Optimal. Leaf size=135 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (-2 a B e+A b e+b B d)}{7 b^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)}{6 b^3}+\frac {B e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^3} \]
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Rubi [A] time = 0.20, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {770, 77} \begin {gather*} \frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (-2 a B e+A b e+b B d)}{7 b^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)}{6 b^3}+\frac {B e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^3} \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) \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) \, 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) \left (a b+b^2 x\right )^5}{b^2}+\frac {(b B d+A b e-2 a B e) \left (a b+b^2 x\right )^6}{b^3}+\frac {B e \left (a b+b^2 x\right )^7}{b^4}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {(A b-a B) (b d-a e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^3}+\frac {(b B d+A b e-2 a B e) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^3}+\frac {B e (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^3}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 214, normalized size = 1.59 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (28 a^5 (3 A (2 d+e x)+B x (3 d+2 e x))+70 a^4 b x (A (6 d+4 e x)+B x (4 d+3 e x))+28 a^3 b^2 x^2 (5 A (4 d+3 e x)+3 B x (5 d+4 e x))+28 a^2 b^3 x^3 (3 A (5 d+4 e x)+2 B x (6 d+5 e x))+4 a b^4 x^4 (7 A (6 d+5 e x)+5 B x (7 d+6 e x))+b^5 x^5 (4 A (7 d+6 e x)+3 B x (8 d+7 e x))\right )}{168 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 2.92, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x) \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 = 239, normalized size = 1.77 \begin {gather*} \frac {1}{8} \, B b^{5} e x^{8} + A a^{5} d x + \frac {1}{7} \, {\left (B b^{5} d + {\left (5 \, B a b^{4} + A b^{5}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left ({\left (5 \, B a b^{4} + A b^{5}\right )} d + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e\right )} x^{6} + {\left ({\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d + 2 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e\right )} x^{5} + \frac {5}{4} \, {\left (2 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d + {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d + {\left (B a^{5} + 5 \, A a^{4} b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{5} e + {\left (B a^{5} + 5 \, A a^{4} b\right )} d\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 425, normalized size = 3.15 \begin {gather*} \frac {1}{8} \, B b^{5} x^{8} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, B b^{5} d x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{7} \, B a b^{4} x^{7} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, A b^{5} x^{7} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{6} \, B a b^{4} d x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, A b^{5} d x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, B a^{2} b^{3} x^{6} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{6} \, A a b^{4} x^{6} e \mathrm {sgn}\left (b x + a\right ) + 2 \, B a^{2} b^{3} d x^{5} \mathrm {sgn}\left (b x + a\right ) + A a b^{4} d x^{5} \mathrm {sgn}\left (b x + a\right ) + 2 \, B a^{3} b^{2} x^{5} e \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{2} b^{3} x^{5} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, B a^{3} b^{2} d x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, A a^{2} b^{3} d x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, B a^{4} b x^{4} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, A a^{3} b^{2} x^{4} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, B a^{4} b d x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, A a^{3} b^{2} d x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, B a^{5} x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, A a^{4} b x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B a^{5} d x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, A a^{4} b d x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, A a^{5} x^{2} e \mathrm {sgn}\left (b x + a\right ) + A a^{5} d 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 = 284, normalized size = 2.10 \begin {gather*} \frac {\left (21 B e \,b^{5} x^{7}+24 x^{6} A \,b^{5} e +120 x^{6} B e a \,b^{4}+24 x^{6} B \,b^{5} d +140 x^{5} A a \,b^{4} e +28 x^{5} A d \,b^{5}+280 x^{5} B e \,a^{2} b^{3}+140 x^{5} B a \,b^{4} d +336 A \,a^{2} b^{3} e \,x^{4}+168 A a \,b^{4} d \,x^{4}+336 B \,a^{3} b^{2} e \,x^{4}+336 B \,a^{2} b^{3} d \,x^{4}+420 x^{3} A \,a^{3} b^{2} e +420 x^{3} A d \,a^{2} b^{3}+210 x^{3} B e \,a^{4} b +420 x^{3} B \,a^{3} b^{2} d +280 x^{2} A \,a^{4} b e +560 x^{2} A d \,a^{3} b^{2}+56 x^{2} B e \,a^{5}+280 x^{2} B \,a^{4} b d +84 x A \,a^{5} e +420 x A d \,a^{4} b +84 x B \,a^{5} d +168 A d \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{168 \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 = 0.57, size = 254, normalized size = 1.88 \begin {gather*} \frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A d x + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{2} e x}{6 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a d}{6 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{3} e}{6 \, b^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (B d + A e\right )} a x}{6 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B e x}{8 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (B d + A e\right )} a^{2}}{6 \, b^{2}} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a e}{56 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (B d + A 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 )\,{\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 ) \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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