Optimal. Leaf size=247 \[ \frac {1040}{7} c^2 d^7 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}+\frac {520 c d^{15/2} \left (b^2-4 a c\right )^{9/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{7 \sqrt {a+b x+c x^2}}+\frac {624}{7} c^2 d^5 \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}-\frac {52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt {a+b x+c x^2}}-\frac {2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.20, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {686, 692, 691, 689, 221} \[ \frac {1040}{7} c^2 d^7 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}+\frac {520 c d^{15/2} \left (b^2-4 a c\right )^{9/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{7 \sqrt {a+b x+c x^2}}+\frac {624}{7} c^2 d^5 \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}-\frac {52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt {a+b x+c x^2}}-\frac {2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 221
Rule 686
Rule 689
Rule 691
Rule 692
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}}+\frac {1}{3} \left (26 c d^2\right ) \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac {2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt {a+b x+c x^2}}+\left (156 c^2 d^4\right ) \int \frac {(b d+2 c d x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt {a+b x+c x^2}}+\frac {624}{7} c^2 d^5 (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}+\frac {1}{7} \left (780 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1040}{7} c^2 \left (b^2-4 a c\right ) d^7 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {624}{7} c^2 d^5 (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}+\frac {1}{7} \left (260 c^2 \left (b^2-4 a c\right )^2 d^8\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1040}{7} c^2 \left (b^2-4 a c\right ) d^7 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {624}{7} c^2 d^5 (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}+\frac {\left (260 c^2 \left (b^2-4 a c\right )^2 d^8 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac {1}{\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}{7 \sqrt {a+b x+c x^2}}\\ &=-\frac {2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1040}{7} c^2 \left (b^2-4 a c\right ) d^7 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {624}{7} c^2 d^5 (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}+\frac {\left (520 c \left (b^2-4 a c\right )^2 d^7 \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 )}{7 \sqrt {a+b x+c x^2}}\\ &=-\frac {2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1040}{7} c^2 \left (b^2-4 a c\right ) d^7 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {624}{7} c^2 d^5 (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}+\frac {520 c \left (b^2-4 a c\right )^{9/4} d^{15/2} \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 )}{7 \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.32, size = 252, normalized size = 1.02 \[ \frac {2 d^7 \sqrt {d (b+2 c x)} \left (16 b^2 c^2 \left (156 a^2+117 a c x^2+116 c^2 x^4\right )+32 b c^3 x \left (-273 a^2-104 a c x^2+36 c^2 x^4\right )-32 c^3 \left (195 a^3+273 a^2 c x^2+52 a c^2 x^4-12 c^3 x^6\right )+2 b^4 c \left (219 c x^2-91 a\right )+16 b^3 c^2 x \left (221 a+112 c x^2\right )+780 c \left (b^2-4 a c\right )^2 (a+x (b+c x)) \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )-7 b^6-266 b^5 c x\right )}{21 (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (128 \, c^{7} d^{7} x^{7} + 448 \, b c^{6} d^{7} x^{6} + 672 \, b^{2} c^{5} d^{7} x^{5} + 560 \, b^{3} c^{4} d^{7} x^{4} + 280 \, b^{4} c^{3} d^{7} x^{3} + 84 \, b^{5} c^{2} d^{7} x^{2} + 14 \, b^{6} c d^{7} x + b^{7} d^{7}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x + {\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (2 \, c d x + b d\right )}^{\frac {15}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 1473, normalized size = 5.96 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (2 \, c d x + b d\right )}^{\frac {15}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (b\,d+2\,c\,d\,x\right )}^{15/2}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]
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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