Optimal. Leaf size=308 \[ -\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (-3 a B e-A b e+4 b B d)}{9 e^5 (a+b x)}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{5 e^5 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3 (B d-A e)}{3 e^5 (a+b x)}+\frac {2 b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^5 (a+b x)} \]
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Rubi [A] time = 0.14, antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {770, 77} \begin {gather*} -\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (-3 a B e-A b e+4 b B d)}{9 e^5 (a+b x)}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{5 e^5 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3 (B d-A e)}{3 e^5 (a+b x)}+\frac {2 b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^5 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int (A+B x) \sqrt {d+e x} \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) \sqrt {d+e x} \, 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 {b^3 (b d-a e)^3 (-B d+A e) \sqrt {d+e x}}{e^4}+\frac {b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{3/2}}{e^4}-\frac {3 b^4 (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{5/2}}{e^4}+\frac {b^5 (-4 b B d+A b e+3 a B e) (d+e x)^{7/2}}{e^4}+\frac {b^6 B (d+e x)^{9/2}}{e^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {2 (b d-a e)^3 (B d-A e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac {2 b^3 B (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 163, normalized size = 0.53 \begin {gather*} \frac {2 \left ((a+b x)^2\right )^{3/2} (d+e x)^{3/2} \left (-385 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+1485 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-693 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)+1155 (b d-a e)^3 (B d-A e)+315 b^3 B (d+e x)^4\right )}{3465 e^5 (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 52.82, size = 374, normalized size = 1.21 \begin {gather*} \frac {2 (d+e x)^{3/2} \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (1155 a^3 A e^4+693 a^3 B e^3 (d+e x)-1155 a^3 B d e^3+2079 a^2 A b e^3 (d+e x)-3465 a^2 A b d e^3+3465 a^2 b B d^2 e^2-4158 a^2 b B d e^2 (d+e x)+1485 a^2 b B e^2 (d+e x)^2+3465 a A b^2 d^2 e^2-4158 a A b^2 d e^2 (d+e x)+1485 a A b^2 e^2 (d+e x)^2-3465 a b^2 B d^3 e+6237 a b^2 B d^2 e (d+e x)-4455 a b^2 B d e (d+e x)^2+1155 a b^2 B e (d+e x)^3-1155 A b^3 d^3 e+2079 A b^3 d^2 e (d+e x)-1485 A b^3 d e (d+e x)^2+385 A b^3 e (d+e x)^3+1155 b^3 B d^4-2772 b^3 B d^3 (d+e x)+2970 b^3 B d^2 (d+e x)^2-1540 b^3 B d (d+e x)^3+315 b^3 B (d+e x)^4\right )}{3465 e^4 (a e+b e x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 353, normalized size = 1.15 \begin {gather*} \frac {2 \, {\left (315 \, B b^{3} e^{5} x^{5} + 128 \, B b^{3} d^{5} + 1155 \, A a^{3} d e^{4} - 176 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e + 792 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{2} - 462 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{3} + 35 \, {\left (B b^{3} d e^{4} + 11 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{5}\right )} x^{4} - 5 \, {\left (8 \, B b^{3} d^{2} e^{3} - 11 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{4} - 297 \, {\left (B a^{2} b + A a b^{2}\right )} e^{5}\right )} x^{3} + 3 \, {\left (16 \, B b^{3} d^{3} e^{2} - 22 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{3} + 99 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{4} + 231 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{5}\right )} x^{2} - {\left (64 \, B b^{3} d^{4} e - 1155 \, A a^{3} e^{5} - 88 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{2} + 396 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{3} - 231 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{4}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 898, normalized size = 2.92
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 317, normalized size = 1.03 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (315 b^{3} B \,x^{4} e^{4}+385 A \,b^{3} e^{4} x^{3}+1155 B a \,b^{2} e^{4} x^{3}-280 B \,b^{3} d \,e^{3} x^{3}+1485 A a \,b^{2} e^{4} x^{2}-330 A \,b^{3} d \,e^{3} x^{2}+1485 B \,a^{2} b \,e^{4} x^{2}-990 B a \,b^{2} d \,e^{3} x^{2}+240 B \,b^{3} d^{2} e^{2} x^{2}+2079 A \,a^{2} b \,e^{4} x -1188 A a \,b^{2} d \,e^{3} x +264 A \,b^{3} d^{2} e^{2} x +693 B \,a^{3} e^{4} x -1188 B \,a^{2} b d \,e^{3} x +792 B a \,b^{2} d^{2} e^{2} x -192 B \,b^{3} d^{3} e x +1155 A \,a^{3} e^{4}-1386 A \,a^{2} b d \,e^{3}+792 A a \,b^{2} d^{2} e^{2}-176 A \,b^{3} d^{3} e -462 B \,a^{3} d \,e^{3}+792 B \,a^{2} b \,d^{2} e^{2}-528 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{3465 \left (b x +a \right )^{3} e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 384, normalized size = 1.25 \begin {gather*} \frac {2 \, {\left (35 \, b^{3} e^{4} x^{4} - 16 \, b^{3} d^{4} + 72 \, a b^{2} d^{3} e - 126 \, a^{2} b d^{2} e^{2} + 105 \, a^{3} d e^{3} + 5 \, {\left (b^{3} d e^{3} + 27 \, a b^{2} e^{4}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{2} e^{2} - 9 \, a b^{2} d e^{3} - 63 \, a^{2} b e^{4}\right )} x^{2} + {\left (8 \, b^{3} d^{3} e - 36 \, a b^{2} d^{2} e^{2} + 63 \, a^{2} b d e^{3} + 105 \, a^{3} e^{4}\right )} x\right )} \sqrt {e x + d} A}{315 \, e^{4}} + \frac {2 \, {\left (315 \, b^{3} e^{5} x^{5} + 128 \, b^{3} d^{5} - 528 \, a b^{2} d^{4} e + 792 \, a^{2} b d^{3} e^{2} - 462 \, a^{3} d^{2} e^{3} + 35 \, {\left (b^{3} d e^{4} + 33 \, a b^{2} e^{5}\right )} x^{4} - 5 \, {\left (8 \, b^{3} d^{2} e^{3} - 33 \, a b^{2} d e^{4} - 297 \, a^{2} b e^{5}\right )} x^{3} + 3 \, {\left (16 \, b^{3} d^{3} e^{2} - 66 \, a b^{2} d^{2} e^{3} + 99 \, a^{2} b d e^{4} + 231 \, a^{3} e^{5}\right )} x^{2} - {\left (64 \, b^{3} d^{4} e - 264 \, a b^{2} d^{3} e^{2} + 396 \, a^{2} b d^{2} e^{3} - 231 \, a^{3} d e^{4}\right )} x\right )} \sqrt {e x + d} B}{3465 \, e^{5}} \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 )\,\sqrt {d+e\,x}\,{\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 ) \sqrt {d + e x} \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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