Optimal. Leaf size=504 \[ -\frac {4 \left (8 b^4-36 a b^2 c+21 a^2 c^2\right ) d^3 x \sqrt {a+b x+c x^2}}{315 c^{7/2} \sqrt {d x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {2 d^2 \sqrt {d x} \left (b \left (8 b^2+3 a c\right )+3 c \left (8 b^2-7 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{315 c^3}-\frac {4 b d^2 \sqrt {d x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c}+\frac {4 \sqrt [4]{a} \left (8 b^4-36 a b^2 c+21 a^2 c^2\right ) d^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{315 c^{15/4} \sqrt {d x} \sqrt {a+b x+c x^2}}-\frac {\sqrt [4]{a} \left (16 b^4-72 a b^2 c+42 a^2 c^2+\sqrt {a} b \sqrt {c} \left (8 b^2-27 a c\right )\right ) d^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{315 c^{15/4} \sqrt {d x} \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.43, antiderivative size = 504, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {756, 846, 828,
855, 853, 1211, 1117, 1209} \begin {gather*} -\frac {\sqrt [4]{a} d^3 \sqrt {x} \left (42 a^2 c^2-72 a b^2 c+\sqrt {a} b \sqrt {c} \left (8 b^2-27 a c\right )+16 b^4\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{315 c^{15/4} \sqrt {d x} \sqrt {a+b x+c x^2}}+\frac {4 \sqrt [4]{a} d^3 \sqrt {x} \left (21 a^2 c^2-36 a b^2 c+8 b^4\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{315 c^{15/4} \sqrt {d x} \sqrt {a+b x+c x^2}}-\frac {4 d^3 x \left (21 a^2 c^2-36 a b^2 c+8 b^4\right ) \sqrt {a+b x+c x^2}}{315 c^{7/2} \sqrt {d x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {2 d^2 \sqrt {d x} \left (3 c x \left (8 b^2-7 a c\right )+b \left (3 a c+8 b^2\right )\right ) \sqrt {a+b x+c x^2}}{315 c^3}-\frac {4 b d^2 \sqrt {d x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 756
Rule 828
Rule 846
Rule 853
Rule 855
Rule 1117
Rule 1209
Rule 1211
Rubi steps
\begin {align*} \int (d x)^{5/2} \sqrt {a+b x+c x^2} \, dx &=\frac {2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c}+\frac {2 \int \sqrt {d x} \left (-\frac {3 a d^2}{2}-3 b d^2 x\right ) \sqrt {a+b x+c x^2} \, dx}{9 c}\\ &=-\frac {4 b d^2 \sqrt {d x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c}+\frac {4 \int \frac {\left (\frac {3}{2} a b d^3+\frac {3}{4} \left (8 b^2-7 a c\right ) d^3 x\right ) \sqrt {a+b x+c x^2}}{\sqrt {d x}} \, dx}{63 c^2}\\ &=\frac {2 d^2 \sqrt {d x} \left (b \left (8 b^2+3 a c\right )+3 c \left (8 b^2-7 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{315 c^3}-\frac {4 b d^2 \sqrt {d x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c}-\frac {8 \int \frac {\frac {3}{8} a b \left (8 b^2-27 a c\right ) d^5+\frac {3}{4} \left (8 b^4-36 a b^2 c+21 a^2 c^2\right ) d^5 x}{\sqrt {d x} \sqrt {a+b x+c x^2}} \, dx}{945 c^3 d^2}\\ &=\frac {2 d^2 \sqrt {d x} \left (b \left (8 b^2+3 a c\right )+3 c \left (8 b^2-7 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{315 c^3}-\frac {4 b d^2 \sqrt {d x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c}-\frac {\left (8 \sqrt {x}\right ) \int \frac {\frac {3}{8} a b \left (8 b^2-27 a c\right ) d^5+\frac {3}{4} \left (8 b^4-36 a b^2 c+21 a^2 c^2\right ) d^5 x}{\sqrt {x} \sqrt {a+b x+c x^2}} \, dx}{945 c^3 d^2 \sqrt {d x}}\\ &=\frac {2 d^2 \sqrt {d x} \left (b \left (8 b^2+3 a c\right )+3 c \left (8 b^2-7 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{315 c^3}-\frac {4 b d^2 \sqrt {d x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c}-\frac {\left (16 \sqrt {x}\right ) \text {Subst}\left (\int \frac {\frac {3}{8} a b \left (8 b^2-27 a c\right ) d^5+\frac {3}{4} \left (8 b^4-36 a b^2 c+21 a^2 c^2\right ) d^5 x^2}{\sqrt {a+b x^2+c x^4}} \, dx,x,\sqrt {x}\right )}{945 c^3 d^2 \sqrt {d x}}\\ &=\frac {2 d^2 \sqrt {d x} \left (b \left (8 b^2+3 a c\right )+3 c \left (8 b^2-7 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{315 c^3}-\frac {4 b d^2 \sqrt {d x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c}+\frac {\left (4 \sqrt {a} \left (8 b^4-36 a b^2 c+21 a^2 c^2\right ) d^3 \sqrt {x}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx,x,\sqrt {x}\right )}{315 c^{7/2} \sqrt {d x}}-\frac {\left (2 \sqrt {a} \left (\sqrt {a} b \left (8 b^2-27 a c\right )+\frac {2 \left (8 b^4-36 a b^2 c+21 a^2 c^2\right )}{\sqrt {c}}\right ) d^3 \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx,x,\sqrt {x}\right )}{315 c^3 \sqrt {d x}}\\ &=-\frac {4 \left (8 b^4-36 a b^2 c+21 a^2 c^2\right ) d^3 x \sqrt {a+b x+c x^2}}{315 c^{7/2} \sqrt {d x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {2 d^2 \sqrt {d x} \left (b \left (8 b^2+3 a c\right )+3 c \left (8 b^2-7 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{315 c^3}-\frac {4 b d^2 \sqrt {d x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c}+\frac {4 \sqrt [4]{a} \left (8 b^4-36 a b^2 c+21 a^2 c^2\right ) d^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{315 c^{15/4} \sqrt {d x} \sqrt {a+b x+c x^2}}-\frac {\sqrt [4]{a} \left (\sqrt {a} b \left (8 b^2-27 a c\right )+\frac {2 \left (8 b^4-36 a b^2 c+21 a^2 c^2\right )}{\sqrt {c}}\right ) d^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{315 c^{13/4} \sqrt {d x} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 22.23, size = 594, normalized size = 1.18 \begin {gather*} \frac {(d x)^{5/2} \left (-\frac {4 \left (8 b^4-36 a b^2 c+21 a^2 c^2\right ) (a+x (b+c x))}{\sqrt {x}}+2 c \sqrt {x} (a+x (b+c x)) \left (8 b^3-6 b^2 c x+b c \left (-27 a+5 c x^2\right )+7 c^2 x \left (2 a+5 c x^2\right )\right )+\frac {i \left (8 b^4-36 a b^2 c+21 a^2 c^2\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}-\frac {i \left (-8 b^5+44 a b^3 c-48 a^2 b c^2+8 b^4 \sqrt {b^2-4 a c}-36 a b^2 c \sqrt {b^2-4 a c}+21 a^2 c^2 \sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}\right )}{315 c^4 x^{5/2} \sqrt {a+x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2061\) vs.
\(2(476)=952\).
time = 0.99, size = 2062, normalized size = 4.09 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.68, size = 267, normalized size = 0.53 \begin {gather*} \frac {2 \, {\left ({\left (16 \, b^{5} - 96 \, a b^{3} c + 123 \, a^{2} b c^{2}\right )} \sqrt {c d} d^{2} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) + 6 \, {\left (8 \, b^{4} c - 36 \, a b^{2} c^{2} + 21 \, a^{2} c^{3}\right )} \sqrt {c d} d^{2} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) + 3 \, {\left (35 \, c^{5} d^{2} x^{3} + 5 \, b c^{4} d^{2} x^{2} - 2 \, {\left (3 \, b^{2} c^{3} - 7 \, a c^{4}\right )} d^{2} x + {\left (8 \, b^{3} c^{2} - 27 \, a b c^{3}\right )} d^{2}\right )} \sqrt {c x^{2} + b x + a} \sqrt {d x}\right )}}{945 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d x\right )^{\frac {5}{2}} \sqrt {a + b x + c x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (d\,x\right )}^{5/2}\,\sqrt {c\,x^2+b\,x+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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