Optimal. Leaf size=218 \[ -\frac {5 (b c-a d)^2 (7 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} d^{9/2}}+\frac {5 \sqrt {a+b x} \sqrt {c+d x} (b c-a d) (7 b c-a d)}{8 d^4}-\frac {5 (a+b x)^{3/2} \sqrt {c+d x} (7 b c-a d)}{12 d^3}+\frac {(a+b x)^{5/2} \sqrt {c+d x} (7 b c-a d)}{3 d^2 (b c-a d)}-\frac {2 c (a+b x)^{7/2}}{d \sqrt {c+d x} (b c-a d)} \]
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Rubi [A] time = 0.13, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {78, 50, 63, 217, 206} \begin {gather*} \frac {(a+b x)^{5/2} \sqrt {c+d x} (7 b c-a d)}{3 d^2 (b c-a d)}-\frac {5 (a+b x)^{3/2} \sqrt {c+d x} (7 b c-a d)}{12 d^3}+\frac {5 \sqrt {a+b x} \sqrt {c+d x} (b c-a d) (7 b c-a d)}{8 d^4}-\frac {5 (b c-a d)^2 (7 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} d^{9/2}}-\frac {2 c (a+b x)^{7/2}}{d \sqrt {c+d x} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x (a+b x)^{5/2}}{(c+d x)^{3/2}} \, dx &=-\frac {2 c (a+b x)^{7/2}}{d (b c-a d) \sqrt {c+d x}}+\frac {(7 b c-a d) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx}{d (b c-a d)}\\ &=-\frac {2 c (a+b x)^{7/2}}{d (b c-a d) \sqrt {c+d x}}+\frac {(7 b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{3 d^2 (b c-a d)}-\frac {(5 (7 b c-a d)) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{6 d^2}\\ &=-\frac {2 c (a+b x)^{7/2}}{d (b c-a d) \sqrt {c+d x}}-\frac {5 (7 b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^3}+\frac {(7 b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{3 d^2 (b c-a d)}+\frac {(5 (b c-a d) (7 b c-a d)) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{8 d^3}\\ &=-\frac {2 c (a+b x)^{7/2}}{d (b c-a d) \sqrt {c+d x}}+\frac {5 (b c-a d) (7 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{8 d^4}-\frac {5 (7 b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^3}+\frac {(7 b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{3 d^2 (b c-a d)}-\frac {\left (5 (b c-a d)^2 (7 b c-a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 d^4}\\ &=-\frac {2 c (a+b x)^{7/2}}{d (b c-a d) \sqrt {c+d x}}+\frac {5 (b c-a d) (7 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{8 d^4}-\frac {5 (7 b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^3}+\frac {(7 b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{3 d^2 (b c-a d)}-\frac {\left (5 (b c-a d)^2 (7 b c-a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{8 b d^4}\\ &=-\frac {2 c (a+b x)^{7/2}}{d (b c-a d) \sqrt {c+d x}}+\frac {5 (b c-a d) (7 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{8 d^4}-\frac {5 (7 b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^3}+\frac {(7 b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{3 d^2 (b c-a d)}-\frac {\left (5 (b c-a d)^2 (7 b c-a d)\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 b d^4}\\ &=-\frac {2 c (a+b x)^{7/2}}{d (b c-a d) \sqrt {c+d x}}+\frac {5 (b c-a d) (7 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{8 d^4}-\frac {5 (7 b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^3}+\frac {(7 b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{3 d^2 (b c-a d)}-\frac {5 (b c-a d)^2 (7 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} d^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 221, normalized size = 1.01 \begin {gather*} \frac {\frac {\sqrt {d} \left (3 a^3 d^2 (27 c+11 d x)+a^2 b d \left (-190 c^2+13 c d x+59 d^2 x^2\right )+a b^2 \left (105 c^3-155 c^2 d x-82 c d^2 x^2+34 d^3 x^3\right )+b^3 x \left (105 c^3+35 c^2 d x-14 c d^2 x^2+8 d^3 x^3\right )\right )}{\sqrt {a+b x}}-\frac {15 (b c-a d)^{5/2} (7 b c-a d) \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{b}}{24 d^{9/2} \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.60, size = 232, normalized size = 1.06 \begin {gather*} \frac {\sqrt {a+b x} (a d-b c)^2 \left (\frac {280 b^2 c d (a+b x)}{c+d x}+15 a b^2 d+\frac {33 a d^3 (a+b x)^2}{(c+d x)^2}+\frac {48 c d^3 (a+b x)^3}{(c+d x)^3}-\frac {40 a b d^2 (a+b x)}{c+d x}-\frac {231 b c d^2 (a+b x)^2}{(c+d x)^2}-105 b^3 c\right )}{24 d^4 \sqrt {c+d x} \left (\frac {d (a+b x)}{c+d x}-b\right )^3}-\frac {5 (b c-a d)^2 (7 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} d^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.84, size = 598, normalized size = 2.74 \begin {gather*} \left [-\frac {15 \, {\left (7 \, b^{3} c^{4} - 15 \, a b^{2} c^{3} d + 9 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3} + {\left (7 \, b^{3} c^{3} d - 15 \, a b^{2} c^{2} d^{2} + 9 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (8 \, b^{3} d^{4} x^{3} + 105 \, b^{3} c^{3} d - 190 \, a b^{2} c^{2} d^{2} + 81 \, a^{2} b c d^{3} - 2 \, {\left (7 \, b^{3} c d^{3} - 13 \, a b^{2} d^{4}\right )} x^{2} + {\left (35 \, b^{3} c^{2} d^{2} - 68 \, a b^{2} c d^{3} + 33 \, a^{2} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (b d^{6} x + b c d^{5}\right )}}, \frac {15 \, {\left (7 \, b^{3} c^{4} - 15 \, a b^{2} c^{3} d + 9 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3} + {\left (7 \, b^{3} c^{3} d - 15 \, a b^{2} c^{2} d^{2} + 9 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (8 \, b^{3} d^{4} x^{3} + 105 \, b^{3} c^{3} d - 190 \, a b^{2} c^{2} d^{2} + 81 \, a^{2} b c d^{3} - 2 \, {\left (7 \, b^{3} c d^{3} - 13 \, a b^{2} d^{4}\right )} x^{2} + {\left (35 \, b^{3} c^{2} d^{2} - 68 \, a b^{2} c d^{3} + 33 \, a^{2} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (b d^{6} x + b c d^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.48, size = 292, normalized size = 1.34 \begin {gather*} \frac {{\left ({\left (b x + a\right )} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} {\left | b \right |}}{b d} - \frac {7 \, b^{2} c d^{5} {\left | b \right |} - a b d^{6} {\left | b \right |}}{b^{2} d^{7}}\right )} + \frac {5 \, {\left (7 \, b^{3} c^{2} d^{4} {\left | b \right |} - 8 \, a b^{2} c d^{5} {\left | b \right |} + a^{2} b d^{6} {\left | b \right |}\right )}}{b^{2} d^{7}}\right )} + \frac {15 \, {\left (7 \, b^{4} c^{3} d^{3} {\left | b \right |} - 15 \, a b^{3} c^{2} d^{4} {\left | b \right |} + 9 \, a^{2} b^{2} c d^{5} {\left | b \right |} - a^{3} b d^{6} {\left | b \right |}\right )}}{b^{2} d^{7}}\right )} \sqrt {b x + a}}{24 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {5 \, {\left (7 \, b^{3} c^{3} {\left | b \right |} - 15 \, a b^{2} c^{2} d {\left | b \right |} + 9 \, a^{2} b c d^{2} {\left | b \right |} - a^{3} d^{3} {\left | b \right |}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{8 \, \sqrt {b d} b d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 689, normalized size = 3.16 \begin {gather*} \frac {\sqrt {b x +a}\, \left (15 a^{3} d^{4} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-135 a^{2} b c \,d^{3} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+225 a \,b^{2} c^{2} d^{2} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-105 b^{3} c^{3} d x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+15 a^{3} c \,d^{3} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-135 a^{2} b \,c^{2} d^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+225 a \,b^{2} c^{3} d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-105 b^{3} c^{4} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+16 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} d^{3} x^{3}+52 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a b \,d^{3} x^{2}-28 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} c \,d^{2} x^{2}+66 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} d^{3} x -136 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a b c \,d^{2} x +70 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} c^{2} d x +162 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} c \,d^{2}-380 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a b \,c^{2} d +210 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} c^{3}\right )}{48 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {d x +c}\, d^{4}} \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 [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x\,{\left (a+b\,x\right )}^{5/2}}{{\left (c+d\,x\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 \frac {x \left (a + b x\right )^{\frac {5}{2}}}{\left (c + d x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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