Optimal. Leaf size=220 \[ -\frac {a^2 (6 b c-11 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}}+\frac {a^2 \sqrt {c+d x} (6 b c-11 a d) (b c-a d)}{b^6}+\frac {a^2 (c+d x)^{3/2} (6 b c-11 a d)}{3 b^5}-\frac {(c+d x)^{5/2} \left (-693 a^2 d^2-5 b d x (10 b c-99 a d)+180 a b c d+20 b^2 c^2\right )}{315 b^4 d^2}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}+\frac {11 x^2 (c+d x)^{5/2}}{9 b^2} \]
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Rubi [A] time = 0.22, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {97, 153, 147, 50, 63, 208} \[ -\frac {(c+d x)^{5/2} \left (-693 a^2 d^2-5 b d x (10 b c-99 a d)+180 a b c d+20 b^2 c^2\right )}{315 b^4 d^2}+\frac {a^2 (c+d x)^{3/2} (6 b c-11 a d)}{3 b^5}+\frac {a^2 \sqrt {c+d x} (6 b c-11 a d) (b c-a d)}{b^6}-\frac {a^2 (6 b c-11 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}+\frac {11 x^2 (c+d x)^{5/2}}{9 b^2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 97
Rule 147
Rule 153
Rule 208
Rubi steps
\begin {align*} \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^2} \, dx &=-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}+\frac {\int \frac {x^2 (c+d x)^{3/2} \left (3 c+\frac {11 d x}{2}\right )}{a+b x} \, dx}{b}\\ &=\frac {11 x^2 (c+d x)^{5/2}}{9 b^2}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}+\frac {2 \int \frac {x (c+d x)^{3/2} \left (-11 a c d+\frac {1}{4} d (10 b c-99 a d) x\right )}{a+b x} \, dx}{9 b^2 d}\\ &=\frac {11 x^2 (c+d x)^{5/2}}{9 b^2}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}-\frac {(c+d x)^{5/2} \left (20 b^2 c^2+180 a b c d-693 a^2 d^2-5 b d (10 b c-99 a d) x\right )}{315 b^4 d^2}+\frac {\left (a^2 (6 b c-11 a d)\right ) \int \frac {(c+d x)^{3/2}}{a+b x} \, dx}{2 b^4}\\ &=\frac {a^2 (6 b c-11 a d) (c+d x)^{3/2}}{3 b^5}+\frac {11 x^2 (c+d x)^{5/2}}{9 b^2}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}-\frac {(c+d x)^{5/2} \left (20 b^2 c^2+180 a b c d-693 a^2 d^2-5 b d (10 b c-99 a d) x\right )}{315 b^4 d^2}+\frac {\left (a^2 (6 b c-11 a d) (b c-a d)\right ) \int \frac {\sqrt {c+d x}}{a+b x} \, dx}{2 b^5}\\ &=\frac {a^2 (6 b c-11 a d) (b c-a d) \sqrt {c+d x}}{b^6}+\frac {a^2 (6 b c-11 a d) (c+d x)^{3/2}}{3 b^5}+\frac {11 x^2 (c+d x)^{5/2}}{9 b^2}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}-\frac {(c+d x)^{5/2} \left (20 b^2 c^2+180 a b c d-693 a^2 d^2-5 b d (10 b c-99 a d) x\right )}{315 b^4 d^2}+\frac {\left (a^2 (6 b c-11 a d) (b c-a d)^2\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 b^6}\\ &=\frac {a^2 (6 b c-11 a d) (b c-a d) \sqrt {c+d x}}{b^6}+\frac {a^2 (6 b c-11 a d) (c+d x)^{3/2}}{3 b^5}+\frac {11 x^2 (c+d x)^{5/2}}{9 b^2}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}-\frac {(c+d x)^{5/2} \left (20 b^2 c^2+180 a b c d-693 a^2 d^2-5 b d (10 b c-99 a d) x\right )}{315 b^4 d^2}+\frac {\left (a^2 (6 b c-11 a d) (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{b^6 d}\\ &=\frac {a^2 (6 b c-11 a d) (b c-a d) \sqrt {c+d x}}{b^6}+\frac {a^2 (6 b c-11 a d) (c+d x)^{3/2}}{3 b^5}+\frac {11 x^2 (c+d x)^{5/2}}{9 b^2}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}-\frac {(c+d x)^{5/2} \left (20 b^2 c^2+180 a b c d-693 a^2 d^2-5 b d (10 b c-99 a d) x\right )}{315 b^4 d^2}-\frac {a^2 (6 b c-11 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}}\\ \end {align*}
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Mathematica [A] time = 0.82, size = 246, normalized size = 1.12 \[ \frac {21 a^2 d^2 (a+b x) (6 b c-11 a d) \left (\sqrt {b} \sqrt {c+d x} \left (15 a^2 d^2-5 a b d (7 c+d x)+b^2 \left (23 c^2+11 c d x+3 d^2 x^2\right )\right )-15 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )\right )+5 b^{7/2} (c+d x)^{7/2} \left (99 a^3 d^2+2 a^2 b d (11 d x-16 c)-2 a b^2 c (2 c+9 d x)-4 b^3 c^2 x\right )+70 b^{11/2} d x^2 (c+d x)^{7/2} (b c-a d)}{315 b^{13/2} d^2 (a+b x) (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.72, size = 788, normalized size = 3.58 \[ \left [\frac {315 \, {\left (6 \, a^{3} b^{2} c^{2} d^{2} - 17 \, a^{4} b c d^{3} + 11 \, a^{5} d^{4} + {\left (6 \, a^{2} b^{3} c^{2} d^{2} - 17 \, a^{3} b^{2} c d^{3} + 11 \, a^{4} b d^{4}\right )} x\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (70 \, b^{5} d^{4} x^{5} - 20 \, a b^{4} c^{4} - 180 \, a^{2} b^{3} c^{3} d + 3213 \, a^{3} b^{2} c^{2} d^{2} - 6510 \, a^{4} b c d^{3} + 3465 \, a^{5} d^{4} + 10 \, {\left (19 \, b^{5} c d^{3} - 11 \, a b^{4} d^{4}\right )} x^{4} + 2 \, {\left (75 \, b^{5} c^{2} d^{2} - 175 \, a b^{4} c d^{3} + 99 \, a^{2} b^{3} d^{4}\right )} x^{3} + 2 \, {\left (5 \, b^{5} c^{3} d - 195 \, a b^{4} c^{2} d^{2} + 423 \, a^{2} b^{3} c d^{3} - 231 \, a^{3} b^{2} d^{4}\right )} x^{2} - 2 \, {\left (10 \, b^{5} c^{4} + 85 \, a b^{4} c^{3} d - 1179 \, a^{2} b^{3} c^{2} d^{2} + 2247 \, a^{3} b^{2} c d^{3} - 1155 \, a^{4} b d^{4}\right )} x\right )} \sqrt {d x + c}}{630 \, {\left (b^{7} d^{2} x + a b^{6} d^{2}\right )}}, -\frac {315 \, {\left (6 \, a^{3} b^{2} c^{2} d^{2} - 17 \, a^{4} b c d^{3} + 11 \, a^{5} d^{4} + {\left (6 \, a^{2} b^{3} c^{2} d^{2} - 17 \, a^{3} b^{2} c d^{3} + 11 \, a^{4} b d^{4}\right )} x\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) - {\left (70 \, b^{5} d^{4} x^{5} - 20 \, a b^{4} c^{4} - 180 \, a^{2} b^{3} c^{3} d + 3213 \, a^{3} b^{2} c^{2} d^{2} - 6510 \, a^{4} b c d^{3} + 3465 \, a^{5} d^{4} + 10 \, {\left (19 \, b^{5} c d^{3} - 11 \, a b^{4} d^{4}\right )} x^{4} + 2 \, {\left (75 \, b^{5} c^{2} d^{2} - 175 \, a b^{4} c d^{3} + 99 \, a^{2} b^{3} d^{4}\right )} x^{3} + 2 \, {\left (5 \, b^{5} c^{3} d - 195 \, a b^{4} c^{2} d^{2} + 423 \, a^{2} b^{3} c d^{3} - 231 \, a^{3} b^{2} d^{4}\right )} x^{2} - 2 \, {\left (10 \, b^{5} c^{4} + 85 \, a b^{4} c^{3} d - 1179 \, a^{2} b^{3} c^{2} d^{2} + 2247 \, a^{3} b^{2} c d^{3} - 1155 \, a^{4} b d^{4}\right )} x\right )} \sqrt {d x + c}}{315 \, {\left (b^{7} d^{2} x + a b^{6} d^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.34, size = 323, normalized size = 1.47 \[ \frac {{\left (6 \, a^{2} b^{3} c^{3} - 23 \, a^{3} b^{2} c^{2} d + 28 \, a^{4} b c d^{2} - 11 \, a^{5} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{6}} + \frac {\sqrt {d x + c} a^{3} b^{2} c^{2} d - 2 \, \sqrt {d x + c} a^{4} b c d^{2} + \sqrt {d x + c} a^{5} d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} b^{6}} + \frac {2 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{16} d^{16} - 45 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{16} c d^{16} - 90 \, {\left (d x + c\right )}^{\frac {7}{2}} a b^{15} d^{17} + 189 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} b^{14} d^{18} + 315 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{14} c d^{18} + 945 \, \sqrt {d x + c} a^{2} b^{14} c^{2} d^{18} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b^{13} d^{19} - 2520 \, \sqrt {d x + c} a^{3} b^{13} c d^{19} + 1575 \, \sqrt {d x + c} a^{4} b^{12} d^{20}\right )}}{315 \, b^{18} d^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 415, normalized size = 1.89 \[ -\frac {11 a^{5} d^{3} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, b^{6}}+\frac {28 a^{4} c \,d^{2} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, b^{5}}-\frac {23 a^{3} c^{2} d \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, b^{4}}+\frac {6 a^{2} c^{3} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, b^{3}}+\frac {\sqrt {d x +c}\, a^{5} d^{3}}{\left (b d x +a d \right ) b^{6}}-\frac {2 \sqrt {d x +c}\, a^{4} c \,d^{2}}{\left (b d x +a d \right ) b^{5}}+\frac {\sqrt {d x +c}\, a^{3} c^{2} d}{\left (b d x +a d \right ) b^{4}}+\frac {10 \sqrt {d x +c}\, a^{4} d^{2}}{b^{6}}-\frac {16 \sqrt {d x +c}\, a^{3} c d}{b^{5}}+\frac {6 \sqrt {d x +c}\, a^{2} c^{2}}{b^{4}}-\frac {8 \left (d x +c \right )^{\frac {3}{2}} a^{3} d}{3 b^{5}}+\frac {2 \left (d x +c \right )^{\frac {3}{2}} a^{2} c}{b^{4}}+\frac {6 \left (d x +c \right )^{\frac {5}{2}} a^{2}}{5 b^{4}}-\frac {4 \left (d x +c \right )^{\frac {7}{2}} a}{7 b^{3} d}-\frac {2 \left (d x +c \right )^{\frac {7}{2}} c}{7 b^{2} d^{2}}+\frac {2 \left (d x +c \right )^{\frac {9}{2}}}{9 b^{2} d^{2}} \]
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 = 0.16, size = 679, normalized size = 3.09 \[ {\left (c+d\,x\right )}^{5/2}\,\left (\frac {6\,c^2}{5\,b^2\,d^2}-\frac {2\,{\left (a\,d-b\,c\right )}^2}{5\,b^4\,d^2}+\frac {2\,\left (\frac {6\,c}{b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d^2}\right )\,\left (a\,d-b\,c\right )}{5\,b}\right )-\left (\frac {{\left (a\,d-b\,c\right )}^2\,\left (\frac {6\,c^2}{b^2\,d^2}-\frac {2\,{\left (a\,d-b\,c\right )}^2}{b^4\,d^2}+\frac {2\,\left (\frac {6\,c}{b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d^2}\right )\,\left (a\,d-b\,c\right )}{b}\right )}{b^2}-\frac {2\,\left (a\,d-b\,c\right )\,\left (\frac {2\,c^3}{b^2\,d^2}-\frac {\left (\frac {6\,c}{b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d^2}\right )\,{\left (a\,d-b\,c\right )}^2}{b^2}+\frac {2\,\left (a\,d-b\,c\right )\,\left (\frac {6\,c^2}{b^2\,d^2}-\frac {2\,{\left (a\,d-b\,c\right )}^2}{b^4\,d^2}+\frac {2\,\left (\frac {6\,c}{b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d^2}\right )\,\left (a\,d-b\,c\right )}{b}\right )}{b}\right )}{b}\right )\,\sqrt {c+d\,x}-{\left (c+d\,x\right )}^{3/2}\,\left (\frac {2\,c^3}{3\,b^2\,d^2}-\frac {\left (\frac {6\,c}{b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d^2}\right )\,{\left (a\,d-b\,c\right )}^2}{3\,b^2}+\frac {2\,\left (a\,d-b\,c\right )\,\left (\frac {6\,c^2}{b^2\,d^2}-\frac {2\,{\left (a\,d-b\,c\right )}^2}{b^4\,d^2}+\frac {2\,\left (\frac {6\,c}{b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d^2}\right )\,\left (a\,d-b\,c\right )}{b}\right )}{3\,b}\right )-\left (\frac {6\,c}{7\,b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{7\,b^3\,d^2}\right )\,{\left (c+d\,x\right )}^{7/2}+\frac {2\,{\left (c+d\,x\right )}^{9/2}}{9\,b^2\,d^2}+\frac {\sqrt {c+d\,x}\,\left (a^5\,d^3-2\,a^4\,b\,c\,d^2+a^3\,b^2\,c^2\,d\right )}{b^7\,\left (c+d\,x\right )-b^7\,c+a\,b^6\,d}-\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{3/2}\,\left (11\,a\,d-6\,b\,c\right )\,\sqrt {c+d\,x}}{11\,a^5\,d^3-28\,a^4\,b\,c\,d^2+23\,a^3\,b^2\,c^2\,d-6\,a^2\,b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^{3/2}\,\left (11\,a\,d-6\,b\,c\right )}{b^{13/2}} \]
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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