Optimal. Leaf size=205 \[ \frac{60 d^{11/2} \left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{7 \sqrt{a+b x+c x^2}}+\frac{120}{7} c d^5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}+\frac{72}{7} c d^3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}-\frac{2 d (b d+2 c d x)^{9/2}}{\sqrt{a+b x+c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.175973, antiderivative size = 205, normalized size of antiderivative = 1., 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 = {686, 692, 691, 689, 221} \[ \frac{120}{7} c d^5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}+\frac{60 d^{11/2} \left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{7 \sqrt{a+b x+c x^2}}+\frac{72}{7} c d^3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}-\frac{2 d (b d+2 c d x)^{9/2}}{\sqrt{a+b x+c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 686
Rule 692
Rule 691
Rule 689
Rule 221
Rubi steps
\begin{align*} \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)^{9/2}}{\sqrt{a+b x+c x^2}}+\left (18 c d^2\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)^{9/2}}{\sqrt{a+b x+c x^2}}+\frac{72}{7} c d^3 (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}+\frac{1}{7} \left (90 c \left (b^2-4 a c\right ) 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}}{\sqrt{a+b x+c x^2}}+\frac{120}{7} c \left (b^2-4 a c\right ) d^5 \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}+\frac{72}{7} c d^3 (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}+\frac{1}{7} \left (30 c \left (b^2-4 a c\right )^2 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}}{\sqrt{a+b x+c x^2}}+\frac{120}{7} c \left (b^2-4 a c\right ) d^5 \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}+\frac{72}{7} c d^3 (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}+\frac{\left (30 c \left (b^2-4 a c\right )^2 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}{7 \sqrt{a+b x+c x^2}}\\ &=-\frac{2 d (b d+2 c d x)^{9/2}}{\sqrt{a+b x+c x^2}}+\frac{120}{7} c \left (b^2-4 a c\right ) d^5 \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}+\frac{72}{7} c d^3 (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}+\frac{\left (60 \left (b^2-4 a c\right )^2 d^5 \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)^{9/2}}{\sqrt{a+b x+c x^2}}+\frac{120}{7} c \left (b^2-4 a c\right ) d^5 \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}+\frac{72}{7} c d^3 (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}+\frac{60 \left (b^2-4 a c\right )^{9/4} 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 )}{7 \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.197787, size = 172, normalized size = 0.84 \[ \frac{2 d^5 \sqrt{d (b+2 c x)} \left (16 c^2 \left (-15 a^2-6 a c x^2+2 c^2 x^4\right )+30 \left (b^2-4 a c\right )^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )+24 b^2 c \left (4 a+3 c x^2\right )+32 b c^2 x \left (2 c x^2-3 a\right )+40 b^3 c x-7 b^4\right )}{7 \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.343, size = 569, normalized size = 2.8 \begin{align*}{\frac{2\,{d}^{5}}{14\,{c}^{2}{x}^{3}+21\,bc{x}^{2}+14\,acx+7\,{b}^{2}x+7\,ab}\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a} \left ( 64\,{x}^{5}{c}^{5}+240\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{a}^{2}{c}^{2}-120\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}a{b}^{2}c+15\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{b}^{4}+160\,{x}^{4}b{c}^{4}-192\,{x}^{3}a{c}^{4}+208\,{x}^{3}{b}^{2}{c}^{3}-288\,{x}^{2}ab{c}^{3}+152\,{x}^{2}{b}^{3}{c}^{2}-480\,x{a}^{2}{c}^{3}+96\,xa{b}^{2}{c}^{2}+26\,x{b}^{4}c-240\,{a}^{2}b{c}^{2}+96\,a{b}^{3}c-7\,{b}^{5} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{11}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (32 \, c^{5} d^{5} x^{5} + 80 \, b c^{4} d^{5} x^{4} + 80 \, b^{2} c^{3} d^{5} x^{3} + 40 \, b^{3} c^{2} d^{5} x^{2} + 10 \, b^{4} c d^{5} x + b^{5} d^{5}\right )} \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{11}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]