Optimal. Leaf size=198 \[ \frac {\sqrt {d} \left (-a^2 d^2+10 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2}}-\frac {c^{3/2} (5 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{x}+\frac {3}{2} d \sqrt {a+b x} (c+d x)^{3/2}+\frac {d \sqrt {a+b x} \sqrt {c+d x} (a d+11 b c)}{4 b} \]
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Rubi [A] time = 0.20, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {97, 154, 157, 63, 217, 206, 93, 208} \[ \frac {\sqrt {d} \left (-a^2 d^2+10 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2}}-\frac {c^{3/2} (5 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{x}+\frac {3}{2} d \sqrt {a+b x} (c+d x)^{3/2}+\frac {d \sqrt {a+b x} \sqrt {c+d x} (a d+11 b c)}{4 b} \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 97
Rule 154
Rule 157
Rule 206
Rule 208
Rule 217
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^2} \, dx &=-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{x}+\int \frac {(c+d x)^{3/2} \left (\frac {1}{2} (b c+5 a d)+3 b d x\right )}{x \sqrt {a+b x}} \, dx\\ &=\frac {3}{2} d \sqrt {a+b x} (c+d x)^{3/2}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{x}+\frac {\int \frac {\sqrt {c+d x} \left (b c (b c+5 a d)+\frac {1}{2} b d (11 b c+a d) x\right )}{x \sqrt {a+b x}} \, dx}{2 b}\\ &=\frac {d (11 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b}+\frac {3}{2} d \sqrt {a+b x} (c+d x)^{3/2}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{x}+\frac {\int \frac {b^2 c^2 (b c+5 a d)+\frac {1}{4} b d \left (15 b^2 c^2+10 a b c d-a^2 d^2\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b^2}\\ &=\frac {d (11 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b}+\frac {3}{2} d \sqrt {a+b x} (c+d x)^{3/2}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{x}+\frac {1}{2} \left (c^2 (b c+5 a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {\left (d \left (15 b^2 c^2+10 a b c d-a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b}\\ &=\frac {d (11 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b}+\frac {3}{2} d \sqrt {a+b x} (c+d x)^{3/2}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{x}+\left (c^2 (b c+5 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )+\frac {\left (d \left (15 b^2 c^2+10 a b c d-a^2 d^2\right )\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 )}{4 b^2}\\ &=\frac {d (11 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b}+\frac {3}{2} d \sqrt {a+b x} (c+d x)^{3/2}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{x}-\frac {c^{3/2} (b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}+\frac {\left (d \left (15 b^2 c^2+10 a b c d-a^2 d^2\right )\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 )}{4 b^2}\\ &=\frac {d (11 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b}+\frac {3}{2} d \sqrt {a+b x} (c+d x)^{3/2}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{x}-\frac {c^{3/2} (b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}+\frac {\sqrt {d} \left (15 b^2 c^2+10 a b c d-a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 1.31, size = 208, normalized size = 1.05 \[ \frac {\frac {\sqrt {d} \sqrt {c+d x} \left (-a^2 d^2+10 a b c d+15 b^2 c^2\right ) \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {b c-a d} \sqrt {\frac {b (c+d x)}{b c-a d}}}-\frac {4 b c^{3/2} (5 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (a d^2 x+b \left (-4 c^2+9 c d x+2 d^2 x^2\right )\right )}{x}}{4 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 5.87, size = 1079, normalized size = 5.45 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.06, size = 597, normalized size = 3.02 \[ \frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )} d^{2} {\left | b \right |}}{b^{2}} + \frac {9 \, b^{3} c d^{3} {\left | b \right |} - a b^{2} d^{4} {\left | b \right |}}{b^{4} d^{2}}\right )} - \frac {8 \, {\left (\sqrt {b d} b^{2} c^{3} {\left | b \right |} + 5 \, \sqrt {b d} a b c^{2} d {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b} - \frac {16 \, {\left (\sqrt {b d} b^{4} c^{4} {\left | b \right |} - 2 \, \sqrt {b d} a b^{3} c^{3} d {\left | b \right |} + \sqrt {b d} a^{2} b^{2} c^{2} d^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c^{3} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b c^{2} d {\left | b \right |}\right )}}{b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}} - \frac {{\left (15 \, \sqrt {b d} b^{2} c^{2} {\left | b \right |} + 10 \, \sqrt {b d} a b c d {\left | b \right |} - \sqrt {b d} a^{2} d^{2} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{b^{2}}}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 503, normalized size = 2.54 \[ -\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (\sqrt {a c}\, a^{2} d^{3} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+20 \sqrt {b d}\, a b \,c^{2} d x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-10 \sqrt {a c}\, a b c \,d^{2} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+4 \sqrt {b d}\, b^{2} c^{3} x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-15 \sqrt {a c}\, b^{2} c^{2} d x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-4 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b \,d^{2} x^{2}-2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a \,d^{2} x -18 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b c d x +8 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b \,c^{2}\right )}{8 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b x} \]
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 \[ \int \frac {\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2}}{x^2} \,d x \]
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 \[ \int \frac {\sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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