Optimal. Leaf size=310 \[ -\frac {\left (b^2-4 a c\right )^{3/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{12 c^4 d^{11/2} \sqrt {a+b x+c x^2}}+\frac {\left (b^2-4 a c\right )^{3/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{12 c^4 d^{11/2} \sqrt {a+b x+c x^2}}-\frac {\sqrt {a+b x+c x^2}}{12 c^3 d^5 \sqrt {b d+2 c d x}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3 (b d+2 c d x)^{5/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{9 c d (b d+2 c d x)^{9/2}} \]
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Rubi [A] time = 0.29, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {684, 691, 690, 307, 221, 1199, 424} \[ -\frac {\left (b^2-4 a c\right )^{3/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{12 c^4 d^{11/2} \sqrt {a+b x+c x^2}}+\frac {\left (b^2-4 a c\right )^{3/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{12 c^4 d^{11/2} \sqrt {a+b x+c x^2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3 (b d+2 c d x)^{5/2}}-\frac {\sqrt {a+b x+c x^2}}{12 c^3 d^5 \sqrt {b d+2 c d x}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{9 c d (b d+2 c d x)^{9/2}} \]
Antiderivative was successfully verified.
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Rule 221
Rule 307
Rule 424
Rule 684
Rule 690
Rule 691
Rule 1199
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{11/2}} \, dx &=-\frac {\left (a+b x+c x^2\right )^{5/2}}{9 c d (b d+2 c d x)^{9/2}}+\frac {5 \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{7/2}} \, dx}{18 c d^2}\\ &=-\frac {\left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3 (b d+2 c d x)^{5/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{9 c d (b d+2 c d x)^{9/2}}+\frac {\int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{3/2}} \, dx}{12 c^2 d^4}\\ &=-\frac {\sqrt {a+b x+c x^2}}{12 c^3 d^5 \sqrt {b d+2 c d x}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3 (b d+2 c d x)^{5/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{9 c d (b d+2 c d x)^{9/2}}+\frac {\int \frac {\sqrt {b d+2 c d x}}{\sqrt {a+b x+c x^2}} \, dx}{24 c^3 d^6}\\ &=-\frac {\sqrt {a+b x+c x^2}}{12 c^3 d^5 \sqrt {b d+2 c d x}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3 (b d+2 c d x)^{5/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{9 c d (b d+2 c d x)^{9/2}}+\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {\sqrt {b d+2 c d x}}{\sqrt {-\frac {a c}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {c^2 x^2}{b^2-4 a c}}} \, dx}{24 c^3 d^6 \sqrt {a+b x+c x^2}}\\ &=-\frac {\sqrt {a+b x+c x^2}}{12 c^3 d^5 \sqrt {b d+2 c d x}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3 (b d+2 c d x)^{5/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{9 c d (b d+2 c d x)^{9/2}}+\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{12 c^4 d^7 \sqrt {a+b x+c x^2}}\\ &=-\frac {\sqrt {a+b x+c x^2}}{12 c^3 d^5 \sqrt {b d+2 c d x}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3 (b d+2 c d x)^{5/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{9 c d (b d+2 c d x)^{9/2}}-\frac {\left (\sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{12 c^4 d^6 \sqrt {a+b x+c x^2}}+\frac {\left (\sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b^2-4 a c} d}}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{12 c^4 d^6 \sqrt {a+b x+c x^2}}\\ &=-\frac {\sqrt {a+b x+c x^2}}{12 c^3 d^5 \sqrt {b d+2 c d x}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3 (b d+2 c d x)^{5/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{9 c d (b d+2 c d x)^{9/2}}-\frac {\left (b^2-4 a c\right )^{3/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{12 c^4 d^{11/2} \sqrt {a+b x+c x^2}}+\frac {\left (\sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x^2}{\sqrt {b^2-4 a c} d}}}{\sqrt {1-\frac {x^2}{\sqrt {b^2-4 a c} d}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{12 c^4 d^6 \sqrt {a+b x+c x^2}}\\ &=-\frac {\sqrt {a+b x+c x^2}}{12 c^3 d^5 \sqrt {b d+2 c d x}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3 (b d+2 c d x)^{5/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{9 c d (b d+2 c d x)^{9/2}}+\frac {\left (b^2-4 a c\right )^{3/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{12 c^4 d^{11/2} \sqrt {a+b x+c x^2}}-\frac {\left (b^2-4 a c\right )^{3/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{12 c^4 d^{11/2} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 109, normalized size = 0.35 \[ -\frac {\left (b^2-4 a c\right )^2 \sqrt {a+x (b+c x)} \sqrt {d (b+2 c x)} \, _2F_1\left (-\frac {5}{2},-\frac {9}{4};-\frac {5}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{288 c^3 d^6 (b+2 c x)^5 \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{64 \, c^{6} d^{6} x^{6} + 192 \, b c^{5} d^{6} x^{5} + 240 \, b^{2} c^{4} d^{6} x^{4} + 160 \, b^{3} c^{3} d^{6} x^{3} + 60 \, b^{4} c^{2} d^{6} x^{2} + 12 \, b^{5} c d^{6} x + b^{6} d^{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 1489, normalized size = 4.80 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac {11}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^{11/2}} \,d x \]
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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