Optimal. Leaf size=174 \[ \frac {\sqrt {a+b x} \left (c \left (3 a^2 d^2-22 a b c d+15 b^2 c^2\right )+d x (5 b c-3 a d) (b c-a d)\right )}{3 b d^3 \sqrt {c+d x} (b c-a d)^2}-\frac {(a d+5 b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{7/2}}-\frac {2 c x^2 \sqrt {a+b x}}{3 d (c+d x)^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.15, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {98, 143, 63, 217, 206} \[ \frac {\sqrt {a+b x} \left (c \left (3 a^2 d^2-22 a b c d+15 b^2 c^2\right )+d x (5 b c-3 a d) (b c-a d)\right )}{3 b d^3 \sqrt {c+d x} (b c-a d)^2}-\frac {(a d+5 b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{7/2}}-\frac {2 c x^2 \sqrt {a+b x}}{3 d (c+d x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 143
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx &=-\frac {2 c x^2 \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {2 \int \frac {x \left (2 a c+\frac {1}{2} (5 b c-3 a d) x\right )}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 d (b c-a d)}\\ &=-\frac {2 c x^2 \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {\sqrt {a+b x} \left (c \left (15 b^2 c^2-22 a b c d+3 a^2 d^2\right )+d (5 b c-3 a d) (b c-a d) x\right )}{3 b d^3 (b c-a d)^2 \sqrt {c+d x}}-\frac {(5 b c+a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b d^3}\\ &=-\frac {2 c x^2 \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {\sqrt {a+b x} \left (c \left (15 b^2 c^2-22 a b c d+3 a^2 d^2\right )+d (5 b c-3 a d) (b c-a d) x\right )}{3 b d^3 (b c-a d)^2 \sqrt {c+d x}}-\frac {(5 b c+a d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b^2 d^3}\\ &=-\frac {2 c x^2 \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {\sqrt {a+b x} \left (c \left (15 b^2 c^2-22 a b c d+3 a^2 d^2\right )+d (5 b c-3 a d) (b c-a d) x\right )}{3 b d^3 (b c-a d)^2 \sqrt {c+d x}}-\frac {(5 b c+a d) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{b^2 d^3}\\ &=-\frac {2 c x^2 \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {\sqrt {a+b x} \left (c \left (15 b^2 c^2-22 a b c d+3 a^2 d^2\right )+d (5 b c-3 a d) (b c-a d) x\right )}{3 b d^3 (b c-a d)^2 \sqrt {c+d x}}-\frac {(5 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 182, normalized size = 1.05 \[ \frac {\frac {c \sqrt {a+b x} \left (3 a^2 d^2 (c+2 d x)-2 a b c d (11 c+15 d x)+5 b^2 c^2 (3 c+4 d x)\right )}{d^2 (b c-a d)^2}-\frac {3 (b c-a d)^{3/2} (a d+5 b c) \left (\frac {b (c+d x)}{b c-a d}\right )^{3/2} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{b^2 d^{5/2}}+3 x^2 \sqrt {a+b x}}{3 b d (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.51, size = 896, normalized size = 5.15 \[ \left [\frac {3 \, {\left (5 \, b^{3} c^{5} - 9 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3} + {\left (5 \, b^{3} c^{3} d^{2} - 9 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} + a^{3} d^{5}\right )} x^{2} + 2 \, {\left (5 \, b^{3} c^{4} d - 9 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} + a^{3} c 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 (15 \, b^{3} c^{4} d - 22 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} + 3 \, {\left (b^{3} c^{2} d^{3} - 2 \, a b^{2} c d^{4} + a^{2} b d^{5}\right )} x^{2} + 2 \, {\left (10 \, b^{3} c^{3} d^{2} - 15 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left (b^{4} c^{4} d^{4} - 2 \, a b^{3} c^{3} d^{5} + a^{2} b^{2} c^{2} d^{6} + {\left (b^{4} c^{2} d^{6} - 2 \, a b^{3} c d^{7} + a^{2} b^{2} d^{8}\right )} x^{2} + 2 \, {\left (b^{4} c^{3} d^{5} - 2 \, a b^{3} c^{2} d^{6} + a^{2} b^{2} c d^{7}\right )} x\right )}}, \frac {3 \, {\left (5 \, b^{3} c^{5} - 9 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3} + {\left (5 \, b^{3} c^{3} d^{2} - 9 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} + a^{3} d^{5}\right )} x^{2} + 2 \, {\left (5 \, b^{3} c^{4} d - 9 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} + a^{3} c 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 (15 \, b^{3} c^{4} d - 22 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} + 3 \, {\left (b^{3} c^{2} d^{3} - 2 \, a b^{2} c d^{4} + a^{2} b d^{5}\right )} x^{2} + 2 \, {\left (10 \, b^{3} c^{3} d^{2} - 15 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (b^{4} c^{4} d^{4} - 2 \, a b^{3} c^{3} d^{5} + a^{2} b^{2} c^{2} d^{6} + {\left (b^{4} c^{2} d^{6} - 2 \, a b^{3} c d^{7} + a^{2} b^{2} d^{8}\right )} x^{2} + 2 \, {\left (b^{4} c^{3} d^{5} - 2 \, a b^{3} c^{2} d^{6} + a^{2} b^{2} c d^{7}\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.62, size = 373, normalized size = 2.14 \[ \frac {{\left ({\left (b x + a\right )} {\left (\frac {3 \, {\left (b^{6} c^{2} d^{4} {\left | b \right |} - 2 \, a b^{5} c d^{5} {\left | b \right |} + a^{2} b^{4} d^{6} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}} + \frac {2 \, {\left (10 \, b^{7} c^{3} d^{3} {\left | b \right |} - 18 \, a b^{6} c^{2} d^{4} {\left | b \right |} + 9 \, a^{2} b^{5} c d^{5} {\left | b \right |} - 3 \, a^{3} b^{4} d^{6} {\left | b \right |}\right )}}{b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}}\right )} + \frac {3 \, {\left (5 \, b^{8} c^{4} d^{2} {\left | b \right |} - 14 \, a b^{7} c^{3} d^{3} {\left | b \right |} + 12 \, a^{2} b^{6} c^{2} d^{4} {\left | b \right |} - 4 \, a^{3} b^{5} c d^{5} {\left | b \right |} + a^{4} b^{4} d^{6} {\left | b \right |}\right )}}{b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}}\right )} \sqrt {b x + a}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {{\left (5 \, b c {\left | b \right |} + a d {\left | b \right |}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 928, normalized size = 5.33 \[ -\frac {\sqrt {b x +a}\, \left (3 a^{3} d^{5} x^{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 )+9 a^{2} b c \,d^{4} x^{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 )-27 a \,b^{2} c^{2} d^{3} x^{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 )+15 b^{3} c^{3} d^{2} x^{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 )+6 a^{3} c \,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 )+18 a^{2} b \,c^{2} 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 )-54 a \,b^{2} c^{3} 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 )+30 b^{3} c^{4} 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 )+3 a^{3} c^{2} 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 )+9 a^{2} b \,c^{3} 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 )-27 a \,b^{2} c^{4} 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 )+15 b^{3} c^{5} \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 )-6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} d^{4} x^{2}+12 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c \,d^{3} x^{2}-6 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} d^{2} x^{2}-12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} c \,d^{3} x +60 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} d^{2} x -40 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{3} d x -6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} c^{2} d^{2}+44 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{3} d -30 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{4}\right )}{6 \sqrt {b d}\, \left (a d -b c \right )^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \left (d x +c \right )^{\frac {3}{2}} b \,d^{3}} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {x^3}{\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x^{3}}{\sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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