Optimal. Leaf size=435 \[ \frac {3003 b^2 e^4 (a+b x)}{64 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^7}+\frac {1001 b e^4 (a+b x)}{64 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^6}+\frac {3003 e^4 (a+b x)}{320 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}+\frac {429 e^3}{64 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}-\frac {143 e^2}{96 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}+\frac {13 e}{24 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac {1}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}-\frac {3003 b^{5/2} e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{15/2}} \]
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Rubi [A] time = 0.26, antiderivative size = 435, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {646, 51, 63, 208} \[ \frac {3003 b^2 e^4 (a+b x)}{64 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^7}+\frac {1001 b e^4 (a+b x)}{64 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^6}+\frac {3003 e^4 (a+b x)}{320 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}+\frac {429 e^3}{64 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}-\frac {143 e^2}{96 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}-\frac {3003 b^{5/2} e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{15/2}}+\frac {13 e}{24 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac {1}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 646
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{7/2} \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 {1}{\left (a b+b^2 x\right )^5 (d+e x)^{7/2}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (13 b^3 e \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^4 (d+e x)^{7/2}} \, dx}{8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (143 b^2 e^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^3 (d+e x)^{7/2}} \, dx}{48 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (429 b e^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^2 (d+e x)^{7/2}} \, dx}{64 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {429 e^3}{64 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \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 {1}{\left (a b+b^2 x\right ) (d+e x)^{7/2}} \, dx}{128 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {429 e^3}{64 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x)}{320 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 b e^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{5/2}} \, dx}{128 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {429 e^3}{64 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x)}{320 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 b e^4 (a+b x)}{64 (b d-a e)^6 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 b^2 e^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{3/2}} \, dx}{128 (b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {429 e^3}{64 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x)}{320 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 b e^4 (a+b x)}{64 (b d-a e)^6 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 b^2 e^4 (a+b x)}{64 (b d-a e)^7 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 b^3 e^4 \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 d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {429 e^3}{64 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x)}{320 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 b e^4 (a+b x)}{64 (b d-a e)^6 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 b^2 e^4 (a+b x)}{64 (b d-a e)^7 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 b^3 e^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 d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {429 e^3}{64 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x)}{320 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 b e^4 (a+b x)}{64 (b d-a e)^6 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 b^2 e^4 (a+b x)}{64 (b d-a e)^7 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3003 b^{5/2} e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 (b d-a e)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 67, normalized size = 0.15 \[ \frac {2 e^4 (a+b x) \, _2F_1\left (-\frac {5}{2},5;-\frac {3}{2};\frac {b (d+e x)}{b d-a e}\right )}{5 \sqrt {(a+b x)^2} (d+e x)^{5/2} (b d-a e)^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.85, size = 3368, normalized size = 7.74 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.66, size = 1114, normalized size = 2.56 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 951, normalized size = 2.19 \[ -\frac {\left (45045 \sqrt {\left (a e -b d \right ) b}\, b^{6} e^{6} x^{6}+165165 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} e^{6} x^{5}+105105 \sqrt {\left (a e -b d \right ) b}\, b^{6} d \,e^{5} x^{5}+219219 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{4} e^{6} x^{4}+387387 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} d \,e^{5} x^{4}+45045 \left (e x +d \right )^{\frac {5}{2}} b^{7} e^{4} x^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+69069 \sqrt {\left (a e -b d \right ) b}\, b^{6} d^{2} e^{4} x^{4}+119691 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{3} e^{6} x^{3}+517803 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{4} d \,e^{5} x^{3}+180180 \left (e x +d \right )^{\frac {5}{2}} a \,b^{6} e^{4} x^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+256971 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} d^{2} e^{4} x^{3}+6435 \sqrt {\left (a e -b d \right ) b}\, b^{6} d^{3} e^{3} x^{3}+18304 \sqrt {\left (a e -b d \right ) b}\, a^{4} b^{2} e^{6} x^{2}+285857 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{3} d \,e^{5} x^{2}+270270 \left (e x +d \right )^{\frac {5}{2}} a^{2} b^{5} e^{4} x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+347919 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{4} d^{2} e^{4} x^{2}+25025 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} d^{3} e^{3} x^{2}-1430 \sqrt {\left (a e -b d \right ) b}\, b^{6} d^{4} e^{2} x^{2}-1664 \sqrt {\left (a e -b d \right ) b}\, a^{5} b \,e^{6} x +44928 \sqrt {\left (a e -b d \right ) b}\, a^{4} b^{2} d \,e^{5} x +180180 \left (e x +d \right )^{\frac {5}{2}} a^{3} b^{4} e^{4} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+196001 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{3} d^{2} e^{4} x +35945 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{4} d^{3} e^{3} x -5460 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} d^{4} e^{2} x +520 \sqrt {\left (a e -b d \right ) b}\, b^{6} d^{5} e x +384 \sqrt {\left (a e -b d \right ) b}\, a^{6} e^{6}-3968 \sqrt {\left (a e -b d \right ) b}\, a^{5} b d \,e^{5}+45045 \left (e x +d \right )^{\frac {5}{2}} a^{4} b^{3} e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+32384 \sqrt {\left (a e -b d \right ) b}\, a^{4} b^{2} d^{2} e^{4}+22155 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{3} d^{3} e^{3}-7630 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{4} d^{4} e^{2}+1960 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} d^{5} e -240 \sqrt {\left (a e -b d \right ) b}\, b^{6} d^{6}\right ) \left (b x +a \right )}{960 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, \left (a e -b d \right )^{7} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}} \]
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 \[ \int \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (d+e\,x\right )}^{7/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
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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