Optimal. Leaf size=259 \[ \frac {e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (-4 a B e+A b e+3 b B d)}{7 b^5}+\frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e) (-2 a B e+A b e+b B d)}{2 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{5 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3 (A b-a B) (b d-a e)^3}{4 b^5}+\frac {B e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^5} \]
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Rubi [A] time = 0.32, antiderivative size = 259, 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^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (-4 a B e+A b e+3 b B d)}{7 b^5}+\frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e) (-2 a B e+A b e+b B d)}{2 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{5 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3 (A b-a B) (b d-a e)^3}{4 b^5}+\frac {B e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^5} \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)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 (A+B x) (d+e x)^3 \, dx}{b^2 \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)^3 \left (a b+b^2 x\right )^3}{b^4}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e) \left (a b+b^2 x\right )^4}{b^5}+\frac {3 e (b d-a e) (b B d+A b e-2 a B e) \left (a b+b^2 x\right )^5}{b^6}+\frac {e^2 (3 b B d+A b e-4 a B e) \left (a b+b^2 x\right )^6}{b^7}+\frac {B e^3 \left (a b+b^2 x\right )^7}{b^8}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {(A b-a B) (b d-a e)^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^5}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 b^5}+\frac {e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^5}+\frac {e^2 (3 b B d+A b e-4 a B e) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^5}+\frac {B e^3 (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^5}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 320, normalized size = 1.24 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (14 a^3 \left (5 A \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+B x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right )+14 a^2 b x \left (3 A \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+B x \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )\right )+2 a b^2 x^2 \left (7 A \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+3 B x \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )\right )+b^3 x^3 \left (2 A \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+B x \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )\right )\right )}{280 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 3.91, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.43, size = 325, normalized size = 1.25 \begin {gather*} \frac {1}{8} \, B b^{3} e^{3} x^{8} + A a^{3} d^{3} x + \frac {1}{7} \, {\left (3 \, B b^{3} d e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (B b^{3} d^{2} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{2} + {\left (B a^{2} b + A a b^{2}\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{3} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (A a^{3} e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e + 3 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{2}\right )} x^{4} + {\left (A a^{3} d e^{2} + {\left (B a^{2} b + A a b^{2}\right )} d^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{3} d^{2} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 594, normalized size = 2.29 \begin {gather*} \frac {1}{8} \, B b^{3} x^{8} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{7} \, B b^{3} d x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B b^{3} d^{2} x^{6} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, B b^{3} d^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{7} \, B a b^{2} x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, A b^{3} x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, B a b^{2} d x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, A b^{3} d x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{5} \, B a b^{2} d^{2} x^{5} e \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, A b^{3} d^{2} x^{5} e \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, B a b^{2} d^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, A b^{3} d^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B a^{2} b x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, A a b^{2} x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{5} \, B a^{2} b d x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{5} \, A a b^{2} d x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{4} \, B a^{2} b d^{2} x^{4} e \mathrm {sgn}\left (b x + a\right ) + \frac {9}{4} \, A a b^{2} d^{2} x^{4} e \mathrm {sgn}\left (b x + a\right ) + B a^{2} b d^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + A a b^{2} d^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, B a^{3} x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, A a^{2} b x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, B a^{3} d x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{4} \, A a^{2} b d x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + B a^{3} d^{2} x^{3} e \mathrm {sgn}\left (b x + a\right ) + 3 \, A a^{2} b d^{2} x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B a^{3} d^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, A a^{2} b d^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, A a^{3} x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + A a^{3} d x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, A a^{3} d^{2} x^{2} e \mathrm {sgn}\left (b x + a\right ) + A a^{3} d^{3} 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 = 428, normalized size = 1.65 \begin {gather*} \frac {\left (35 B \,e^{3} b^{3} x^{7}+40 x^{6} A \,b^{3} e^{3}+120 x^{6} B \,e^{3} a \,b^{2}+120 x^{6} B \,b^{3} d \,e^{2}+140 x^{5} A a \,b^{2} e^{3}+140 x^{5} A \,b^{3} d \,e^{2}+140 x^{5} B \,e^{3} a^{2} b +420 x^{5} B a \,b^{2} d \,e^{2}+140 x^{5} B \,b^{3} d^{2} e +168 x^{4} A \,a^{2} b \,e^{3}+504 x^{4} A a \,b^{2} d \,e^{2}+168 x^{4} A \,b^{3} d^{2} e +56 x^{4} B \,a^{3} e^{3}+504 x^{4} B \,a^{2} b d \,e^{2}+504 x^{4} B a \,b^{2} d^{2} e +56 x^{4} B \,b^{3} d^{3}+70 x^{3} A \,a^{3} e^{3}+630 x^{3} A \,a^{2} b d \,e^{2}+630 x^{3} A a \,b^{2} d^{2} e +70 x^{3} A \,d^{3} b^{3}+210 x^{3} B \,a^{3} d \,e^{2}+630 x^{3} B \,a^{2} b \,d^{2} e +210 x^{3} B a \,b^{2} d^{3}+280 A \,a^{3} d \,e^{2} x^{2}+840 A \,a^{2} b \,d^{2} e \,x^{2}+280 A a \,b^{2} d^{3} x^{2}+280 B \,a^{3} d^{2} e \,x^{2}+280 B \,a^{2} b \,d^{3} x^{2}+420 x A \,a^{3} d^{2} e +420 x A \,d^{3} a^{2} b +140 x \,a^{3} B \,d^{3}+280 A \,d^{3} a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} x}{280 \left (b x +a \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.56, size = 698, normalized size = 2.69 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B e^{3} x^{3}}{8 \, b^{2}} + \frac {1}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A d^{3} x + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a^{4} e^{3} x}{4 \, b^{4}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a e^{3} x^{2}}{56 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A a d^{3}}{4 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a^{5} e^{3}}{4 \, b^{5}} + \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{2} e^{3} x}{56 \, b^{4}} - \frac {69 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{3} e^{3}}{280 \, b^{5}} - \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} x}{4 \, b^{3}} + \frac {3 \, {\left (B d^{2} e + A d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} x}{4 \, b^{2}} - \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a x}{4 \, b} + \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} x^{2}}{7 \, b^{2}} - \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{4}}{4 \, b^{4}} + \frac {3 \, {\left (B d^{2} e + A d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3}}{4 \, b^{3}} - \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2}}{4 \, b^{2}} - \frac {3 \, {\left (3 \, B d e^{2} + A e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a x}{14 \, b^{3}} + \frac {{\left (B d^{2} e + A d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} x}{2 \, b^{2}} + \frac {17 \, {\left (3 \, B d e^{2} + A e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2}}{70 \, b^{4}} - \frac {7 \, {\left (B d^{2} e + A d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a}{10 \, b^{3}} + \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}{5 \, 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 = -0.00 \begin {gather*} \int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,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 )^{3} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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