∫ (d+ex)3∕2 ___________________

Optimal. Leaf size=441 \[ -\frac {\sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (12 b c d-7 b^2 e+4 a c e+12 c (2 c d-b e) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \sqrt {c} \left (16 c^2 d^2+b \left (3 b-2 \sqrt {b^2-4 a c}\right ) e^2-4 c e \left (4 b d-\sqrt {b^2-4 a c} d-a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {3 \sqrt {c} \left (16 c^2 d^2+b \left (3 b+2 \sqrt {b^2-4 a c}\right ) e^2-4 c e \left (4 b d+\sqrt {b^2-4 a c} d-a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

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Rubi [A]
time = 1.36, antiderivative size = 441, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {752, 836, 840, 1180, 214} \begin {gather*} -\frac {3 \sqrt {c} \left (-4 c e \left (-d \sqrt {b^2-4 a c}-a e+4 b d\right )+b e^2 \left (3 b-2 \sqrt {b^2-4 a c}\right )+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {3 \sqrt {c} \left (-4 c e \left (d \sqrt {b^2-4 a c}-a e+4 b d\right )+b e^2 \left (2 \sqrt {b^2-4 a c}+3 b\right )+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (4 a c e-7 b^2 e+12 c x (2 c d-b e)+12 b c d\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \end {gather*}

\begin {align*} \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {\sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} \left (12 c d^2-7 b d e+2 a e^2\right )+\frac {5}{2} e (2 c d-b e) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=-\frac {\sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (12 b c d-7 b^2 e+4 a c e+12 c (2 c d-b e) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\int \frac {\frac {3}{4} \left (c d^2-b d e+a e^2\right ) \left (16 c^2 d^2-12 b c d e+b^2 e^2+4 a c e^2\right )+3 c e (2 c d-b e) \left (c d^2-b d e+a e^2\right ) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {\sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (12 b c d-7 b^2 e+4 a c e+12 c (2 c d-b e) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {-3 c d e (2 c d-b e) \left (c d^2-b d e+a e^2\right )+\frac {3}{4} e \left (c d^2-b d e+a e^2\right ) \left (16 c^2 d^2-12 b c d e+b^2 e^2+4 a c e^2\right )+3 c e (2 c d-b e) \left (c d^2-b d e+a e^2\right ) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {\sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (12 b c d-7 b^2 e+4 a c e+12 c (2 c d-b e) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\left (3 c \left (16 c^2 d^2+b \left (3 b-2 \sqrt {b^2-4 a c}\right ) e^2-4 c e \left (4 b d-\sqrt {b^2-4 a c} d-a e\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 \left (b^2-4 a c\right )^{5/2}}-\frac {\left (3 c \left (16 c^2 d^2+b \left (3 b+2 \sqrt {b^2-4 a c}\right ) e^2-4 c e \left (4 b d+\sqrt {b^2-4 a c} d-a e\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 \left (b^2-4 a c\right )^{5/2}}\\ &=-\frac {\sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (12 b c d-7 b^2 e+4 a c e+12 c (2 c d-b e) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \sqrt {c} \left (16 c^2 d^2+b \left (3 b-2 \sqrt {b^2-4 a c}\right ) e^2-4 c e \left (4 b d-\sqrt {b^2-4 a c} d-a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {3 \sqrt {c} \left (16 c^2 d^2+b \left (3 b+2 \sqrt {b^2-4 a c}\right ) e^2-4 c e \left (4 b d+\sqrt {b^2-4 a c} d-a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(4707\) vs. \(2(441)=882\).
time = 16.19, size = 4707, normalized size = 10.67 \begin {gather*} \text {Result too large to show} \end {gather*}

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method	result	size

derivativedivides	\(2 e^{5} \left (\frac {-\frac {3 c^{2} \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{2 e^{4} \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}+\frac {c \left (4 a c \,e^{2}-19 b^{2} e^{2}+72 b c d e -72 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}-\frac {\left (b e -2 c d \right ) \left (16 a c \,e^{2}+5 b^{2} e^{2}-36 b c d e +36 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}-\frac {3 \left (4 e^{4} a^{2} c +a \,b^{2} e^{4}-12 a b c d \,e^{3}+12 d^{2} e^{2} c^{2} a -b^{3} d \,e^{3}+9 b^{2} c \,d^{2} e^{2}-16 d^{3} e b \,c^{2}+8 d^{4} c^{3}\right ) \sqrt {e x +d}}{8 e^{4} \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}}{\left (\left (e x +d \right )^{2} c +b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}\right )^{2}}+\frac {3 c \left (\frac {\left (-4 a c \,e^{2}-3 b^{2} e^{2}+16 b c d e -16 c^{2} d^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (4 a c \,e^{2}+3 b^{2} e^{2}-16 b c d e +16 c^{2} d^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 e^{4} \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}\right )\)	\(703\)
default	\(2 e^{5} \left (\frac {-\frac {3 c^{2} \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{2 e^{4} \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}+\frac {c \left (4 a c \,e^{2}-19 b^{2} e^{2}+72 b c d e -72 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}-\frac {\left (b e -2 c d \right ) \left (16 a c \,e^{2}+5 b^{2} e^{2}-36 b c d e +36 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}-\frac {3 \left (4 e^{4} a^{2} c +a \,b^{2} e^{4}-12 a b c d \,e^{3}+12 d^{2} e^{2} c^{2} a -b^{3} d \,e^{3}+9 b^{2} c \,d^{2} e^{2}-16 d^{3} e b \,c^{2}+8 d^{4} c^{3}\right ) \sqrt {e x +d}}{8 e^{4} \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}}{\left (\left (e x +d \right )^{2} c +b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}\right )^{2}}+\frac {3 c \left (\frac {\left (-4 a c \,e^{2}-3 b^{2} e^{2}+16 b c d e -16 c^{2} d^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (4 a c \,e^{2}+3 b^{2} e^{2}-16 b c d e +16 c^{2} d^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 e^{4} \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}\right )\)	\(703\)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 10990 vs. \(2 (396) = 792\).
time = 3.18, size = 10990, normalized size = 24.92 \begin {gather*} \text {Too large to display} \end {gather*}

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1452 vs. \(2 (396) = 792\).
time = 2.97, size = 1452, normalized size = 3.29 \begin {gather*} -\frac {3 \, {\left (16 \, \sqrt {b^{2} - 4 \, a c} c^{3} d^{2} + 4 \, {\left (b^{2} c^{2} - 4 \, a c^{3} - 4 \, \sqrt {b^{2} - 4 \, a c} b c^{2}\right )} d e - {\left (2 \, b^{3} c - 8 \, a b c^{2} - {\left (3 \, b^{2} c + 4 \, a c^{2}\right )} \sqrt {b^{2} - 4 \, a c}\right )} e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, b^{4} c d - 16 \, a b^{2} c^{2} d + 32 \, a^{2} c^{3} d - b^{5} e + 8 \, a b^{3} c e - 16 \, a^{2} b c^{2} e + \sqrt {{\left (2 \, b^{4} c d - 16 \, a b^{2} c^{2} d + 32 \, a^{2} c^{3} d - b^{5} e + 8 \, a b^{3} c e - 16 \, a^{2} b c^{2} e\right )}^{2} - 4 \, {\left (b^{4} c d^{2} - 8 \, a b^{2} c^{2} d^{2} + 16 \, a^{2} c^{3} d^{2} - b^{5} d e + 8 \, a b^{3} c d e - 16 \, a^{2} b c^{2} d e + a b^{4} e^{2} - 8 \, a^{2} b^{2} c e^{2} + 16 \, a^{3} c^{2} e^{2}\right )} {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )}}}{b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}}}}\right )}{4 \, {\left (2 \, {\left (b^{6} c - 12 \, a b^{4} c^{2} + 48 \, a^{2} b^{2} c^{3} - 64 \, a^{3} c^{4}\right )} d - {\left (b^{7} - 12 \, a b^{5} c + 48 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3} - {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {b^{2} - 4 \, a c}\right )} e\right )} {\left | c \right |}} + \frac {3 \, {\left (16 \, \sqrt {b^{2} - 4 \, a c} c^{3} d^{2} - 4 \, {\left (b^{2} c^{2} - 4 \, a c^{3} + 4 \, \sqrt {b^{2} - 4 \, a c} b c^{2}\right )} d e + {\left (2 \, b^{3} c - 8 \, a b c^{2} + {\left (3 \, b^{2} c + 4 \, a c^{2}\right )} \sqrt {b^{2} - 4 \, a c}\right )} e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, b^{4} c d - 16 \, a b^{2} c^{2} d + 32 \, a^{2} c^{3} d - b^{5} e + 8 \, a b^{3} c e - 16 \, a^{2} b c^{2} e - \sqrt {{\left (2 \, b^{4} c d - 16 \, a b^{2} c^{2} d + 32 \, a^{2} c^{3} d - b^{5} e + 8 \, a b^{3} c e - 16 \, a^{2} b c^{2} e\right )}^{2} - 4 \, {\left (b^{4} c d^{2} - 8 \, a b^{2} c^{2} d^{2} + 16 \, a^{2} c^{3} d^{2} - b^{5} d e + 8 \, a b^{3} c d e - 16 \, a^{2} b c^{2} d e + a b^{4} e^{2} - 8 \, a^{2} b^{2} c e^{2} + 16 \, a^{3} c^{2} e^{2}\right )} {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )}}}{b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}}}}\right )}{4 \, {\left (2 \, {\left (b^{6} c - 12 \, a b^{4} c^{2} + 48 \, a^{2} b^{2} c^{3} - 64 \, a^{3} c^{4}\right )} d - {\left (b^{7} - 12 \, a b^{5} c + 48 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3} + {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {b^{2} - 4 \, a c}\right )} e\right )} {\left | c \right |}} + \frac {24 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d e - 72 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{2} e + 72 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{3} e - 24 \, \sqrt {x e + d} c^{3} d^{4} e - 12 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{2} e^{2} + 72 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{2} d e^{2} - 108 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{2} d^{2} e^{2} + 48 \, \sqrt {x e + d} b c^{2} d^{3} e^{2} - 19 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c e^{3} + 4 \, {\left (x e + d\right )}^{\frac {5}{2}} a c^{2} e^{3} + 46 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c d e^{3} + 32 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} d e^{3} - 27 \, \sqrt {x e + d} b^{2} c d^{2} e^{3} - 36 \, \sqrt {x e + d} a c^{2} d^{2} e^{3} - 5 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} e^{4} - 16 \, {\left (x e + d\right )}^{\frac {3}{2}} a b c e^{4} + 3 \, \sqrt {x e + d} b^{3} d e^{4} + 36 \, \sqrt {x e + d} a b c d e^{4} - 3 \, \sqrt {x e + d} a b^{2} e^{5} - 12 \, \sqrt {x e + d} a^{2} c e^{5}}{4 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e + a e^{2}\right )}^{2}} \end {gather*}

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Mupad [B]
time = 21.97, size = 2500, normalized size = 5.67 \begin {gather*} \text {Too large to display} \end {gather*}

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3.24.4 \(\int \frac {(d+e x)^{3/2}}{(a+b x+c x^2)^3} \, dx\) [2304]