Optimal. Leaf size=191 \[ \frac {(b c-a d) \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{7/2} d^{5/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^2 d^2+2 a b c d+b^2 c^2\right )}{8 b^3 d^2}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 a d+3 b c)}{12 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{3/2}}{3 b d} \]
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Rubi [A] time = 0.16, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {90, 80, 50, 63, 217, 206} \[ \frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^2 d^2+2 a b c d+b^2 c^2\right )}{8 b^3 d^2}+\frac {(b c-a d) \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{7/2} d^{5/2}}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 a d+3 b c)}{12 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{3/2}}{3 b d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 90
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt {c+d x}}{\sqrt {a+b x}} \, dx &=\frac {x \sqrt {a+b x} (c+d x)^{3/2}}{3 b d}+\frac {\int \frac {\sqrt {c+d x} \left (-a c-\frac {1}{2} (3 b c+5 a d) x\right )}{\sqrt {a+b x}} \, dx}{3 b d}\\ &=-\frac {(3 b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{3/2}}{3 b d}+\frac {\left (b^2 c^2+2 a b c d+5 a^2 d^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx}{8 b^2 d^2}\\ &=\frac {\left (b^2 c^2+2 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^3 d^2}-\frac {(3 b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{3/2}}{3 b d}+\frac {\left ((b c-a d) \left (b^2 c^2+2 a b c d+5 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 b^3 d^2}\\ &=\frac {\left (b^2 c^2+2 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^3 d^2}-\frac {(3 b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{3/2}}{3 b d}+\frac {\left ((b c-a d) \left (b^2 c^2+2 a b c d+5 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 )}{8 b^4 d^2}\\ &=\frac {\left (b^2 c^2+2 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^3 d^2}-\frac {(3 b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{3/2}}{3 b d}+\frac {\left ((b c-a d) \left (b^2 c^2+2 a b c d+5 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 )}{8 b^4 d^2}\\ &=\frac {\left (b^2 c^2+2 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^3 d^2}-\frac {(3 b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{3/2}}{3 b d}+\frac {(b c-a d) \left (b^2 c^2+2 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{7/2} d^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 167, normalized size = 0.87 \[ \frac {\sqrt {c+d x} \left (\sqrt {d} \sqrt {a+b x} \left (15 a^2 d^2-2 a b d (2 c+5 d x)+b^2 \left (-3 c^2+2 c d x+8 d^2 x^2\right )\right )+\frac {3 \sqrt {b c-a d} \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {\frac {b (c+d x)}{b c-a d}}}\right )}{24 b^3 d^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 408, normalized size = 2.14 \[ \left [-\frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (8 \, b^{3} d^{3} x^{2} - 3 \, b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + 15 \, a^{2} b d^{3} + 2 \, {\left (b^{3} c d^{2} - 5 \, a b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, b^{4} d^{3}}, -\frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (8 \, b^{3} d^{3} x^{2} - 3 \, b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + 15 \, a^{2} b d^{3} + 2 \, {\left (b^{3} c d^{2} - 5 \, a b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, b^{4} d^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.07, size = 207, normalized size = 1.08 \[ \frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} + \frac {b^{6} c d^{3} - 13 \, a b^{5} d^{4}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{7} c^{2} d^{2} + 2 \, a b^{6} c d^{3} - 11 \, a^{2} b^{5} d^{4}\right )}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b d^{2}}\right )} {\left | b \right |}}{24 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 395, normalized size = 2.07 \[ -\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (15 a^{3} d^{3} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-9 a^{2} b c \,d^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-3 a \,b^{2} c^{2} d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-3 b^{3} c^{3} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-16 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} d^{2} x^{2}+20 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,d^{2} x -4 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c d x -30 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2}+8 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d +6 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2}\right )}{48 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} d^{2}} \]
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 [B] time = 50.20, size = 924, normalized size = 4.84 \[ -\frac {\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (-\frac {5\,a^3\,b^2\,d^3}{4}+\frac {3\,a^2\,b^3\,c\,d^2}{4}+\frac {a\,b^4\,c^2\,d}{4}+\frac {b^5\,c^3}{4}\right )}{d^8\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (\frac {33\,a^3\,d^3}{2}+\frac {313\,a^2\,b\,c\,d^2}{2}+\frac {275\,a\,b^2\,c^2\,d}{2}+\frac {19\,b^3\,c^3}{2}\right )}{d^6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (-\frac {85\,a^3\,b\,d^3}{12}+\frac {17\,a^2\,b^2\,c\,d^2}{4}+\frac {91\,a\,b^3\,c^2\,d}{4}+\frac {17\,b^4\,c^3}{12}\right )}{d^7\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{11}\,\left (-\frac {5\,a^3\,d^3}{4}+\frac {3\,a^2\,b\,c\,d^2}{4}+\frac {a\,b^2\,c^2\,d}{4}+\frac {b^3\,c^3}{4}\right )}{b^3\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{11}}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^9\,\left (-\frac {85\,a^3\,d^3}{12}+\frac {17\,a^2\,b\,c\,d^2}{4}+\frac {91\,a\,b^2\,c^2\,d}{4}+\frac {17\,b^3\,c^3}{12}\right )}{b^2\,d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^9}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (\frac {33\,a^3\,d^3}{2}+\frac {313\,a^2\,b\,c\,d^2}{2}+\frac {275\,a\,b^2\,c^2\,d}{2}+\frac {19\,b^3\,c^3}{2}\right )}{b\,d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^7}+\frac {\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6\,\left (128\,a^2\,d^2+\frac {704\,a\,b\,c\,d}{3}+64\,b^2\,c^2\right )}{d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}+\frac {\sqrt {a}\,\sqrt {c}\,\left (32\,b\,c^2+96\,a\,d\,c\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}+\frac {\sqrt {a}\,\sqrt {c}\,\left (32\,b^3\,c^2+96\,a\,d\,b^2\,c\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{d^6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{12}}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{12}}+\frac {b^6}{d^6}-\frac {6\,b^5\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}+\frac {15\,b^4\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}-\frac {20\,b^3\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}+\frac {15\,b^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}-\frac {6\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{10}}{d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{10}}}-\frac {\mathrm {atanh}\left (\frac {\sqrt {d}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}\right )\,\left (a\,d-b\,c\right )\,\left (5\,a^2\,d^2+2\,a\,b\,c\,d+b^2\,c^2\right )}{4\,b^{7/2}\,d^{5/2}} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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