Optimal. Leaf size=299 \[ -\frac {616 c d^{13/2} \left (b^2-4 a c\right )^{7/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{5 \sqrt {a+b x+c x^2}}+\frac {616 c d^{13/2} \left (b^2-4 a c\right )^{7/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{5 \sqrt {a+b x+c x^2}}+\frac {1232}{15} c^2 d^5 \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}-\frac {44 c d^3 (b d+2 c d x)^{7/2}}{3 \sqrt {a+b x+c x^2}}-\frac {2 d (b d+2 c d x)^{11/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.29, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {686, 692, 691, 690, 307, 221, 1199, 424} \[ -\frac {616 c d^{13/2} \left (b^2-4 a c\right )^{7/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{5 \sqrt {a+b x+c x^2}}+\frac {616 c d^{13/2} \left (b^2-4 a c\right )^{7/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{5 \sqrt {a+b x+c x^2}}+\frac {1232}{15} c^2 d^5 \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}-\frac {44 c d^3 (b d+2 c d x)^{7/2}}{3 \sqrt {a+b x+c x^2}}-\frac {2 d (b d+2 c d x)^{11/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 221
Rule 307
Rule 424
Rule 686
Rule 690
Rule 691
Rule 692
Rule 1199
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^{13/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 d (b d+2 c d x)^{11/2}}{3 \left (a+b x+c x^2\right )^{3/2}}+\frac {1}{3} \left (22 c d^2\right ) \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac {2 d (b d+2 c d x)^{11/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {44 c d^3 (b d+2 c d x)^{7/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1}{3} \left (308 c^2 d^4\right ) \int \frac {(b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {2 d (b d+2 c d x)^{11/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {44 c d^3 (b d+2 c d x)^{7/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1232}{15} c^2 d^5 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}+\frac {1}{5} \left (308 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac {\sqrt {b d+2 c d x}}{\sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {2 d (b d+2 c d x)^{11/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {44 c d^3 (b d+2 c d x)^{7/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1232}{15} c^2 d^5 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}+\frac {\left (308 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 {\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}{5 \sqrt {a+b x+c x^2}}\\ &=-\frac {2 d (b d+2 c d x)^{11/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {44 c d^3 (b d+2 c d x)^{7/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1232}{15} c^2 d^5 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}+\frac {\left (616 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 ) \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 )}{5 \sqrt {a+b x+c x^2}}\\ &=-\frac {2 d (b d+2 c d x)^{11/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {44 c d^3 (b d+2 c d x)^{7/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1232}{15} c^2 d^5 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}-\frac {\left (616 c \left (b^2-4 a c\right )^{3/2} d^6 \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 )}{5 \sqrt {a+b x+c x^2}}+\frac {\left (616 c \left (b^2-4 a c\right )^{3/2} d^6 \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 )}{5 \sqrt {a+b x+c x^2}}\\ &=-\frac {2 d (b d+2 c d x)^{11/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {44 c d^3 (b d+2 c d x)^{7/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1232}{15} c^2 d^5 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}-\frac {616 c \left (b^2-4 a c\right )^{7/4} d^{13/2} \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 )}{5 \sqrt {a+b x+c x^2}}+\frac {\left (616 c \left (b^2-4 a c\right )^{3/2} d^6 \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 )}{5 \sqrt {a+b x+c x^2}}\\ &=-\frac {2 d (b d+2 c d x)^{11/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {44 c d^3 (b d+2 c d x)^{7/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1232}{15} c^2 d^5 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}+\frac {616 c \left (b^2-4 a c\right )^{7/4} d^{13/2} \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 )}{5 \sqrt {a+b x+c x^2}}-\frac {616 c \left (b^2-4 a c\right )^{7/4} d^{13/2} \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 )}{5 \sqrt {a+b x+c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.22, size = 202, normalized size = 0.68 \[ \frac {4 d^5 (d (b+2 c x))^{3/2} \left (616 c \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}} \left (4 a^2 c+a \left (-b^2+4 b c x+4 c^2 x^2\right )-b^2 x (b+c x)\right ) \, _2F_1\left (\frac {3}{4},\frac {5}{2};\frac {7}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )+16 c^2 \left (-77 a^2-33 a c x^2+3 c^2 x^4\right )+4 b^2 c \left (121 a+51 c x^2\right )+48 b c^2 x \left (2 c x^2-11 a\right )-41 b^4+156 b^3 c x\right )}{15 (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (64 \, c^{6} d^{6} x^{6} + 192 \, b c^{5} d^{6} x^{5} + 240 \, b^{2} c^{4} d^{6} x^{4} + 160 \, b^{3} c^{3} d^{6} x^{3} + 60 \, b^{4} c^{2} d^{6} x^{2} + 12 \, b^{5} c d^{6} x + b^{6} d^{6}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x + {\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (2 \, c d x + b d\right )}^{\frac {13}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.13, size = 1328, normalized size = 4.44 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (2 \, c d x + b d\right )}^{\frac {13}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (b\,d+2\,c\,d\,x\right )}^{13/2}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________