Optimal. Leaf size=229 \[ -\frac{5 \left (b^2-4 a c\right )^{13/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{308 c^4 \sqrt{d} \sqrt{a+b x+c x^2}}+\frac{5 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{308 c^3 d}-\frac{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \sqrt{b d+2 c d x}}{154 c^2 d}+\frac{\left (a+b x+c x^2\right )^{5/2} \sqrt{b d+2 c d x}}{11 c d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.187023, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {685, 691, 689, 221} \[ \frac{5 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{308 c^3 d}-\frac{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \sqrt{b d+2 c d x}}{154 c^2 d}-\frac{5 \left (b^2-4 a c\right )^{13/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{308 c^4 \sqrt{d} \sqrt{a+b x+c x^2}}+\frac{\left (a+b x+c x^2\right )^{5/2} \sqrt{b d+2 c d x}}{11 c d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 685
Rule 691
Rule 689
Rule 221
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{5/2}}{\sqrt{b d+2 c d x}} \, dx &=\frac{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}}{11 c d}-\frac{\left (5 \left (b^2-4 a c\right )\right ) \int \frac{\left (a+b x+c x^2\right )^{3/2}}{\sqrt{b d+2 c d x}} \, dx}{22 c}\\ &=-\frac{5 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{154 c^2 d}+\frac{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}}{11 c d}+\frac{\left (15 \left (b^2-4 a c\right )^2\right ) \int \frac{\sqrt{a+b x+c x^2}}{\sqrt{b d+2 c d x}} \, dx}{308 c^2}\\ &=\frac{5 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{308 c^3 d}-\frac{5 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{154 c^2 d}+\frac{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}}{11 c d}-\frac{\left (5 \left (b^2-4 a c\right )^3\right ) \int \frac{1}{\sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}} \, dx}{616 c^3}\\ &=\frac{5 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{308 c^3 d}-\frac{5 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{154 c^2 d}+\frac{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}}{11 c d}-\frac{\left (5 \left (b^2-4 a c\right )^3 \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}{616 c^3 \sqrt{a+b x+c x^2}}\\ &=\frac{5 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{308 c^3 d}-\frac{5 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{154 c^2 d}+\frac{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}}{11 c d}-\frac{\left (5 \left (b^2-4 a c\right )^3 \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 )}{308 c^4 d \sqrt{a+b x+c x^2}}\\ &=\frac{5 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{308 c^3 d}-\frac{5 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{154 c^2 d}+\frac{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}}{11 c d}-\frac{5 \left (b^2-4 a c\right )^{13/4} \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 d x}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{308 c^4 \sqrt{d} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0801335, size = 101, normalized size = 0.44 \[ \frac{\left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left (-\frac{5}{2},\frac{1}{4};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{32 c^3 d \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.243, size = 798, normalized size = 3.5 \begin{align*} \text{result too large to display} \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 (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{\sqrt{2 \, c d x + b d}}\,{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 (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 )} \sqrt{c x^{2} + b x + a}}{\sqrt{2 \, c d x + b d}}, 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 (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{\sqrt{2 \, c d x + b d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]