Optimal. Leaf size=136 \[ \frac {32 d^3 (c+d x)^{5/2}}{1155 (a+b x)^{5/2} (b c-a d)^4}-\frac {16 d^2 (c+d x)^{5/2}}{231 (a+b x)^{7/2} (b c-a d)^3}+\frac {4 d (c+d x)^{5/2}}{33 (a+b x)^{9/2} (b c-a d)^2}-\frac {2 (c+d x)^{5/2}}{11 (a+b x)^{11/2} (b c-a d)} \]
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Rubi [A] time = 0.03, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {45, 37} \begin {gather*} \frac {32 d^3 (c+d x)^{5/2}}{1155 (a+b x)^{5/2} (b c-a d)^4}-\frac {16 d^2 (c+d x)^{5/2}}{231 (a+b x)^{7/2} (b c-a d)^3}+\frac {4 d (c+d x)^{5/2}}{33 (a+b x)^{9/2} (b c-a d)^2}-\frac {2 (c+d x)^{5/2}}{11 (a+b x)^{11/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {(c+d x)^{3/2}}{(a+b x)^{13/2}} \, dx &=-\frac {2 (c+d x)^{5/2}}{11 (b c-a d) (a+b x)^{11/2}}-\frac {(6 d) \int \frac {(c+d x)^{3/2}}{(a+b x)^{11/2}} \, dx}{11 (b c-a d)}\\ &=-\frac {2 (c+d x)^{5/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {4 d (c+d x)^{5/2}}{33 (b c-a d)^2 (a+b x)^{9/2}}+\frac {\left (8 d^2\right ) \int \frac {(c+d x)^{3/2}}{(a+b x)^{9/2}} \, dx}{33 (b c-a d)^2}\\ &=-\frac {2 (c+d x)^{5/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {4 d (c+d x)^{5/2}}{33 (b c-a d)^2 (a+b x)^{9/2}}-\frac {16 d^2 (c+d x)^{5/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}-\frac {\left (16 d^3\right ) \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2}} \, dx}{231 (b c-a d)^3}\\ &=-\frac {2 (c+d x)^{5/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {4 d (c+d x)^{5/2}}{33 (b c-a d)^2 (a+b x)^{9/2}}-\frac {16 d^2 (c+d x)^{5/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}+\frac {32 d^3 (c+d x)^{5/2}}{1155 (b c-a d)^4 (a+b x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 118, normalized size = 0.87 \begin {gather*} \frac {2 (c+d x)^{5/2} \left (231 a^3 d^3+99 a^2 b d^2 (2 d x-5 c)+11 a b^2 d \left (35 c^2-20 c d x+8 d^2 x^2\right )+b^3 \left (-105 c^3+70 c^2 d x-40 c d^2 x^2+16 d^3 x^3\right )\right )}{1155 (a+b x)^{11/2} (b c-a d)^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 95, normalized size = 0.70 \begin {gather*} \frac {2 (c+d x)^{11/2} \left (\frac {385 b^2 d (a+b x)}{c+d x}+\frac {231 d^3 (a+b x)^3}{(c+d x)^3}-\frac {495 b d^2 (a+b x)^2}{(c+d x)^2}-105 b^3\right )}{1155 (a+b x)^{11/2} (b c-a d)^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 28.36, size = 649, normalized size = 4.77 \begin {gather*} \frac {2 \, {\left (16 \, b^{3} d^{5} x^{5} - 105 \, b^{3} c^{5} + 385 \, a b^{2} c^{4} d - 495 \, a^{2} b c^{3} d^{2} + 231 \, a^{3} c^{2} d^{3} - 8 \, {\left (b^{3} c d^{4} - 11 \, a b^{2} d^{5}\right )} x^{4} + 2 \, {\left (3 \, b^{3} c^{2} d^{3} - 22 \, a b^{2} c d^{4} + 99 \, a^{2} b d^{5}\right )} x^{3} - {\left (5 \, b^{3} c^{3} d^{2} - 33 \, a b^{2} c^{2} d^{3} + 99 \, a^{2} b c d^{4} - 231 \, a^{3} d^{5}\right )} x^{2} - 2 \, {\left (70 \, b^{3} c^{4} d - 275 \, a b^{2} c^{3} d^{2} + 396 \, a^{2} b c^{2} d^{3} - 231 \, a^{3} c d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{1155 \, {\left (a^{6} b^{4} c^{4} - 4 \, a^{7} b^{3} c^{3} d + 6 \, a^{8} b^{2} c^{2} d^{2} - 4 \, a^{9} b c d^{3} + a^{10} d^{4} + {\left (b^{10} c^{4} - 4 \, a b^{9} c^{3} d + 6 \, a^{2} b^{8} c^{2} d^{2} - 4 \, a^{3} b^{7} c d^{3} + a^{4} b^{6} d^{4}\right )} x^{6} + 6 \, {\left (a b^{9} c^{4} - 4 \, a^{2} b^{8} c^{3} d + 6 \, a^{3} b^{7} c^{2} d^{2} - 4 \, a^{4} b^{6} c d^{3} + a^{5} b^{5} d^{4}\right )} x^{5} + 15 \, {\left (a^{2} b^{8} c^{4} - 4 \, a^{3} b^{7} c^{3} d + 6 \, a^{4} b^{6} c^{2} d^{2} - 4 \, a^{5} b^{5} c d^{3} + a^{6} b^{4} d^{4}\right )} x^{4} + 20 \, {\left (a^{3} b^{7} c^{4} - 4 \, a^{4} b^{6} c^{3} d + 6 \, a^{5} b^{5} c^{2} d^{2} - 4 \, a^{6} b^{4} c d^{3} + a^{7} b^{3} d^{4}\right )} x^{3} + 15 \, {\left (a^{4} b^{6} c^{4} - 4 \, a^{5} b^{5} c^{3} d + 6 \, a^{6} b^{4} c^{2} d^{2} - 4 \, a^{7} b^{3} c d^{3} + a^{8} b^{2} d^{4}\right )} x^{2} + 6 \, {\left (a^{5} b^{5} c^{4} - 4 \, a^{6} b^{4} c^{3} d + 6 \, a^{7} b^{3} c^{2} d^{2} - 4 \, a^{8} b^{2} c d^{3} + a^{9} b d^{4}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.28, size = 1823, normalized size = 13.40
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 = 171, normalized size = 1.26 \begin {gather*} \frac {2 \left (d x +c \right )^{\frac {5}{2}} \left (16 b^{3} x^{3} d^{3}+88 a \,b^{2} d^{3} x^{2}-40 b^{3} c \,d^{2} x^{2}+198 a^{2} b \,d^{3} x -220 a \,b^{2} c \,d^{2} x +70 b^{3} c^{2} d x +231 a^{3} d^{3}-495 a^{2} b c \,d^{2}+385 a \,b^{2} c^{2} d -105 b^{3} c^{3}\right )}{1155 \left (b x +a \right )^{\frac {11}{2}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.33, size = 376, normalized size = 2.76 \begin {gather*} \frac {\sqrt {c+d\,x}\,\left (\frac {x^2\,\left (462\,a^3\,d^5-198\,a^2\,b\,c\,d^4+66\,a\,b^2\,c^2\,d^3-10\,b^3\,c^3\,d^2\right )}{1155\,b^5\,{\left (a\,d-b\,c\right )}^4}-\frac {-462\,a^3\,c^2\,d^3+990\,a^2\,b\,c^3\,d^2-770\,a\,b^2\,c^4\,d+210\,b^3\,c^5}{1155\,b^5\,{\left (a\,d-b\,c\right )}^4}+\frac {x\,\left (924\,a^3\,c\,d^4-1584\,a^2\,b\,c^2\,d^3+1100\,a\,b^2\,c^3\,d^2-280\,b^3\,c^4\,d\right )}{1155\,b^5\,{\left (a\,d-b\,c\right )}^4}+\frac {32\,d^5\,x^5}{1155\,b^2\,{\left (a\,d-b\,c\right )}^4}+\frac {16\,d^4\,x^4\,\left (11\,a\,d-b\,c\right )}{1155\,b^3\,{\left (a\,d-b\,c\right )}^4}+\frac {4\,d^3\,x^3\,\left (99\,a^2\,d^2-22\,a\,b\,c\,d+3\,b^2\,c^2\right )}{1155\,b^4\,{\left (a\,d-b\,c\right )}^4}\right )}{x^5\,\sqrt {a+b\,x}+\frac {a^5\,\sqrt {a+b\,x}}{b^5}+\frac {10\,a^2\,x^3\,\sqrt {a+b\,x}}{b^2}+\frac {10\,a^3\,x^2\,\sqrt {a+b\,x}}{b^3}+\frac {5\,a\,x^4\,\sqrt {a+b\,x}}{b}+\frac {5\,a^4\,x\,\sqrt {a+b\,x}}{b^4}} \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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