Optimal. Leaf size=196 \[ -\frac {2 d (b d+2 c d x)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+80 c^2 d^5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {40 c \left (b^2-4 a c\right )^{5/4} d^{11/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{\sqrt {a+b x+c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.12, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {700, 706, 705,
703, 227} \begin {gather*} \frac {40 c d^{11/2} \left (b^2-4 a c\right )^{5/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{\sqrt {a+b x+c x^2}}+80 c^2 d^5 \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}-\frac {2 d (b d+2 c d x)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 227
Rule 700
Rule 703
Rule 705
Rule 706
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 d (b d+2 c d x)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}+\left (6 c d^2\right ) \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac {2 d (b d+2 c d x)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+\left (60 c^2 d^4\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)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+80 c^2 d^5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\left (20 c^2 \left (b^2-4 a c\right ) d^6\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)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+80 c^2 d^5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {\left (20 c^2 \left (b^2-4 a c\right ) d^6 \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}{\sqrt {a+b x+c x^2}}\\ &=-\frac {2 d (b d+2 c d x)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+80 c^2 d^5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {\left (40 c \left (b^2-4 a c\right ) d^5 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {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 )}{\sqrt {a+b x+c x^2}}\\ &=-\frac {2 d (b d+2 c d x)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+80 c^2 d^5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {40 c \left (b^2-4 a c\right )^{5/4} d^{11/2} \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 )}{\sqrt {a+b x+c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.15, size = 201, normalized size = 1.03 \begin {gather*} \frac {2 d^5 \sqrt {d (b+2 c x)} \left (-b^4-26 b^3 c x+6 b^2 c \left (-3 a+c x^2\right )+8 b c^2 x \left (21 a+8 c x^2\right )+8 c^2 \left (15 a^2+21 a c x^2+4 c^2 x^4\right )-60 c \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \left (4 a^2 c-b^2 x (b+c x)+a \left (-b^2+4 b c x+4 c^2 x^2\right )\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )\right )}{3 (a+x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(957\) vs.
\(2(168)=336\).
time = 0.93, size = 958, normalized size = 4.89 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.94, size = 297, normalized size = 1.52 \begin {gather*} \frac {2 \, {\left (30 \, \sqrt {2} {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{5} x^{4} + 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{5} x^{3} + {\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} d^{5} x^{2} + 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} d^{5} x + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{5}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + {\left (32 \, c^{4} d^{5} x^{4} + 64 \, b c^{3} d^{5} x^{3} + 6 \, {\left (b^{2} c^{2} + 28 \, a c^{3}\right )} d^{5} x^{2} - 2 \, {\left (13 \, b^{3} c - 84 \, a b c^{2}\right )} d^{5} x - {\left (b^{4} + 18 \, a b^{2} c - 120 \, a^{2} c^{2}\right )} d^{5}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,d+2\,c\,d\,x\right )}^{11/2}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________