Optimal. Leaf size=152 \[ \frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^3 (a+b x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^3 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{9 e^3 (a+b x)} \]
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Rubi [A] time = 0.07, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {770, 21, 43} \begin {gather*} \frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^3 (a+b x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^3 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{9 e^3 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 43
Rule 770
Rubi steps
\begin {align*} \int (a+b x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right ) (d+e x)^{7/2} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^2 (d+e x)^{7/2} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^2 (d+e x)^{7/2}}{e^2}-\frac {2 b (b d-a e) (d+e x)^{9/2}}{e^2}+\frac {b^2 (d+e x)^{11/2}}{e^2}\right ) \, dx}{a b+b^2 x}\\ &=\frac {2 (b d-a e)^2 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^3 (a+b x)}-\frac {4 b (b d-a e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^3 (a+b x)}+\frac {2 b^2 (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^3 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 79, normalized size = 0.52 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} (d+e x)^{9/2} \left (143 a^2 e^2+26 a b e (9 e x-2 d)+b^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )}{1287 e^3 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 51.00, size = 100, normalized size = 0.66 \begin {gather*} \frac {2 (d+e x)^{9/2} \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (143 a^2 e^2+234 a b e (d+e x)-286 a b d e+143 b^2 d^2+99 b^2 (d+e x)^2-234 b^2 d (d+e x)\right )}{1287 e^2 (a e+b e x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 212, normalized size = 1.39 \begin {gather*} \frac {2 \, {\left (99 \, b^{2} e^{6} x^{6} + 8 \, b^{2} d^{6} - 52 \, a b d^{5} e + 143 \, a^{2} d^{4} e^{2} + 18 \, {\left (20 \, b^{2} d e^{5} + 13 \, a b e^{6}\right )} x^{5} + {\left (458 \, b^{2} d^{2} e^{4} + 884 \, a b d e^{5} + 143 \, a^{2} e^{6}\right )} x^{4} + 4 \, {\left (53 \, b^{2} d^{3} e^{3} + 299 \, a b d^{2} e^{4} + 143 \, a^{2} d e^{5}\right )} x^{3} + 3 \, {\left (b^{2} d^{4} e^{2} + 208 \, a b d^{3} e^{3} + 286 \, a^{2} d^{2} e^{4}\right )} x^{2} - 2 \, {\left (2 \, b^{2} d^{5} e - 13 \, a b d^{4} e^{2} - 286 \, a^{2} d^{3} e^{3}\right )} x\right )} \sqrt {e x + d}}{1287 \, e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 930, normalized size = 6.12
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 = 79, normalized size = 0.52 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (99 b^{2} x^{2} e^{2}+234 a b \,e^{2} x -36 b^{2} d e x +143 a^{2} e^{2}-52 a b d e +8 b^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{1287 \left (b x +a \right ) e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.66, size = 263, normalized size = 1.73 \begin {gather*} \frac {2 \, {\left (9 \, b e^{5} x^{5} - 2 \, b d^{5} + 11 \, a d^{4} e + {\left (34 \, b d e^{4} + 11 \, a e^{5}\right )} x^{4} + 2 \, {\left (23 \, b d^{2} e^{3} + 22 \, a d e^{4}\right )} x^{3} + 6 \, {\left (4 \, b d^{3} e^{2} + 11 \, a d^{2} e^{3}\right )} x^{2} + {\left (b d^{4} e + 44 \, a d^{3} e^{2}\right )} x\right )} \sqrt {e x + d} a}{99 \, e^{2}} + \frac {2 \, {\left (99 \, b e^{6} x^{6} + 8 \, b d^{6} - 26 \, a d^{5} e + 9 \, {\left (40 \, b d e^{5} + 13 \, a e^{6}\right )} x^{5} + 2 \, {\left (229 \, b d^{2} e^{4} + 221 \, a d e^{5}\right )} x^{4} + 2 \, {\left (106 \, b d^{3} e^{3} + 299 \, a d^{2} e^{4}\right )} x^{3} + 3 \, {\left (b d^{4} e^{2} + 104 \, a d^{3} e^{3}\right )} x^{2} - {\left (4 \, b d^{5} e - 13 \, a d^{4} e^{2}\right )} x\right )} \sqrt {e x + d} b}{1287 \, e^{3}} \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 \sqrt {{\left (a+b\,x\right )}^2}\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{7/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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