Optimal. Leaf size=509 \[ \frac{\sqrt{a+b x+c x^2} (d+e x)^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) \left (e g^2 (m+1) (b d-a e)+c \left (3 d^2 g^2-2 d e f g (m+4)+e^2 f^2 (m+4)\right )\right )}{c e^3 (m+1) (m+4) \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{g \sqrt{a+b x+c x^2} (d+e x)^{m+2} (b e g (2 m+5)+2 c (3 d g-2 e f (m+4))) F_1\left (m+2;-\frac{1}{2},-\frac{1}{2};m+3;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{2 c e^3 (m+2) (m+4) \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{g^2 \left (a+b x+c x^2\right )^{3/2} (d+e x)^{m+1}}{c e (m+4)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.84172, antiderivative size = 506, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {1653, 843, 759, 133} \[ \frac{\sqrt{a+b x+c x^2} (d+e x)^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) \left (g^2 (b d-a e)+\frac{c \left (3 d^2 g^2-2 d e f g (m+4)+e^2 f^2 (m+4)\right )}{e (m+1)}\right )}{c e^2 (m+4) \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{g \sqrt{a+b x+c x^2} (d+e x)^{m+2} (b e g (2 m+5)+6 c d g-4 c e f (m+4)) F_1\left (m+2;-\frac{1}{2},-\frac{1}{2};m+3;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{2 c e^3 (m+2) (m+4) \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{g^2 \left (a+b x+c x^2\right )^{3/2} (d+e x)^{m+1}}{c e (m+4)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1653
Rule 843
Rule 759
Rule 133
Rubi steps
\begin{align*} \int (d+e x)^m (f+g x)^2 \sqrt{a+b x+c x^2} \, dx &=\frac{g^2 (d+e x)^{1+m} \left (a+b x+c x^2\right )^{3/2}}{c e (4+m)}+\frac{\int (d+e x)^m \left (\frac{1}{2} e \left (2 c e f^2 (4+m)-g^2 (3 b d+2 a e (1+m))\right )-\frac{1}{2} e g (6 c d g-4 c e f (4+m)+b e g (5+2 m)) x\right ) \sqrt{a+b x+c x^2} \, dx}{c e^2 (4+m)}\\ &=\frac{g^2 (d+e x)^{1+m} \left (a+b x+c x^2\right )^{3/2}}{c e (4+m)}-\frac{(g (6 c d g-4 c e f (4+m)+b e g (5+2 m))) \int (d+e x)^{1+m} \sqrt{a+b x+c x^2} \, dx}{2 c e^2 (4+m)}+\frac{\left (e (b d-a e) g^2 (1+m)+c \left (3 d^2 g^2+e^2 f^2 (4+m)-2 d e f g (4+m)\right )\right ) \int (d+e x)^m \sqrt{a+b x+c x^2} \, dx}{c e^2 (4+m)}\\ &=\frac{g^2 (d+e x)^{1+m} \left (a+b x+c x^2\right )^{3/2}}{c e (4+m)}-\frac{\left (g (6 c d g-4 c e f (4+m)+b e g (5+2 m)) \sqrt{a+b x+c x^2}\right ) \operatorname{Subst}\left (\int x^{1+m} \sqrt{1-\frac{2 c x}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}} \sqrt{1-\frac{2 c x}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \, dx,x,d+e x\right )}{2 c e^3 (4+m) \sqrt{1-\frac{d+e x}{d-\frac{\left (b-\sqrt{b^2-4 a c}\right ) e}{2 c}}} \sqrt{1-\frac{d+e x}{d-\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c}}}}+\frac{\left (\left (e (b d-a e) g^2 (1+m)+c \left (3 d^2 g^2+e^2 f^2 (4+m)-2 d e f g (4+m)\right )\right ) \sqrt{a+b x+c x^2}\right ) \operatorname{Subst}\left (\int x^m \sqrt{1-\frac{2 c x}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}} \sqrt{1-\frac{2 c x}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \, dx,x,d+e x\right )}{c e^3 (4+m) \sqrt{1-\frac{d+e x}{d-\frac{\left (b-\sqrt{b^2-4 a c}\right ) e}{2 c}}} \sqrt{1-\frac{d+e x}{d-\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c}}}}\\ &=\frac{g^2 (d+e x)^{1+m} \left (a+b x+c x^2\right )^{3/2}}{c e (4+m)}+\frac{\left (e (b d-a e) g^2 (1+m)+c \left (3 d^2 g^2+e^2 f^2 (4+m)-2 d e f g (4+m)\right )\right ) (d+e x)^{1+m} \sqrt{a+b x+c x^2} F_1\left (1+m;-\frac{1}{2},-\frac{1}{2};2+m;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{c e^3 (1+m) (4+m) \sqrt{1-\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}}-\frac{g (6 c d g-4 c e f (4+m)+b e g (5+2 m)) (d+e x)^{2+m} \sqrt{a+b x+c x^2} F_1\left (2+m;-\frac{1}{2},-\frac{1}{2};3+m;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{2 c e^3 (2+m) (4+m) \sqrt{1-\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}}\\ \end{align*}
Mathematica [F] time = 1.45589, size = 0, normalized size = 0. \[ \int (d+e x)^m (f+g x)^2 \sqrt{a+b x+c x^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 1.539, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m} \left ( gx+f \right ) ^{2}\sqrt{c{x}^{2}+bx+a}\, dx \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 \sqrt{c x^{2} + b x + a}{\left (g x + f\right )}^{2}{\left (e x + d\right )}^{m}\,{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 ({\left (g^{2} x^{2} + 2 \, f g x + f^{2}\right )} \sqrt{c x^{2} + b x + a}{\left (e x + d\right )}^{m}, 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 \sqrt{c x^{2} + b x + a}{\left (g x + f\right )}^{2}{\left (e x + d\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]