Optimal. Leaf size=400 \[ -\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3003 e^4 (a+b x) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) \sqrt {d+e x} (b d-a e)^2}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 e^4 (a+b x) (d+e x)^{3/2} (b d-a e)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.24, antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {646, 47, 50, 63, 208} \[ -\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 e^4 (a+b x) (d+e x)^{3/2} (b d-a e)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) \sqrt {d+e x} (b d-a e)^2}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3003 e^4 (a+b x) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
Rule 646
Rubi steps
\begin {align*} \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{13/2}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (13 b^2 e \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{11/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (143 e^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{9/2}}{\left (a b+b^2 x\right )^3} \, dx}{48 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (429 e^3 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{7/2}}{\left (a b+b^2 x\right )^2} \, dx}{64 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 e^4 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{a b+b^2 x} \, dx}{128 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 e^4 \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{128 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {1001 e^4 (b d-a e) (a+b x) (d+e x)^{3/2}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 e^4 \left (b^2 d-a b e\right )^2 \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{128 b^8 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {3003 e^4 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 e^4 (b d-a e) (a+b x) (d+e x)^{3/2}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 e^4 \left (b^2 d-a b e\right )^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{128 b^{10} \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {3003 e^4 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 e^4 (b d-a e) (a+b x) (d+e x)^{3/2}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 e^3 \left (b^2 d-a b e\right )^3 \left (a b+b^2 x\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{64 b^{10} \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {3003 e^4 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 e^4 (b d-a e) (a+b x) (d+e x)^{3/2}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3003 e^4 (b d-a e)^{5/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 67, normalized size = 0.17 \[ -\frac {2 e^4 (a+b x) (d+e x)^{15/2} \, _2F_1\left (5,\frac {15}{2};\frac {17}{2};\frac {b (d+e x)}{b d-a e}\right )}{15 \sqrt {(a+b x)^2} (b d-a e)^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.03, size = 1286, normalized size = 3.22 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.45, size = 700, normalized size = 1.75 \[ \frac {3003 \, {\left (b^{3} d^{3} e^{4} - 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} b d e^{6} - a^{3} e^{7}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, \sqrt {-b^{2} d + a b e} b^{7} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} - \frac {4431 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{6} d^{3} e^{4} - 11767 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{6} d^{4} e^{4} + 10633 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{6} d^{5} e^{4} - 3249 \, \sqrt {x e + d} b^{6} d^{6} e^{4} - 13293 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{5} d^{2} e^{5} + 47068 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{5} d^{3} e^{5} - 53165 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{5} d^{4} e^{5} + 19494 \, \sqrt {x e + d} a b^{5} d^{5} e^{5} + 13293 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{2} b^{4} d e^{6} - 70602 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{4} d^{2} e^{6} + 106330 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{4} d^{3} e^{6} - 48735 \, \sqrt {x e + d} a^{2} b^{4} d^{4} e^{6} - 4431 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{3} b^{3} e^{7} + 47068 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{3} b^{3} d e^{7} - 106330 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{3} d^{2} e^{7} + 64980 \, \sqrt {x e + d} a^{3} b^{3} d^{3} e^{7} - 11767 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{4} b^{2} e^{8} + 53165 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{4} b^{2} d e^{8} - 48735 \, \sqrt {x e + d} a^{4} b^{2} d^{2} e^{8} - 10633 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{5} b e^{9} + 19494 \, \sqrt {x e + d} a^{5} b d e^{9} - 3249 \, \sqrt {x e + d} a^{6} e^{10}}{192 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{7} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{20} e^{4} + 25 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{20} d e^{4} + 225 \, \sqrt {x e + d} b^{20} d^{2} e^{4} - 25 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{19} e^{5} - 450 \, \sqrt {x e + d} a b^{19} d e^{5} + 225 \, \sqrt {x e + d} a^{2} b^{18} e^{6}\right )}}{15 \, b^{25} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 2192, normalized size = 5.48 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{\frac {13}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d+e\,x\right )}^{13/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
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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