} + A b^{5}\right )} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (B b^{5} d^{2} + 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left ({\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} + 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{2}\right )} x^{6} + {\left ({\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} + 4 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e + {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} + 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e + {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{5} e^{2} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} + 2 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{5} d e + {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{2}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 679, normalized size = 3.43 \begin {gather*} \frac {1}{9} \, B b^{5} x^{9} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, B b^{5} d x^{8} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, B b^{5} d^{2} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{8} \, B a b^{4} x^{8} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{8} \, A b^{5} x^{8} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, B a b^{4} d x^{7} e \mathrm {sgn}\left (b x + a\right ) + \frac {2}{7} \, A b^{5} d x^{7} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{6} \, B a b^{4} d^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, A b^{5} d^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, B a^{2} b^{3} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{7} \, A a b^{4} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, B a^{2} b^{3} d x^{6} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, A a b^{4} d x^{6} e \mathrm {sgn}\left (b x + a\right ) + 2 \, B a^{2} b^{3} d^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + A a b^{4} d^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, B a^{3} b^{2} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, A a^{2} b^{3} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, B a^{3} b^{2} d x^{5} e \mathrm {sgn}\left (b x + a\right ) + 4 \, A a^{2} b^{3} d x^{5} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, B a^{3} b^{2} d^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, A a^{2} b^{3} d^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + B a^{4} b x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{3} b^{2} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, B a^{4} b d x^{4} e \mathrm {sgn}\left (b x + a\right ) + 5 \, A a^{3} b^{2} d x^{4} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, B a^{4} b d^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, A a^{3} b^{2} d^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, B a^{5} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, A a^{4} b x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, B a^{5} d x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, A a^{4} b d x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B a^{5} d^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, A a^{4} b d^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, A a^{5} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + A a^{5} d x^{2} e \mathrm {sgn}\left (b x + a\right ) + A a^{5} d^{2} x \mathrm {sgn}\left (b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 480, normalized size = 2.42 \begin {gather*} \frac {\left (56 B \,e^{2} b^{5} x^{8}+63 x^{7} A \,b^{5} e^{2}+315 x^{7} B \,e^{2} a \,b^{4}+126 x^{7} B \,b^{5} d e +360 x^{6} A a \,b^{4} e^{2}+144 x^{6} A \,b^{5} d e +720 x^{6} B \,e^{2} a^{2} b^{3}+720 x^{6} B a \,b^{4} d e +72 x^{6} B \,b^{5} d^{2}+840 x^{5} A \,a^{2} b^{3} e^{2}+840 x^{5} A a \,b^{4} d e +84 x^{5} A \,d^{2} b^{5}+840 x^{5} B \,e^{2} a^{3} b^{2}+1680 x^{5} B \,a^{2} b^{3} d e +420 x^{5} B a \,b^{4} d^{2}+1008 A \,a^{3} b^{2} e^{2} x^{4}+2016 A \,a^{2} b^{3} d e \,x^{4}+504 A a \,b^{4} d^{2} x^{4}+504 B \,a^{4} b \,e^{2} x^{4}+2016 B \,a^{3} b^{2} d e \,x^{4}+1008 B \,a^{2} b^{3} d^{2} x^{4}+630 x^{3} A \,a^{4} b \,e^{2}+2520 x^{3} A \,a^{3} b^{2} d e +1260 x^{3} A \,d^{2} a^{2} b^{3}+126 x^{3} B \,e^{2} a^{5}+1260 x^{3} B \,a^{4} b d e +1260 x^{3} B \,a^{3} b^{2} d^{2}+168 x^{2} A \,a^{5} e^{2}+1680 x^{2} A \,a^{4} b d e +1680 x^{2} A \,d^{2} a^{3} b^{2}+336 x^{2} B \,a^{5} d e +840 x^{2} B \,a^{4} b \,d^{2}+504 x A \,a^{5} d e +1260 x A \,d^{2} a^{4} b +252 x B \,a^{5} d^{2}+504 A \,d^{2} a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{504 \left (b x +a \right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.61, size = 456, normalized size = 2.30 \begin {gather*} \frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A d^{2} x - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{3} e^{2} x}{6 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B e^{2} x^{2}}{9 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a d^{2}}{6 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{4} e^{2}}{6 \, b^{4}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a e^{2} x}{72 \, b^{3}} + \frac {83 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{2} e^{2}}{504 \, b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (2 \, B d e + A e^{2}\right )} a^{2} x}{6 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (B d^{2} + 2 \, A d e\right )} a x}{6 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (2 \, B d e + A e^{2}\right )} a^{3}}{6 \, b^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (B d^{2} + 2 \, A d e\right )} a^{2}}{6 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (2 \, B d e + A e^{2}\right )} x}{8 \, b^{2}} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (2 \, B d e + A e^{2}\right )} a}{56 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (B d^{2} + 2 \, A d e\right )}}{7 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,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 (A + B x\right ) \left (d + e x\right )^{2} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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