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 = 0.84, 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 133
Rule 759
Rule 843
Rule 1653
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 = 1.39, size = 0, normalized size = 0.00 \[ \int (d+e x)^m (f+g x)^2 \sqrt {a+b x+c x^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (g^{2} x^{2} + 2 \, f g x + f^{2}\right )} \sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c x^{2} + b x + a} {\left (g x + f\right )}^{2} {\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \left (g x +f \right )^{2} \sqrt {c \,x^{2}+b x +a}\, \left (e x +d \right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c x^{2} + b x + a} {\left (g x + f\right )}^{2} {\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (f+g\,x\right )}^2\,{\left (d+e\,x\right )}^m\,\sqrt {c\,x^2+b\,x+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d + e x\right )^{m} \left (f + g x\right )^{2} \sqrt {a + b x + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________