Optimal. Leaf size=128 \[ -\frac {2 b (d+e x)^{13/2} (-2 a B e-A b e+3 b B d)}{13 e^4}+\frac {2 (d+e x)^{11/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{11 e^4}-\frac {2 (d+e x)^{9/2} (b d-a e)^2 (B d-A e)}{9 e^4}+\frac {2 b^2 B (d+e x)^{15/2}}{15 e^4} \]
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Rubi [A] time = 0.07, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {77} \begin {gather*} -\frac {2 b (d+e x)^{13/2} (-2 a B e-A b e+3 b B d)}{13 e^4}+\frac {2 (d+e x)^{11/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{11 e^4}-\frac {2 (d+e x)^{9/2} (b d-a e)^2 (B d-A e)}{9 e^4}+\frac {2 b^2 B (d+e x)^{15/2}}{15 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int (a+b x)^2 (A+B x) (d+e x)^{7/2} \, dx &=\int \left (\frac {(-b d+a e)^2 (-B d+A e) (d+e x)^{7/2}}{e^3}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^{9/2}}{e^3}+\frac {b (-3 b B d+A b e+2 a B e) (d+e x)^{11/2}}{e^3}+\frac {b^2 B (d+e x)^{13/2}}{e^3}\right ) \, dx\\ &=-\frac {2 (b d-a e)^2 (B d-A e) (d+e x)^{9/2}}{9 e^4}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{11/2}}{11 e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{13/2}}{13 e^4}+\frac {2 b^2 B (d+e x)^{15/2}}{15 e^4}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 107, normalized size = 0.84 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (-495 b (d+e x)^2 (-2 a B e-A b e+3 b B d)+585 (d+e x) (b d-a e) (-a B e-2 A b e+3 b B d)-715 (b d-a e)^2 (B d-A e)+429 b^2 B (d+e x)^3\right )}{6435 e^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 193, normalized size = 1.51 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (715 a^2 A e^3+585 a^2 B e^2 (d+e x)-715 a^2 B d e^2+1170 a A b e^2 (d+e x)-1430 a A b d e^2+1430 a b B d^2 e-2340 a b B d e (d+e x)+990 a b B e (d+e x)^2+715 A b^2 d^2 e-1170 A b^2 d e (d+e x)+495 A b^2 e (d+e x)^2-715 b^2 B d^3+1755 b^2 B d^2 (d+e x)-1485 b^2 B d (d+e x)^2+429 b^2 B (d+e x)^3\right )}{6435 e^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.37, size = 424, normalized size = 3.31 \begin {gather*} \frac {2 \, {\left (429 \, B b^{2} e^{7} x^{7} - 16 \, B b^{2} d^{7} + 715 \, A a^{2} d^{4} e^{3} + 40 \, {\left (2 \, B a b + A b^{2}\right )} d^{6} e - 130 \, {\left (B a^{2} + 2 \, A a b\right )} d^{5} e^{2} + 33 \, {\left (46 \, B b^{2} d e^{6} + 15 \, {\left (2 \, B a b + A b^{2}\right )} e^{7}\right )} x^{6} + 9 \, {\left (206 \, B b^{2} d^{2} e^{5} + 200 \, {\left (2 \, B a b + A b^{2}\right )} d e^{6} + 65 \, {\left (B a^{2} + 2 \, A a b\right )} e^{7}\right )} x^{5} + 5 \, {\left (160 \, B b^{2} d^{3} e^{4} + 143 \, A a^{2} e^{7} + 458 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{5} + 442 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{6}\right )} x^{4} + 5 \, {\left (B b^{2} d^{4} e^{3} + 572 \, A a^{2} d e^{6} + 212 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e^{4} + 598 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{5}\right )} x^{3} - 3 \, {\left (2 \, B b^{2} d^{5} e^{2} - 1430 \, A a^{2} d^{2} e^{5} - 5 \, {\left (2 \, B a b + A b^{2}\right )} d^{4} e^{3} - 520 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{4}\right )} x^{2} + {\left (8 \, B b^{2} d^{6} e + 2860 \, A a^{2} d^{3} e^{4} - 20 \, {\left (2 \, B a b + A b^{2}\right )} d^{5} e^{2} + 65 \, {\left (B a^{2} + 2 \, A a b\right )} d^{4} e^{3}\right )} x\right )} \sqrt {e x + d}}{6435 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.49, size = 1913, normalized size = 14.95
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 169, normalized size = 1.32 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (429 b^{2} B \,x^{3} e^{3}+495 A \,b^{2} e^{3} x^{2}+990 B a b \,e^{3} x^{2}-198 B \,b^{2} d \,e^{2} x^{2}+1170 A a b \,e^{3} x -180 A \,b^{2} d \,e^{2} x +585 B \,a^{2} e^{3} x -360 B a b d \,e^{2} x +72 B \,b^{2} d^{2} e x +715 a^{2} A \,e^{3}-260 A a b d \,e^{2}+40 A \,b^{2} d^{2} e -130 B \,a^{2} d \,e^{2}+80 B a b \,d^{2} e -16 B \,b^{2} d^{3}\right )}{6435 e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 159, normalized size = 1.24 \begin {gather*} \frac {2 \, {\left (429 \, {\left (e x + d\right )}^{\frac {15}{2}} B b^{2} - 495 \, {\left (3 \, B b^{2} d - {\left (2 \, B a b + A b^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 585 \, {\left (3 \, B b^{2} d^{2} - 2 \, {\left (2 \, B a b + A b^{2}\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 715 \, {\left (B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{6435 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 115, normalized size = 0.90 \begin {gather*} \frac {{\left (d+e\,x\right )}^{13/2}\,\left (2\,A\,b^2\,e-6\,B\,b^2\,d+4\,B\,a\,b\,e\right )}{13\,e^4}+\frac {2\,B\,b^2\,{\left (d+e\,x\right )}^{15/2}}{15\,e^4}+\frac {2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,b\,e+B\,a\,e-3\,B\,b\,d\right )}{11\,e^4}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.81, size = 1020, normalized size = 7.97 \begin {gather*} \begin {cases} \frac {2 A a^{2} d^{4} \sqrt {d + e x}}{9 e} + \frac {8 A a^{2} d^{3} x \sqrt {d + e x}}{9} + \frac {4 A a^{2} d^{2} e x^{2} \sqrt {d + e x}}{3} + \frac {8 A a^{2} d e^{2} x^{3} \sqrt {d + e x}}{9} + \frac {2 A a^{2} e^{3} x^{4} \sqrt {d + e x}}{9} - \frac {8 A a b d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {4 A a b d^{4} x \sqrt {d + e x}}{99 e} + \frac {32 A a b d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {184 A a b d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {136 A a b d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {4 A a b e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {16 A b^{2} d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {8 A b^{2} d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {2 A b^{2} d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {424 A b^{2} d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {916 A b^{2} d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {80 A b^{2} d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {2 A b^{2} e^{3} x^{6} \sqrt {d + e x}}{13} - \frac {4 B a^{2} d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {2 B a^{2} d^{4} x \sqrt {d + e x}}{99 e} + \frac {16 B a^{2} d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {92 B a^{2} d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {68 B a^{2} d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {2 B a^{2} e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {32 B a b d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {16 B a b d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {4 B a b d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {848 B a b d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {1832 B a b d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {160 B a b d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {4 B a b e^{3} x^{6} \sqrt {d + e x}}{13} - \frac {32 B b^{2} d^{7} \sqrt {d + e x}}{6435 e^{4}} + \frac {16 B b^{2} d^{6} x \sqrt {d + e x}}{6435 e^{3}} - \frac {4 B b^{2} d^{5} x^{2} \sqrt {d + e x}}{2145 e^{2}} + \frac {2 B b^{2} d^{4} x^{3} \sqrt {d + e x}}{1287 e} + \frac {320 B b^{2} d^{3} x^{4} \sqrt {d + e x}}{1287} + \frac {412 B b^{2} d^{2} e x^{5} \sqrt {d + e x}}{715} + \frac {92 B b^{2} d e^{2} x^{6} \sqrt {d + e x}}{195} + \frac {2 B b^{2} e^{3} x^{7} \sqrt {d + e x}}{15} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (A a^{2} x + A a b x^{2} + \frac {A b^{2} x^{3}}{3} + \frac {B a^{2} x^{2}}{2} + \frac {2 B a b x^{3}}{3} + \frac {B b^{2} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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