Optimal. Leaf size=244 \[ \frac {2 b (e+f x)^{5/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{5 d^3 f^3}-\frac {2 b^2 (e+f x)^{7/2} (-3 a d f+b c f+2 b d e)}{7 d^2 f^3}+\frac {2 (b c-a d)^3 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{11/2}}-\frac {2 \sqrt {e+f x} (b c-a d)^3 (d e-c f)}{d^5}-\frac {2 (e+f x)^{3/2} (b c-a d)^3}{3 d^4}+\frac {2 b^3 (e+f x)^{9/2}}{9 d f^3} \]
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Rubi [A] time = 0.22, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {88, 50, 63, 208} \[ \frac {2 b (e+f x)^{5/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{5 d^3 f^3}-\frac {2 b^2 (e+f x)^{7/2} (-3 a d f+b c f+2 b d e)}{7 d^2 f^3}-\frac {2 (e+f x)^{3/2} (b c-a d)^3}{3 d^4}-\frac {2 \sqrt {e+f x} (b c-a d)^3 (d e-c f)}{d^5}+\frac {2 (b c-a d)^3 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{11/2}}+\frac {2 b^3 (e+f x)^{9/2}}{9 d f^3} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 88
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x)^3 (e+f x)^{3/2}}{c+d x} \, dx &=\int \left (\frac {b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{3/2}}{d^3 f^2}+\frac {(-b c+a d)^3 (e+f x)^{3/2}}{d^3 (c+d x)}-\frac {b^2 (2 b d e+b c f-3 a d f) (e+f x)^{5/2}}{d^2 f^2}+\frac {b^3 (e+f x)^{7/2}}{d f^2}\right ) \, dx\\ &=\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{5/2}}{5 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{7/2}}{7 d^2 f^3}+\frac {2 b^3 (e+f x)^{9/2}}{9 d f^3}-\frac {(b c-a d)^3 \int \frac {(e+f x)^{3/2}}{c+d x} \, dx}{d^3}\\ &=-\frac {2 (b c-a d)^3 (e+f x)^{3/2}}{3 d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{5/2}}{5 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{7/2}}{7 d^2 f^3}+\frac {2 b^3 (e+f x)^{9/2}}{9 d f^3}-\frac {\left ((b c-a d)^3 (d e-c f)\right ) \int \frac {\sqrt {e+f x}}{c+d x} \, dx}{d^4}\\ &=-\frac {2 (b c-a d)^3 (d e-c f) \sqrt {e+f x}}{d^5}-\frac {2 (b c-a d)^3 (e+f x)^{3/2}}{3 d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{5/2}}{5 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{7/2}}{7 d^2 f^3}+\frac {2 b^3 (e+f x)^{9/2}}{9 d f^3}-\frac {\left ((b c-a d)^3 (d e-c f)^2\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}} \, dx}{d^5}\\ &=-\frac {2 (b c-a d)^3 (d e-c f) \sqrt {e+f x}}{d^5}-\frac {2 (b c-a d)^3 (e+f x)^{3/2}}{3 d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{5/2}}{5 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{7/2}}{7 d^2 f^3}+\frac {2 b^3 (e+f x)^{9/2}}{9 d f^3}-\frac {\left (2 (b c-a d)^3 (d e-c f)^2\right ) \operatorname {Subst}\left (\int \frac {1}{c-\frac {d e}{f}+\frac {d x^2}{f}} \, dx,x,\sqrt {e+f x}\right )}{d^5 f}\\ &=-\frac {2 (b c-a d)^3 (d e-c f) \sqrt {e+f x}}{d^5}-\frac {2 (b c-a d)^3 (e+f x)^{3/2}}{3 d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{5/2}}{5 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{7/2}}{7 d^2 f^3}+\frac {2 b^3 (e+f x)^{9/2}}{9 d f^3}+\frac {2 (b c-a d)^3 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 232, normalized size = 0.95 \[ \frac {2 \left (\frac {63 b d (e+f x)^{5/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{f^3}-\frac {45 b^2 d^2 (e+f x)^{7/2} (-3 a d f+b c f+2 b d e)}{f^3}+\frac {315 (a d-b c)^3 (d e-c f) \left (\sqrt {d} \sqrt {e+f x}-\sqrt {d e-c f} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )\right )}{d^{3/2}}-105 (e+f x)^{3/2} (b c-a d)^3+\frac {35 b^3 d^3 (e+f x)^{9/2}}{f^3}\right )}{315 d^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.78, size = 1165, normalized size = 4.77 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.40, size = 683, normalized size = 2.80 \[ -\frac {2 \, {\left (b^{3} c^{5} f^{2} - 3 \, a b^{2} c^{4} d f^{2} + 3 \, a^{2} b c^{3} d^{2} f^{2} - a^{3} c^{2} d^{3} f^{2} - 2 \, b^{3} c^{4} d f e + 6 \, a b^{2} c^{3} d^{2} f e - 6 \, a^{2} b c^{2} d^{3} f e + 2 \, a^{3} c d^{4} f e + b^{3} c^{3} d^{2} e^{2} - 3 \, a b^{2} c^{2} d^{3} e^{2} + 3 \, a^{2} b c d^{4} e^{2} - a^{3} d^{5} e^{2}\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {c d f - d^{2} e}}\right )}{\sqrt {c d f - d^{2} e} d^{5}} + \frac {2 \, {\left (35 \, {\left (f x + e\right )}^{\frac {9}{2}} b^{3} d^{8} f^{24} - 45 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{3} c d^{7} f^{25} + 135 \, {\left (f x + e\right )}^{\frac {7}{2}} a b^{2} d^{8} f^{25} + 63 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{3} c^{2} d^{6} f^{26} - 189 \, {\left (f x + e\right )}^{\frac {5}{2}} a b^{2} c d^{7} f^{26} + 189 \, {\left (f x + e\right )}^{\frac {5}{2}} a^{2} b d^{8} f^{26} - 105 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c^{3} d^{5} f^{27} + 315 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} c^{2} d^{6} f^{27} - 315 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} b c d^{7} f^{27} + 105 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{3} d^{8} f^{27} + 315 \, \sqrt {f x + e} b^{3} c^{4} d^{4} f^{28} - 945 \, \sqrt {f x + e} a b^{2} c^{3} d^{5} f^{28} + 945 \, \sqrt {f x + e} a^{2} b c^{2} d^{6} f^{28} - 315 \, \sqrt {f x + e} a^{3} c d^{7} f^{28} - 90 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{3} d^{8} f^{24} e + 63 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{3} c d^{7} f^{25} e - 189 \, {\left (f x + e\right )}^{\frac {5}{2}} a b^{2} d^{8} f^{25} e - 315 \, \sqrt {f x + e} b^{3} c^{3} d^{5} f^{27} e + 945 \, \sqrt {f x + e} a b^{2} c^{2} d^{6} f^{27} e - 945 \, \sqrt {f x + e} a^{2} b c d^{7} f^{27} e + 315 \, \sqrt {f x + e} a^{3} d^{8} f^{27} e + 63 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{3} d^{8} f^{24} e^{2}\right )}}{315 \, d^{9} f^{27}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 984, normalized size = 4.03 \[ \text {result too large to display} \]
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 = 1.28, size = 672, normalized size = 2.75 \[ {\left (e+f\,x\right )}^{5/2}\,\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{5\,d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{5\,d\,f^3}\right )-{\left (e+f\,x\right )}^{7/2}\,\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{7\,d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{7\,d^2\,f^6}\right )+{\left (e+f\,x\right )}^{3/2}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^3}{3\,d\,f^3}-\frac {\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{d\,f^3}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{3\,d\,f^3}\right )+\frac {2\,b^3\,{\left (e+f\,x\right )}^{9/2}}{9\,d\,f^3}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^3\,{\left (c\,f-d\,e\right )}^{3/2}}{a^3\,c^2\,d^3\,f^2-2\,a^3\,c\,d^4\,e\,f+a^3\,d^5\,e^2-3\,a^2\,b\,c^3\,d^2\,f^2+6\,a^2\,b\,c^2\,d^3\,e\,f-3\,a^2\,b\,c\,d^4\,e^2+3\,a\,b^2\,c^4\,d\,f^2-6\,a\,b^2\,c^3\,d^2\,e\,f+3\,a\,b^2\,c^2\,d^3\,e^2-b^3\,c^5\,f^2+2\,b^3\,c^4\,d\,e\,f-b^3\,c^3\,d^2\,e^2}\right )\,{\left (a\,d-b\,c\right )}^3\,{\left (c\,f-d\,e\right )}^{3/2}}{d^{11/2}}-\frac {\sqrt {e+f\,x}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^3}{d\,f^3}-\frac {\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{d\,f^3}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 119.26, size = 381, normalized size = 1.56 \[ \frac {2 b^{3} \left (e + f x\right )^{\frac {9}{2}}}{9 d f^{3}} + \frac {\left (e + f x\right )^{\frac {7}{2}} \left (6 a b^{2} d f - 2 b^{3} c f - 4 b^{3} d e\right )}{7 d^{2} f^{3}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \left (6 a^{2} b d^{2} f^{2} - 6 a b^{2} c d f^{2} - 6 a b^{2} d^{2} e f + 2 b^{3} c^{2} f^{2} + 2 b^{3} c d e f + 2 b^{3} d^{2} e^{2}\right )}{5 d^{3} f^{3}} + \frac {\left (e + f x\right )^{\frac {3}{2}} \left (2 a^{3} d^{3} - 6 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}\right )}{3 d^{4}} + \frac {\sqrt {e + f x} \left (- 2 a^{3} c d^{3} f + 2 a^{3} d^{4} e + 6 a^{2} b c^{2} d^{2} f - 6 a^{2} b c d^{3} e - 6 a b^{2} c^{3} d f + 6 a b^{2} c^{2} d^{2} e + 2 b^{3} c^{4} f - 2 b^{3} c^{3} d e\right )}{d^{5}} + \frac {2 \left (a d - b c\right )^{3} \left (c f - d e\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{6} \sqrt {\frac {c f - d e}{d}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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